861 inches

Units 1-3Unit 7

Algebra 1 Unit 7

.861 inches

Teacher Guide

A1

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ISBN 978-1-64573-288-4

AGA1.0

Unit Narrative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Finding Unknown Inputs

Lesson 1: Finding Unknown Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Lesson 2: When and Why Do We Write Quadratic Equations? . . . . . . . . . . . . . 43

Solving Quadratic Equations

Lesson 3: Solving Quadratic Equations by Reasoning. . . . . . . . . . . . . . . . . . . . . 58

Lesson 4: Solving Quadratic Equations with the Zero Product Property . . . . . 71

Lesson 5: How Many Solutions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Lesson 6: Rewriting Quadratic Expressions in Factored Form (Part 1) . . . . . . 104



Lesson 9: Solving Quadratic Equations by Using Factored Form . . . . . . . . . . 154

Lesson 10: Rewriting Quadratic Expressions in Factored Form (Part 4) . . . . . 169

Completing the Square

Lesson 11: What are Perfect Squares? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Lesson 12: Completing the Square (Part 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202



Lesson 15: Quadratic Equations with Irrational Solutions . . . . . . . . . . . . . . . . 251

The Quadratic Formula

Lesson 16: The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Lesson 17: Applying the Quadratic Formula (Part 1) . . . . . . . . . . . . . . . . . . . . . 280

Lesson 18: Applying the Quadratic Formula (Part 2) . . . . . . . . . . . . . . . . . . . . . 295

Lesson 19: Deriving the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Lesson 20: Rational and Irrational Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Lesson 21: Sums and Products of Rational and Irrational Numbers . . . . . . . 350

Vertex Form Revisited

Lesson 22: Rewriting Quadratic Expressions in Vertex Form. . . . . . . . . . . . . . 370

Lesson 23: Using Quadratic Expressions in Vertex Form to Solve Problems. 391

Putting It All Together

Lesson 24: Using Quadratic Equations to Model Situations and SolveProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Attributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Unit NarrativeUnit NarrativePrior to this unit, students have studied quadratic functions. They analyzed and representedquadratic functions using expressions, tables, graphs, and descriptions. Students also evaluated thefunctions and interpreted the input and the output values in context. They encountered the terms“standard form,” “factored form,” and “vertex form” and examined the advantages of each form.They also rewrote expressions from factored form and vertex form to standard form.

In this unit, students interpret, write, and solve quadratic equations. They see that writing andsolving quadratic equations enables them to find input values that produce certain output values.Suppose the revenue of a theater is a function of the ticket price for a performance. At what ticketprice would the theater earn $10,000? Previously, students were only able to solve such problemsby observing graphs and estimating, or by guessing and checking. Here, they learn to answer suchquestions algebraically.

Students begin solving quadratic equations by reasoning. For instance, to solve , theythink: Adding 9 to a squared number makes 25. That squared number must be 16, so must be 4or -4. Along the way, students see that quadratic equations can have 2, 1, or 0 solutions.

Next, students learn that equations of the form can be easily solved by applyingthe zero product property, which says that when two factors have a product of 0, one of the factorsmust be 0. When the equations are not in factored form, students rearrange them so that one sideis 0, and rewrite the expressions from standard form to factored form. Students soon recognizethat not all quadratic expressions in standard form can be rewritten into factored form. Even whenit is possible, finding the right two numbers may be tedious, so another strategy is needed.

Students encounter perfect squares and notice that solving a quadratic equation is prettystraightforward when the equation contains a perfect square on one side and a number on theother. They learn that we can put equations into this helpful format by completing the square, that is,by rewriting the equation such that one side is a perfect square. Although this method can be usedto solve any quadratic equation, it is not practical for solving all equations. This challenge motivatesthe quadratic formula.

Once introduced to the formula, students apply it to solve contextual and abstract problems,including those that they couldn’t previously solve. After gaining some experience using theformula, students investigate how it is derived. They find that the formula essentially encapsulatesall the steps of completing the square into a single expression. Just like completing the square, thequadratic formula can be used to solve any equation, but it may not always be the quickest method.Students consider how to use the different methods strategically.

Throughout the unit, students see that solutions to quadratic equations are often irrationalnumbers. Sometimes they are expressed as sums or products of a rational number and an

Unit 7: Quadratic Equations Unit Narrative 3

irrational number (such as or ). Students reason about whether such sums and

products are rational or irrational.

Toward the end of the unit, students revisit the vertex form and recall that it can be used to identifythe maximum or minimum of a quadratic function. Previously students learned to rewriteexpressions from vertex form to standard form. Now they can go in reverse—by completing thesquare. Being able to rewrite expressions in vertex form allows students to effectively solveproblems about maximum and minimum values of quadratic functions.

In the final lesson, students integrate their insights and choose appropriate strategies to solve anapplied problem and a mathematical problem (a system of linear and quadratic equations).

Required MaterialsCopies of blackline masterGraphing technologyExamples of graphing technology are:a handheld graphing calculator, a computer witha graphing calculator application installed, andan internet-enabled device with access to a sitelike desmos.com/calculator or geogebra.org/graphing. For students using the digitalmaterials, a separate graphing calculator toolisn't necessary; interactive applets areembedded throughout, and a graphingcalculator tool is accessible on the studentdigital toolkit page.

Pre-printed slips, cut from copies of theblackline masterScientific calculatorsScissorsSticky notesTools for creating a visual displayAny way for students to create work that can beeasily displayed to the class. Examples: chartpaper and markers, whiteboard space andmarkers, shared online drawing tool, access to adocument camera.

4 Teacher Guide Algebra 1

AssessmentsAssessment : Check Your ReadinessTeacher InstructionsNeither scientific calculator nor graphing calculator should be used in this assessment.

Student InstructionsNo calculators should be used in this assessment.

Problem 1The content assessed in this problem is first encountered in Lesson 6: Rewriting QuadraticExpressions in Factored Form (Part 1).

In this unit, students will rewrite expressions in factored form, which involves finding positive andnegative rational numbers with certain sums and products. This item assesses students’ knowledgeof operations on signed numbers.

If most students struggle with this item, plan to provide additional time for students to reason withexpressions during the lessons on rewriting expressions into factored forms. Encourage students touse rectangle diagrams to organize their work and invite them to check their factored form bymultiplying.

StatementEvaluate each expression.

1.

2.

3.

4.

5.

6.

7.

Solution1. -18

2. 7

Unit 7: Quadratic Equations Assessments 5

3. -11

4. 18

5. 0

6. -9

7. 9

Aligned Standards7.NS.A.1, 7.NS.A.2

Problem 2The content assessed in this problem is first encountered in Lesson 5: How Many Solutions?.

This item gauges students’ understanding of the zeros of functions and their graphical andalgebraic representations. In the unit, students formalize their observation that equations of theform can be solved by finding the -intercepts of the graph of . They also solve theequations algebraically.

If most students struggle with this item, plan to incorporate a review of the connections betweenthe zeros of a function and the -intercepts of the graph after the warm-up. Consider displaying achart that details these connections, and keep it posted during this lesson.

StatementA quadratic function is defined by . Diego is trying to find the zeros of thefunction.

Select all statements that are true in this situation.

A. Diego knows the input of and is trying to find the output.

B. Diego is trying to find the input for a particular output of .

C. Diego is looking for the value of .

D. Diego is looking to find when .

E. The zeros can be found by solving .

F. The zeros can be found by graphing and finding the -intercepts.

Solution["B", "D", "E", "F"]

Aligned StandardsHSF-IF.A.1, HSF-IF.C.7


Problem 3The content assessed in this problem is first encountered in Lesson 4: Solving Quadratic Equationswith the Zero Product Property.

This item assesses students’ understanding of linear equations and their solutions. Students explainwhether a given value is a solution to a linear equation in one variable and, if not, find the correctsolution.

If most students struggle with this item, plan to provide additional practice solving equations of theform mentally as students first learn about the zero product property. Reinforce thatwhen one side of an equation is 0, we are looking for the value of the variable that makes the otherside equal 0 as well.

Statement1. Is a solution to the equation ? Explain or show your reasoning. If it is not a

solution, find the correct solution.

2. Is 8 a solution to the equation ? Explain or show your reasoning. If itis not a solution, find the correct solution.

Solution1. Yes. Sample reasoning: When is substituted for and the

expression is evaluated, the result equals 5.

2. No. Sample reasoning: Substituting 8 for and then evaluating theexpression on each side does not yield a true statement. is25. is 36. 25 does not equal 36. The correct solution is 19.Sample reasoning:

Aligned StandardsHSA-REI.B.3

Problem 4The content assessed in this problem is first encountered in Lesson 15: Quadratic Equations withIrrational Solutions.

In this unit, students write and solve quadratic equations in geometric contexts. This item can showwhether students understand the relationship between the area and side length of a square, and

whether they recognize that the side length of a square with area can be expressed as .

Students who agree with Kiran’s answer of 7 may be confusing area and perimeter. Students whoagree with Lin’s estimate of 5.2 likely understand the geometric relationship and the rational

Unit 7: Quadratic Equations 7

approximation of but may not recognize that a length could be expressed as , or that

is more accurate than 5.2.

If most students do well with this item, it may be possible to skip Lesson 15 Activity 1.

StatementA square has an area of 28 square inches. Kiran says its side length is 7 inches. Noah says its

side length is inches. Lin says it is about 5.2 inches. Which student, if any, do you think iscorrect? Explain your reasoning.

SolutionAgree with Noah. Sample response: The area of a square is the side length squared. is 49, not 28.

is 27.04, which is not far off, but it is nearly 1 square unit away from 28. is exactly 28.

Aligned Standards8.EE.A.2

Problem 5The content assessed in this problem is first encountered in Lesson 1: Finding Unknown Inputs.

In this item, students identify quadratic expressions that describe a geometric relationship.

Students who choose A may intend to subtract all the non-pond area from 100 square meters butdo not subtract all the necessary regions. Students who choose B may correctly think of subtractingthe two regions—each with area — from , but neglect to account for the overlap in theregions. Students who choose D may recognize as representing the side length of the pondbut then find the area of the wrong region.

If most students struggle with this item, plan to be prepared to use the scaffolding questions givenin Lesson 1 Activity 3, and support students by labeling the lengths of the picture. Some studentsmay benefit from using a ruler to see more concretely that when putting 2 segments together tomake a longer segment, the lengths are added, and when taking away a part of a segment, thelengths are subtracted.

StatementA square garden with side length 10 meters has a squarepond in one corner, as shown in the diagram.

Select all expressions that represent the area of the pond, insquare meters.


A.

B.

C.

D.

E.

F.

Solution["C", "E", "F"]

Aligned StandardsHSA-SSE.A.1


In this unit, students will explore rational and irrational solutions to quadratic equations. This itemoffers insight on students’ understanding of rational and irrational numbers.

Students who select only non-integers (C, F, and G) may recall that rational numbers are fractions,but do not recall that whole numbers are also fractions (for example, ). Students who neglect

D may incorrectly think of irrational numbers as numbers with a square root symbol and rational

numbers as those without. (They may therefore see as irrational, even though it equals 4.)

Students who do not choose negative numbers may mistakenly think that rational numbers arealways positive. Those who do not choose G may not recall that a number in decimal form withrepeating digits is a rational number.

If most students struggle with this item, plan to revisit the meaning of rational and irrationalnumbers during the launch of Activity 2 in Lesson 15 rather than in the synthesis. Considerdisplaying a chart that shows examples of rational and irrational numbers as a reference in this andlater lessons.

StatementSelect all the rational numbers.


A. 8

B.

C.

D.

E.

F. -0.75

G.

Solution["A", "C", "D", "F", "G"]

Aligned Standards8.NS.A.1


Students’ work on quadratic equations will include finding rational approximations of irrationalsolutions. In this item, students estimate and compare the square roots of some rational numbers.

If most students struggle with this item, plan to use the warm-up of Lesson 15 to ensure they

understand why , and consider expanding the activity synthesis with additional examples.

StatementPut these numbers in order from least to greatest.

•

•

•

••


Solution

Aligned Standards8.NS.A.2

Problem 8The content assessed in this problem is first encountered in Lesson 3: Solving Quadratic Equationsby Reasoning.

Students solve simple quadratic equations by reasoning and recognizing the structure in eachequation. They identify equations that don’t have a solution and explain why this is the case.

In grade 8, students solved equations of the form where is a positive rational number.They may not recall that such an equation has two solutions. Those who do recall may write apositive number and a negative number, which could be decimal estimations (such as 3.87 and

-3.87 for the solutions of ), or numbers in square root notation ( and ). Studentsare not yet expected to be able to write solutions using square root notation when is negative.

If most students do well with this item, plan to spend less time on the warm-up for Lesson 3 by onlyassigning some of the questions.

StatementDetermine if each equation has a solution. If so, name the solution(s). If not, briefly explainwhy it doesn’t have a solution.

1.

2.

3.

4.

Solution1. Yes. 8 and -8

2. Yes. and

3. No. Sample explanation: A number multiplied by its opposite cannot be positive.

4. No. Sample explanation: Squaring a number cannot result in a negative number.

Aligned Standards8.EE.A.2, HSA-REI.B.4.b


Assessment : Mid-Unit AssessmentTeacher InstructionsGive this assessment after lesson 10.

Make a judgment about whether to allow graphing technology, or just a scientific calculator.Graphing technology would allow students to approach items 1, 4, 5, and 6 in additional ways thataren't covered by the sample responses and the aligned standards.

Problem 1Students find the solutions of equations by reasoning and making use of structure. For example,they recognize that there are two possible numbers that can be squared to give the same positivenumber, and that no number (that they know of) can be squared to make a negative number. (Notethat even though C actually has two complex solutions, it has no real solutions. In this course,students haven't yet learned that there are numbers that are not real, so at this time, they areexpected to say that equations like this have no solutions.)

StatementSelect all equations that have two solutions.

A.

B.

C.

D.

E.

F.

Solution["A", "E", "F"]

Aligned StandardsHSA-REI.B.4.b

Problem 2Students interpret a graph of a quadratic function and its connections to an associated quadraticequation. The item gauges whether students recognize that to solve a quadratic equation of theform “ ” is to find the zeros of the function and the -intercepts of the graph. If there areno -intercepts, that means there are no solutions.


StatementHere is a graph of the equation .

Which statement is true about the solutions to the equation?

A. The equation has no solution.

B. The equation has one solution: 5.

C. The equation has two solutions: -2 and 0.

D. The equation has infinitely many solutions.

SolutionA

Aligned StandardsHSA-REI.D, HSA-REI.D.10

Problem 3Students who choose B may not recognize that the sum of -2 and 18 is positive. Option C gives anexpression in standard form with a negative value for , but the constant term is not -36. Studentswho choose D neglect to see that the would be positive, and that 4 times 9 is 36, not -36.

StatementSelect all expressions that could be equivalent to where is negative.

A.

B.

C.

D.

E.

Solution["A", "E"]



Problem 4Given quadratic expressions in standard form, students write equivalent expressions in factoredform. Students rely on the structure that connects standard form and factored form to do so.

For expressions of the form , they look for two numbers that multiply to make and addup to the value of . When the expression is a difference of a square and a constant term, forexample , where there is no linear term, they look for numbers that are opposites thatmultiply to and add up to 0. Because no opposites have a positive product and a sum of 0,expressions like cannot be written in factored form.

StatementEach expression is given in standard form. Rewrite it in factored form. If it cannot berewritten in factored form, write “cannot be done.”

1.

2.

3.

4.

5.

Solution1.

2.

3. cannot be done

4. cannot be done

5.


Problem 5Some students may be unfamiliar with the term “pen” in the context of enclosure for animals.Clarify the term as needed.

The last question can be answered in a number of ways, including by guessing and checking or bydrawing and reasoning with a diagram. Look for evidence of students solving the given equation.


StatementA rancher has a square cow pen with side length . He decides to change the shape of thepen while still using the same amount of fencing materials. The equation

represents the relationship of the new side lengths (in feet) of the penand its area (in square feet).

1. If 300 represents the area of the new cow pen, what do the expressions andeach represent?

2. Find , the original side length (in feet) of the square pen. Show your reasoning.

Solution1. Sample response:

The new length is and the new width is (or vice versa).

The rancher subtracts 10 feet from two opposite sides of the original square and adds 10feet to the other two opposite sides.

2. 20 feet. Sample reasoning: is equivalent to , which is. Solving for gives either 20 or -20. Only a positive value makes sense here.

Minimal Tier 1 response:

Work is complete and correct.

Sample:

1. He made it 10 feet longer and 10 feet narrower.

2. 20 feet.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correctwork with insufficient explanation or justification.

Sample errors: omission of or incorrect units; incorrect answer to part a; work in partb involves errors after at least attempting to distribute the expression on the left.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.

Sample errors: work in part b does not show an attempt to expand the expression on the left.

◦◦

••

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•

••


Aligned StandardsHSA-REI.B.4.b, HSA-SSE.A.1

Problem 6The first question is already in factored form, so students simply apply the zero product property.The second question requires rewriting the expression on the left in factored form. The lastquestion requires rearranging the equation (so that one side is 0) before rewriting the expressionon the left in factored form.

StatementSolve each equation. Show your reasoning.

1.

2.

3.

Solution1. or . Sample reasoning:

2. or . Sample reasoning:




Sample:

1. , (Minimal work here is all right so long as the solutions are clearly associatedwith each factor: for example, using arrows or writing solutions directly underneath.)

••


2. , or

3. , , or

Tier 2 response:


Sample errors: a solution of rather than in part a; solutions have incorrect signs; only

one solution is given; only one problem part contains an error listed under Tier 3 response.

Tier 3 response:


Sample errors: work contains two or more of the following errors or contains one of theseerrors in multiple problem parts: attempting to solve by dividing both sides by or (or, inpart a, by first distributing the left side); incorrect rewriting in parts b or c; in part c, an attemptto rewrite in factored form without first subtracting 7 from each side.


Problem 7For the first question, it is not essential that students mention “zero product property.” What isimportant is recognizing that when two factors have a product of 0, one of the factors must be 0,and that this fact makes it possible to solve the equation.

Statement1. To solve , Mai rewrote the equation as . Explain how rewriting

this equation in factored form enables Mai to solve the equation.

2. Solve by rewriting the equation. Show your reasoning.

3. Explain why dividing each side of by is not a reliable way to solve theequation.

SolutionSample response:

1. Rewriting the equation as allows us to use the zero product property. Ifmultiplying and makes 0, either is 0, or is 0, which means the solution is either

or .

2. or . Sample reasoning: , or . If and multiply tomake 0, then either or , so or .

•

•

••


3. Dividing by is not valid when , because it would mean dividing by . But needs tobe considered as a possible solution, and in fact, when the expression is evaluated at ,we get 0 on both sides, so 0 is a solution. This solution is lost when we divide by .


Work is complete and correct, with complete explanation or justification.

Sample:

1. Because now it's two things multiplied together that equal zero, so one of those things has toequal zero. could equal 0 or .

2.

3. Because if you might be dividing by zero.

Tier 2 response:


Sample errors: explanation in part a asserts that something has to equal zero but does notexplain (or mention) that there must be two solutions; solution in part b involves dividing eachside by but correctly considers the separate case where ; solutions in part b are and

; explanation in part c correctly points out an overlooked solution but does not explain thatthis is due to division by zero; one of parts a or c has an error listed under Tier 3 response.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.

Sample errors: little headway in work for part b; two of the following errors: explanation inpart a does not involve the zero product property (in name or concept); work for part binvolves dividing both sides by resulting in one solution only; explanation in part c does notinvolve dividing by zero.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understandingand mastery.

Sample errors: little headway in work for part b along with a Tier 3 error types in one of part aor part c; Tier 3 error types in all three problem parts.

Aligned StandardsHSA-REI.A.1, HSA-REI.B.4.b

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•

••

•

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Assessment : End-of-Unit AssessmentTeacher InstructionsUse of a four-function calculator or a scientific calculator is acceptable. Graphing technology shouldnot be used, because it would allow students to approach items 2, 3, 4, 5, and 7 in a way thatwouldn't assess the intended standards.

Student InstructionsYou may use a scientific calculator, but not a graphing calculator.

Problem 1To rewrite or solve equations by completing the square requires recognizing when a quadraticexpression, in various forms, is the square of a linear expression (a perfect square). This itemassesses this ability.

StatementSelect all expressions that are squares of linear expressions.

A.

B.

C.

D.

E.

F.

Solution["B", "D"]


Problem 2Solving an equation by completing the square involves using the structure of perfect squares andmaintaining the equality of the two sides of an equation. This item gauges both understandings.

StatementSelect all equations that are equivalent to .


A.

B.

C.

D.

E.

F.

Solution["C", "F"]

Aligned StandardsHSA-REI.B.4.a

Problem 3This item requires understanding that the maximum or minimum of a quadratic function can beeasily identified when the expression defining the function is in vertex form.

Some students may try rewriting each option in standard form and seeing which one yields anexpression identical to the one that defines . Two expressions (B and D) are equivalent to theoriginal expression, but only D readily reveals the vertex of the function.

StatementA quadratic function is defined by . Which expression also defines andbest reveals the maximum or minimum of the function?

A.

B.

C.

D.

SolutionD

Aligned StandardsHSA-SSE.B.3.b


Problem 4Students complete the square for quadratic expressions where the squared term has a coefficientof 1 and a coefficient other than 1. Students may show the use of the symbol to show positiveand negative square roots.

In the second equation, the coefficients 9 and 12 were intentionally selected to make completingthe square easier without having to first rewrite the expression with 9 as a factor. Students thatchoose a different approach to complete the square (such as transforming the equationto ) could earn full credit even though this option is not shown in the solution.

StatementSolve each equation by completing the square. Show your reasoning.

1.

2.

Solution1. or . Sample reasoning:




Acceptable errors: While ideally students would show more work than in the sample, it iscrucial that one of the steps involves adding the same quantity to each side before rewritingthe expression on one side as a squared factor.

Sample:

••

•


1.

2.

Tier 2 response:


Sample errors: part a is solved by rewriting the expression on the left in factored form( ) and then writing each factor as equal to -20 ( and ); parta involves the expression ; part b involves the expressions andthen ; sign errors in either problem part; errors in either problem part that occurafter sucessfully writing the equations in the form ; little progress on part b withsuccessful work in part a.

Tier 3 response:


Sample errors: in neither problem part is the correct number added or subtracted from eachside; in neither problem part is the perfect-square expression correct.


Problem 5Students solve a quadratic equation using a method of their choice.

The equation can be solved by rewriting in factored form, but it may require some guessing andchecking if the right combination of factors doesn’t come to mind right away. The equation can alsobe solved by completing the square, but the coefficient of the linear term—an odd number—makesit computationally more challenging. Students may prefer using the quadratic formula because itcan be easily applied and the numbers don’t lend themselves to complicated calculations.

It’s not necessary to require students to write “or” in the solution. Either could be appropriate,depending on how the question is interpreted:

The solutions to the equation are and 2.

If , then or .

•

•

••

••


StatementSolve the equation .

Solutionor .


Problem 6This item elicits an explanation about why the sum of a rational number and an irrational numbercannot be rational. (It assumes that students can distinguish rational and irrational numbers.)

StatementA solution to an equation Jada solved is . She is trying to determine whether thatsolution is rational or irrational.

Jada knows that -2 is a rational number and is an irrational number. She also knows thatthe sum of any two rational numbers is always rational.

1. If we add 2 (a rational number) to , what is the sum? Is the sum rational orirrational?

2. Explain why cannot be rational.

Solution1. The sum is , which is irrational.

2. Sample response: If was rational, adding another rational number to it would give a

rational sum. But we just saw that adding 2 to gives an irrational number, which

means must be irrational.



Sample:

1. , irrational.

2. When you add two rational numbers together, you should get another rational number.

Adding and 2 gives you an irrational number.

Tier 2 response:

••



Sample errors: only one of the two answers in part a is correct; an arithmetic error in part athat carries through to a good explanation in part b; explanation in part b notes that

is irrational but does not invoke the principle that the sum of two rationalnumbers is also rational; explanation in part b invokes this principle but does not explain how

it helps us decide if is rational; explanation in part b is along the lines of, “any numberinvolving the square root of a non-square number is irrational.”

Tier 3 response:


Sample errors: correct answers to part a but the answer to part b isn’t close; Tier 2 error typesin both problem parts.

Aligned StandardsHSN-RN.B.3

Problem 7Given a quadratic function that describes a profit model, students write and solve an equation todetermine when the function has a certain value. They also find the maximum value of the functionby rewriting the quadratic expression in vertex form, or by reasoning about the quantities.

StatementA community theater uses the function to model the profit (indollars) expected in a weekend when the tickets to a comedy show are priced at dollarseach.

1. Write and solve an equation to find out the prices at which the theater would earn$1,500 in profit from the comedy show each weekend. Show your reasoning.

2. At what price would the theater make the maximum profit, and what is that maximumprofit? Show your reasoning.

Solution1. or . Sample reasoning: Using the

quadratic formula for , the solution is or .

2. The greatest profit can be expected when the price per ticket is $25. At that price, the profit is$2,400. Sample reasoning:

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Rewriting the equation in vertex form gives us the vertex.

In the previous question, we found the profit to be $1,500 when or . The vertexof the graph representing the function is halfway between the two, at . Evaluating thefunction at gives 2,400.


Work is complete and correct, with complete explanation or justification.

Sample:

1. , . I used the quadratic formula to findand . So the prices are $16 and $40.

2. The vertex is halfway between 16 and 40, which is 25. . So the most money thetheater can make is $2,400, when it charges $25.

Tier 2 response:


Sample errors: work in either problem part is correct but does not explain the significance ofthe solutions to the equations in context; small computational or sign errors in the quadraticformula (with work shown); an incorrect attempt to solve the equation in part a by completingthe square with some good progress shown; errors finding the halfway point between the

-intercepts in part b (with work shown); errors putting the equation in vertex form so long asthe “completing the square” steps are handled correctly; errors completing the square relatedto dividing by the expression by -4 (or rewriting the expression with -4 as a factor); only the

-coordinate of the vertex is found; correct answer to either problem part with no workshown.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.

Sample errors: is not set equal to 1,500; errors completing the square in either problempart that are more severe than Type 2; an error averaging the coordinates of the -interceptsalong with omitting the -coordinate of the vertex; incorrect answer to either problem partwith no work shown; three or more distinct Tier 2 error types.

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Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understandingand mastery.

Sample errors: Tier 3 errors in both problem parts.

Aligned StandardsHSA-CED.A.1, HSA-REI.B.4.b, HSA-SSE.B.3.b

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Lesson 1: Finding Unknown Inputs

GoalsExplain (orally and in writing) the meaning of the solution to a quadratic equation in terms of asituation.

Write a quadratic equation that represents geometric constraints.

Learning TargetsI can explain the meaning of a solution to an equation in terms of a situation.

I can write a quadratic equation that represents a situation.

Lesson NarrativeIn a previous unit, students studied quadratic functions in some depth. They built quadraticexpressions to represent situations and wrote equivalent expressions. They also graphed, analyzed,and evaluated quadratic functions to solve problems. In some cases, they investigated andinterpreted the outputs of the functions. In others, they looked for the input values that producecertain outputs, and they found these values mainly by reasoning with graphs.

This unit picks up on where that unit left off. Sometimes we have a relationship that can beexpressed with a quadratic function, and we want to know what input generates a particularoutput. How do we find out other than by graphing and estimating, or by guessing and checking?

In this lesson, students encounter a problem that cannot be easily solved by familiar strategies,which gives them a chance to persevere in problem solving (MP1). They write a quadratic equationand interpret what a solution means in the given situation. The work here motivates the need tosolve quadratic equations. The formal definition of a quadratic equation will be introduced until thenext lesson, after students have seen some variations of such equations and worked with them incontext.

AlignmentsBuilding On

HSF-IF.B.4: For a function that models a relationship between two quantities, interpret keyfeatures of graphs and tables in terms of the quantities, and sketch graphs showing keyfeatures given a verbal description of the relationship. Key features include: intercepts;intervals where the function is increasing, decreasing, positive, or negative; relative maximumsand minimums; symmetries; end behavior; and periodicity.

Addressing

HSA-CED.A.1: Create equations and inequalities in one variable and use them to solveproblems. Include equations arising from linear and quadratic functions, and simple rationaland exponential functions.

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Building Towards

HSA-CED.A.3: Represent constraints by equations or inequalities, and by systems of equationsand/or inequalities, and interpret solutions as viable or nonviable options in a modelingcontext. For example, represent inequalities describing nutritional and cost constraints oncombinations of different foods.

Instructional Routines

Aspects of Mathematical Modeling

MLR6: Three Reads

MLR7: Compare and Connect

Required Materials

Copies of blackline master Scissors

Required Preparation

Print copies of the blackline master, one copy for every student. Note that the images need to beprinted on 8.5" by 11" paper for the indicated dimensions to be accurate. Printing on paper ofanother size (such as A4 paper) may distort the image. (The goals of the activity don't depend onthe dimensions being exactly right, however.)

Student Learning Goals

Let’s find some new equations to solve.

1.1 What Goes Up Must Come DownWarm Up: 5 minutesThis task reminds students that they can use a graph of a function to gain some information aboutthe situation that the function models.

Every question can be answered, or at least estimated, by analyzing the graph. As students work,look for those who use the given equation to solve or verify the answers. For example, the firstquestion can be answered by evaluating and the second question by evaluating . Invitethem to share their strategy during synthesis.

Building On

HSF-IF.B.4

Launch

Given their earlier work on quadratic functions, students should be familiar with projectiles. Ifneeded, give a brief orientation on the context. Tell students that there are devices that usecompressed air or other means to generate a great amount of force and launch a potato orsimilar-sized object.

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(If desired and time permitting, find and show a short video clip of someone using an air-poweredor a catapult-type device. Warn students that some of these devices can be dangerous and theyshouldn’t try to build one without help from an adult.)

Student Task Statement

A mechanical device is used to launch a potato vertically into the air. The potato is launchedfrom a platform 20 feet above the ground, with an initial vertical velocity of 92 feet persecond.

The function modelsthe height of the potato over the ground, infeet, seconds after launch.

Here is the graph representing the function.

For each question, be prepared to explain your reasoning.

1. What is the height of the potato 1 second after launch?

2. 8 seconds after launch, will the potato still be in the air?

3. Will the potato reach 120 feet? If so, when will it happen?

4. When will the potato hit the ground?

Student Response

1. 96 feet (or another height that approximates 96 feet)

2. no

3. Yes, the potato’s height will be 120 feet at two times: approximately 1.5 seconds and 4.3seconds after launch.

4. after approximately 6 seconds

Activity Synthesis

Focus the discussion on how students used the graph to help them answer the questions.Encourage students to use precise mathematical vocabulary in their explanation. Invite students,especially those who do not rely solely on the graph, to share their responses and reasoning.

Point out that we can gather quite a bit of information about the function from the graph, but theinformation may not be precise.

Unit 7 Lesson 1: Finding Unknown Inputs 29

“Is there a way to get more exact answers rather than estimates?” (Some questions can beanswered by evaluating the function. For questions that are not easy to calculate at this point,we can only estimate from the graph.)

“Which questions in the activity could be answered by calculating?” (the first two questions,about where the potato is after some specified number of seconds) “Which questions werenot as easy to figure out by calculation? (those about when the potato reaches certain heights)

“How can we verify the time when the potato is 120 feet above the ground or when, exactly, ithits the ground?” (We can find the values that make equal 120 by looking at the graphand then evaluating the function at those values of . To find when the potato hits the groundexactly, we would need to find the zeros of by solving the equation .)

Tell students that, in this unit, they will investigate how answers to these questions could becalculated rather than estimated from a graph or approximated by guessing and checking.

1.2 A Trip to the Frame Shop20 minutesThe purpose of this task is to motivate the need to write and solve a quadratic equation in order tosolve a problem.

Students are asked to frame a picture by cutting up a rectangular piece of “framing material” intostrips and arranging it around a picture such that they create a frame with a uniform thickness. Theframing material has a different length-to-width ratio and is smaller than the picture, so studentscannot simply center the framing material on the back of the picture.

Students are not expected to succeed in the task through trial and error. They are meant tostruggle, just enough to want to know a way to solve the problem that is better than by guessingand checking. Some students may try writing an equation to help them figure out the rightmeasurement for the frame, but they don’t yet have the knowledge to solve the quadratic equation.(The solutions to this equation are irrational, so it is also unlikely for students to find them bychance.)

This activity was inspired by the post “When I Got Them to Beg” on http://fawnnguyen.com/got-beg/by Fawn Nguyen, used with permission.

Building Towards

HSA-CED.A.3



Launch

Ask students if they ever had an artwork or a picture framed at a frame shop. If students areunfamiliar with custom framing, explain that it is very expensive—often hundreds of dollars for one

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piece. The framing materials, which are cut to exact specifications, can be costly. The time and laborneeded to properly frame a picture further push up the cost.

Tell students that they are now going to frame a picture, using a sheet of paper as their framingmaterial. The sheet is to be cut such that:

All of it is used. (Framing material is expensive!)

The framing material does not overlap.

The resulting frame has uniform thickness all the way around.

Distribute scissors, along with the pictures and the “framing material” (copies of the blacklinemaster).

Support for Students with Disabilities

Representation: Access for Perception. Display and read the directions aloud. Students who bothlisten to and read the information will benefit from additional processing time. Check forunderstanding by inviting students to rephrase directions in their own words.Supports accessibility for: Language; Memory


Your teacher will give you a picture that is 7 inches by 4 inches, a piece of framing materialmeasuring 4 inches by 2.5 inches, and a pair of scissors.

Cut the framing material to create a rectangular frame for the picture. The frame should havethe same thickness all the way around and have no overlaps. All of the framing materialshould be used (with no leftover pieces). Framing material is very expensive!

You get 3 copies of the framing material, in case you make mistakes and need to recut.

Student Response

Samples of student work (used with permission from Fawn Nguyen):

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Are You Ready for More?

Han says, “The perimeter of the picture is 22 inches. If I cut the framing material into 9 pieces,each one being 2.5 inches by inch, I’ll have more than enough material to surround the

picture because those pieces would mean 22.5 inches for the frame.”

Do you agree with Han? Explain your reasoning.

Student Response

No, Han is not correct. Sample reasoning: While the total length of those pieces is enough tosurround the picture, he will need at least two pieces for each short side and at least 3 pieces foreach long side, which already add up to 10 pieces. (Even if he did use 10 pieces that were 2.5 inchesby inch each, unless he does more cutting, the picture and the frame would not form a rectangle

with constant thickness.)

Activity Synthesis

Ask a few students to show their “frames.” Consider asking questions such as:

“How did you decide how thick the frame should be?” (I tried a couple of different thicknessesto see if they would work. I first tried inch, but that didn't give enough to frame the entire

picture. Then, I tried inch, but that was too thin.)

“How did you know what thickness would be too large or too small?” (It is hard to tell, but Iknow that the framing needs to have a linear measurement of at least 22 inches, enough forthe entire perimeter of the picture, plus some more length for the corners. If the frame is toothin, there will be extra material. If it is too thick, there won't be enough to get the22-plus-some-inch length.)

“How did you determine the thickness such that no framing materials were left unused? Is itpossible to determine this?” (I tried to find a way to evenly spread the area of the framingmaterial around the picture, but I wasn't quite sure how to do that.)

“What frame thickness did you end up with?”

“Were all the strips or pieces the same size?”

Students are likely to share the challenges they encountered along the way. Tell students that, inthis unit, they will learn strategies that are more effective than trial and error or solving problemssuch as this one.

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Support for English Language Learners

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students forthe whole-class discussion. Invite students to quietly circulate and examine at least 2 other“frames” in the room. Give students quiet think time to consider what is the same and what isdifferent about the constructed frames. Next, ask students to find a partner to discuss whatthey noticed. Listen for and amplify observations that include connections across approaches,challenges faced identifying dimensions of framing material, and the need for moresophisticated strategies for solving this problem using quadratic equations.Design Principle(s): Cultivate conversation

1.3 Representing the Framing Problem10 minutesThis activity allows students to formulate a mathematical model around the framing task they sawearlier (MP4). Students are prompted to write an equation but are not expected to solve it at thispoint. In writing an equation and interpreting the solution in context, students practice reasoningquantitatively and abstractly (MP2).

The prompt to write an equation is left relatively open to allow for different approaches. Forinstance, students may think in terms of:

The length and width of the framed picture. The length is 7 inches plus the thickness of theframe, , on either side, which gives . The height is 4 inches plus the same thickness, ,for the top and bottom, which gives . The total area is or 38 squareinches, so the equation is .

The area of the picture plus the area of the frame. The area of the picture is 28 square inches.The area of the frame, in square inches, can be found by decomposing it into four squares (inthe corners) that are by or each, two rectangles that are each (top and bottom), andtwo rectangles that are each (left and right). The equation isor .

As students work, monitor for different strategies students use. Invite students with contrastingapproaches to share later.

Addressing

HSA-CED.A.1



MLR6: Three Reads

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Launch

Tell students that they are now to write an equation to represent the quantities in the framingproblem. Consider arranging students in groups of 2 and asking them to think quietly about thequestions before conferring with their partner.

If students have trouble getting started, suggest that they start by labeling the diagram withrelevant lengths. Then, if needed, use scaffolding questions such as:

“How could we show that the frame is the same thickness all the way around?” (Use the samevariable, for example , to label the thickness of the strip of framing on all four sides.)

“What is the combined area of the picture and the framing material?” (38 square inches,because )

“How could we express the width of the framed picture?” ( ) “What about the height ofthe framed picture?” ( )

“Once the framing material is cut up and arranged around the picture, how could we expressits area?” (Two rectangles with area and two rectangles with area , or

)


Reading: MLR6 Three Reads. Use this routine to support reading comprehension, withoutsolving, for students. Use the first read to orient students to the situation. Ask students todescribe what the situation is about without using numbers (creating a frame for a rectangularpicture). If students understand the situation based on their work in the prior activity, skip tothe next read. After the second read, students list any quantities that can be counted ormeasured, without focusing on specific values (dimensions of the picture and framingmaterials). During the third read, the question or prompt is revealed. Invite students tobrainstorm possible strategies, referencing the relevant quantities named after the secondread. This helps students connect the language in the word problem and the reasoning neededto solve the problem while keeping the cognitive demand of the task.Design Principle: Support sense-making


Representation: Internalize Comprehension. Represent the same information through differentmodalities by using diagrams. Provide students with graph paper and suggest that they drawand label their diagrams with everything they know so far. Invite students to suggest types ofdiagrams that might be helpful to draw, such as the strips of framing material, and the pictureboth with and without the frame.Supports accessibility for: Conceptual processing; Visual-spatial processing

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Anticipated Misconceptions

Some students may struggle to express the overall length and width of the framed picture becauseof trouble combining numbers and variables. Consider drawing a segment composed of 3 pieces oflength 1.5 inches, 10 inches, and 1.5 inches, and prompting students to find the length of the entiresegment. Then, change each 1.5 to an and ask for an expression for the length of the entiresegment ( ).

If students mistake the result of adding and as , consider drawing a square with side lengthand ask students to write expressions for the perimeter and area. Then, ask them to point out thedifference between multiplying to find area and adding to find total length. If needed, draw a fewmore segments that are decomposed into parts, with each part labeled with a number or a variableexpression, and ask students to write expressions for the total length.


Here is a diagram that shows the picture with a frame that is the same thickness all the wayaround. The picture is 7 inches by 4 inches. The frame is created from 10 square inches offraming material (in the form of a rectangle measuring 4 inches by 2.5 inches).

1. Write an equation to represent the relationship between the measurements of thepicture and of the frame, and the area of the framed picture. Be prepared to explainwhat each part of your equation represents.

2. What would a solution to this equation mean in this situation?

Student Response

1. or (or equivalent)

2. A solution would represent the thickness of the frame when all of the framing material is used.

Activity Synthesis

Select students to present their equations and reasoning. If no students bring up one of thestrategies shown in the Activity Narrative, bring it up. Consider asking students whether

and (or other correct equations that come up) areequivalent and how to check if they are.


Discuss what a solution to these equations represents. Make sure students understand that asolution reveals the thickness of the frame when all of the framing material (10 square inches) isused.

Solicit some ideas on how they might go about finding the solution(s). Students may mention:

graphing the equation and finding a point on the graph with a-coordinate of 38

evaluating the expression on one side of the equation at different values of until it has avalue of 38

using a spreadsheet to evaluate the expression at different values of and see what valuesproduce an output of 38

We can see that the equations and are each composedof a quadratic expression. Tell students that these are examples of quadratic equations. They will learnseveral techniques for using algebra to solve equations like this.

Formally, a quadratic equation is defined as one that can be written in the form ofwhere is not 0. This formal definition will be introduced in the next lesson, when students workwith equations of that form and make sense of them in context.

Lesson SynthesisTo highlight reasons for writing and solving an equation, discuss questions such as:

“In this lesson, we saw a few examples where we knew the output of a function and wereinterested in finding the input that produced it. In the launched potato situation, what outputswere known?” (The potato’s heights or distances from the ground.) “What inputs did we wantto know?” (The times at which the potato was at certain heights.)

“In the framing problem, what was the output that we cared about?” (The total area of theframed picture, which was 38 square inches.) “What information was the input?” (Thethickness of the frame that would produce that total area.)

“Why might it be helpful to write an equation to represent a problem such as the one aboutframing?” (An equation helps us see the relationship between the quantities in the problem,including those whose values are unknown. We can solve for those values.)

“In the framing situation, if represents the thickness of the frame, in inches, what would asolution to the equation mean?” (The thickness of the frame that wouldproduce a total area of 50 square inches.)

"Equations such as and are called quadraticequations. Why do you think equations like these are described as quadratic?" (Eachequation has a quadratic expression in it. Each one relates quantities in a quadratic function.)

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(Quadratic equations will be defined more formally in the next lesson. It is fine and expected thatstudents have an informal or partial understanding of what they are at this point.)

1.4 Interpreting a SolutionCool Down: 5 minutesBuilding Towards

HSA-CED.A.3


A framed picture has a total area , in square inches. The thickness of the frame isrepresented by , in inches. The equation relates these two variables.

1. What are the length and width of the picture without the frame?

2. What would a solution to the equation mean in this situation?

Student Response

1. 8 inches and 10 inches

2. A solution would represent the thickness of a frame that results in a total area of 100 squareinches.

Student Lesson SummaryThe height of a softball, in feet, seconds after someone throws it straight up, can be definedby . The input of function is time, and the output is height.

We can find the output of this function at any given input. For instance:

At the beginning of the softball's journey, when , its height is given by .

Two seconds later, when , its height is given by .

The values of and can be found using a graph or by evaluating the expressionat those values of .

What if we know the output of the function and want to find the inputs? For example:

When does the softball hit the ground?

Answering this question means finding the values of that make , or solving.

How long will it take the ball to reach 8 feet?

This means finding one or more values of that make , or solving the equation.

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The equations and are quadratic equations. One wayto solve these equations is by graphing .

To answer the first question, we can look for the horizontal intercepts of the graph,where the vertical coordinate is 0.

To answer the second question, we can look for the horizontal coordinates thatcorrespond to a vertical coordinate of 8.

We can see that there are two solutions to theequation .

The softball has a height of 8 feet twice, whengoing up and when coming down, and theseoccur when is about 0.1 or 1.9.

Often, when we are modeling a situation mathematically, an approximate solution is goodenough. Sometimes, however, we would like to know exact solutions, and it may not bepossible to find them using a graph.

In this unit, we will learn more about quadratic equations and how to solve them exactlyusing algebraic techniques.

Glossaryquadratic expression

Lesson 1 Practice ProblemsProblem 1

StatementA girl throws a paper airplane from her treehouse. The height of the plane is a function oftime and can be modeled by the equation . Height is measured in feet

and time is measured in seconds.

a. Evaluate and explain what this value means in this situation.

b. What would a solution to mean in this situation?

c. What does the equation mean?

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d. What does the model say about the airplane 2.5 seconds after the girl throws it if eachof these statements is true?

Solutiona. . Sample explanation: The girl’s treehouse is 25 feet tall. The initial height of the

plane is 25 feet.

b. is the time when the plane hits the ground.

c. After 9 seconds, the plane is 7 feet above the ground.

d. Sample response: The height of the airplane is 28.125 feet. The plane reaches its maximumheight at 2.5 seconds.

Problem 2StatementA square picture has a frame that is 3 inches thick all the way around. The total side length ofthe picture and frame is inches.

Which expression represents the area of the square picture, without the frame? If you getstuck, try sketching a diagram.

A.

B.

C.

D.

SolutionD


Problem 3StatementThe revenue from a youth league baseballgame depends on the price of per ticket, .

Here is a graph that represents the revenuefunction, .

Select all the true statements.

A. is a little more than 600.

B. is a little less than 5.

C. The maximum possible ticket price is $15.

D. The maximum possible revenue is about $1,125.

E. If tickets cost $10, the predicted revenue is $1,000.

F. If tickets cost $20, the predicted revenue is $1,000.

Solution["A", "D", "E", "F"]

Problem 4StatementA garden designer designed a square decorative pool. The pool issurrounded by a walkway.

On two opposite sides of the pool, the walkway is 8 feet. On the othertwo opposite sides, the walkway is 10 feet.

Here is a diagram of the design.

The final design for the pool and walkway covers a total area of 1,440 square feet.

a. The side length of the square pool is . Write an expression that represents:

i. the total length of the rectangle (including the pool and walkway)


ii. the total width of the rectangle (including the pool and walkway)

iii. the total area of the pool and walkway

b. Write an equation of the form: . What does a solution to theequation mean in this situation?

Solutiona. The expressions for the first two parts may be reversed, as students may interpret the length

and width differently.i. . There are 10 feet of walkway on either side of the pool and the pool is feet long.

ii. . There are 8 feet of walkway on either side of the pool and the pool is feet long.

iii. or (or equivalent)

b. Sample response: . A solution would represent the side length of thepool when the walkway has the given widths and the total area is 1,440 square feet.

Problem 5StatementSuppose and each represent the position number of a letter in the alphabet, butrepresents the letters in the original message and represents the letters in a secret code.The equation is used to encode a message.

a. Write an equation that can be used to decode the secret code into the original message.

b. What does this code say: "OCVJ KU HWP!"?

Solutiona.

b. The original message says: "MATH IS FUN!"

(From Unit 4, Lesson 15.)

Problem 6StatementAn American traveler who is heading to Europe is exchanging some U.S. dollars for Europeaneuros. At the time of his travel, 1 dollar can be exchanged for 0.91 euros.

a. Find the amount of money in euros that the American traveler would get if heexchanged 100 dollars.

b. What if he exchanged 500 dollars?


c. Write an equation that gives the amount of money in euros, , as a function of the dollaramount being exchanged, .

d. Upon returning to America, the traveler has 42 euros to exchange back into U.S. dollars.How many dollars would he get if the exchange rate is still the same?

e. Write an equation that gives the amount of money in dollars, , as a function of the euroamount being exchanged, .

Solutiona. 91 euros

b. 455 euros

c.

d. 46.15 dollars

e.


Problem 7StatementA random sample of people are asked to give a taste score—either "low" or "high"—to twodifferent types of ice cream. The two types of ice cream have identical formulas, except theydiffer in the percentage of sugar in the ice cream.

What values could be used to complete the table so that it suggests there is an associationbetween taste score and percentage of sugar? Explain your reasoning.

12% sugar 15% sugar

low taste score 239

high taste score 126

SolutionSample response: 61 for the first value and 300 for the second value. Using these values means that61 out of 300 people surveyed gave the ice cream with 15% sugar a low taste score, and 300 out of426 people gave the ice cream with 15% sugar a high taste score. This implies that there is anassociation between the percentage of sugar and the taste score for ice cream.



Lesson 2: When and Why Do We Write QuadraticEquations?

GoalsRecognize the limitations of certain strategies used to solve a quadratic equation.

Understand that the factored form of a quadratic expression can help us find the zeros of aquadratic function and solve a quadratic equation.

Write quadratic equations and reason about their solutions in terms of a situation.

Learning TargetsI can recognize the factored form of a quadratic expression and know when it can be usefulfor solving problems.

I can use a graph to find the solutions to a quadratic equation but also know its limitations.

Lesson NarrativeIn this lesson, students revisit some situations that can be modeled with quadratic functions. Theyanalyze and interpret given equations, write equations to represent relationships and constraints(MP4), and work to solve these equations. In doing so, students see that sometimes solutions toquadratic equations cannot be easily or precisely found by graphing or reasoning.

In an earlier unit, students saw that when a function is defined by a quadratic expression infactored form, the zeros of the function could be easily identified. Here, they notice that whena quadratic equation is written as , solving the equation is also relatively simple.This revelation motivates upcoming work on rewriting quadratic expressions into factored form.

Alignments

Building On

HSA-REI.A.1: Explain each step in solving a simple equation as following from the equality ofnumbers asserted at the previous step, starting from the assumption that the originalequation has a solution. Construct a viable argument to justify a solution method.

HSA-REI.B.3: Solve linear equations and inequalities in one variable, including equations withcoefficients represented by letters.

Addressing

HSA-REI.B.4: Solve quadratic equations in one variable.

Building Towards


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Unit 7 Lesson 2 43



MLR2: Collect and Display

MLR8: Discussion Supports

Poll the Class

Think Pair Share


Be prepared to display a graph using technology during the activity synthesis of The Flying PotatoAgain.


Let’s try to solve some quadratic equations.

2.1 How Many Tickets?Warm Up: 5 minutesThis warm-up activates what students know about interpreting equations in context and aboutsolving for a variable. The given equation is linear and is relatively straightforward. The workprepares students to reason about quadratic equations in the lesson.

To find the unknown input in each question, students might:

Try different values of until they find one that yields the specified value of .

Reason backwards (subtract 2.5 from 62.50 and then divide the result by 12) without writingout the steps.

Solve and by performing the same operation to each sideto isolate .

Building On

HSA-REI.B.3


The expression represents the cost to purchase tickets for a play, where is thenumber of tickets. Be prepared to explain your response to each question.

1. A family paid $62.50 for tickets. How many tickets were bought?

2. A teacher paid $278.50 for tickets for her students. How many tickets were bought?

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Student Response

1. 5. Sample reasoning: Subtracting 2.50 from 62.50 gives 60. If is 60, then is 5.

2. 23. Sample reasoning: Subtracting 2.50 from 278.50 gives 276. If is 276, then is 23.

Activity Synthesis

Ask students to share their responses and reasoning. Highlight the different strategies used toanswer the questions.

If no students thought of the situation in terms of a function and wrote an equation, point out thatwe can think about cost as being a function of the number of tickets. In answering the questions,we were looking for the inputs that produce different outputs. This can be done by solvingequations. In the case of linear equations, we can “do the same thing to each side” to isolate thevariable.

2.2 The Flying Potato Again15 minutesThis task prompts students to try different ways to solve a quadratic equation. They are familiarwith solving equations by performing the same operation to each side of an equation, but here theysee that this is not really a workable strategy. Students are also discouraged from using a graph tosolve the equation. Because they do not yet know an efficient way to use algebra to solve

, they need to try different strategies and persevere in problem solving (MP1).

As students work, notice those who use strategies listed in the Activity Synthesis. Ask them to sharetheir approach during discussion.

Building On

HSA-REI.A.1

Building Towards

HSA-CED.A.1

HSA-REI.B.4



Poll the Class

Think Pair Share

Launch

Arrange students in groups of 2. Give them a few minutes of quiet think time and then time tocollaborate on solving the equations.

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Unit 7 Lesson 2 45

No graphing technology should be used in this activity.


Conversing: MLR8 Discussion Supports. In groups of two, students should take turns describingwhat steps they think should be taken next to find the solutions and explaining the reasoningbehind those steps. Display the following sentence frames for all to see: “We should do ____next because . . .”, and “I noticed ___ , so I think . . . .” Encourage students to challenge eachother when they disagree. This will help students clarify their reasoning when persevering inproblem solving quadratic equations.Design Principle(s): Support sense-making; Maximize meta-awareness


The other day, you saw an equation that defines the height of a potato as a function of timeafter it was launched from a mechanical device. Here is a different function modeling theheight of a potato, in feet, seconds after being fired from a different device:

1. What equation would we solve to find the time at which the potato hits the ground?

2. Use any method except graphing to find a solution to this equation.

Student Response

1.

2. Answers vary. Students are likely to find an approximate solution (about 5.7 seconds) by trialand error.

Activity Synthesis

Poll the class on their solutions to the equation and record and display the solutions for all to see.Then, ask some students to share their strategies and any associated challenges. If not mentionedby students, discuss the limitations of these approaches:

Isolating the variable: If we try to solve by performing the same operation to each side of theequation, we quickly get stuck.

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But then what? If we add to each side, we now have a variable on both sides and cannot combineany like terms. We could multiply or divide each side by any constant we wish, but we are no closerto isolating .

Guessing and checking: We can evaluate the quadratic expression at different values of untilthe expression has a value that is 0 or close to 0. For example, when is 4, the expression hasa value of 128. At , it has a value of 64, and at , it has a value of -32. That means isbetween 5 and 6, so we need to try different decimal values in that range.

This process is laborious, and may not get us to a precise solution.

Graphing: Students may suggest that a graph would allow them to solve the problem muchmore quickly. Use graphing technology to demonstrate that if we graph the equation ,an approximate solution given is 5.702, as shown in the image.

If we evaluate , however, we getroughly -0.044864, rather than exactly 0.

A graph is useful for approximating values, butit isn’t always possible to use it to find exactvalues.

Tell students that in this unit they will learn some efficient strategies for solving equations likethese.


Representation: Internalize Comprehension. Use color and annotations to illustrate studentthinking. As students share their reasoning about methods to find a solution, scribe theirthinking on a visible display. Display sentence frames for students to use during the discussion,such as: “This method works/doesn’t work because . . . .” and “Another strategy would be _____because . . . .”Supports accessibility for: Visual-spatial processing; Conceptual processing

2.3 Revenue from Ticket Sales15 minutesThis activity aims to show that it is relatively easy to solve a quadratic equation when one side of theequation is zero and the other side is a quadratic expression in factored form, and that it may be alittle tricky to solve the equation otherwise.

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Unit 7 Lesson 2 47

The activity prompts students to recall what they learned in an earlier unit, that the zeros of afunction correspond to the horizontal intercepts of the graph representing that function, and thatthe zeros are the solutions to an equation of the form .

As students make sense of the equations and ways to use them to solve a contextual problem, theypractice reasoning quantitatively and abstractly (MP2).

Addressing

HSA-REI.B.4

Building Towards

HSA-CED.A.1



Launch

Keep students in groups of 2.

Read the opening paragraph in the task statement. Give students a moment to think about howmuch the school would collect if they sell the tickets at $5 each (it would collect $875). Briefly surveyhow they found out the answer. (Students are likely to have used the factored form because it lendsitself to simpler calculations.)

Ask students to recall the form in which each quadratic expression is written. Then, give them aminute to talk to a partner and recall at least two things about each form and what the form mighttell us about the graph of the function that the expression defines.

Record their responses for all to see. If no students mentioned the connection between either ofthe forms to the horizontal intercepts of the graph or the zeros of the function, ask them about it.

Tell students that they do not need to find the ticket prices in the second question, but only to thinkabout how to go about doing so.


Conversing: MLR2 Collect and Display. As students discuss the differences between the forms ofeach equation with their partner, listen for and collect the language students use: vertex,-intercept, zeros, etc. Write the students’ words and phrases on a visual display and refer to it

during the discussion later in the launch. As the activity continues, update the visual displayand continue to do so throughout the remainder of the lesson. Remind students to borrowlanguage from the display as needed. This will help students read and use mathematicallanguage during their partner and whole-class discussions.Design Principle(s): Maximize meta-awareness; Support sense-making

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If students struggle to connect the expressions that define the function to the questions, ask themwhat the input and output of the function represent. If students struggle with the first question, askthem what values of would yield a value of 0 for the expression.


The expressions and define the same function. The function modelsthe revenue a school would earn from selling raffle tickets at dollars each.

1. At what price or prices would the school collect $0 revenue from raffle sales? Explain orshow your reasoning.

2. The school staff noticed that there are two ticket prices that would both result in arevenue of $500. How would you find out what those two prices are?

Student Response

1. $0 and $40. Sample reasoning: If the equation is true, then either is equal to0, or is equal to 0. If , then is 40, because is 0.

2. Sample response: We could write and , but aside fromtrial and error or graphing, we don’t know a straightforward way to solve either equation.


Can you find the following prices without graphing?

1. If the school charges $10, it will collect $1,500 in revenue. Find another price that wouldgenerate $1,500 in revenue.

2. If the school charges $28, it will collect $1,680 in revenue. Find another price that wouldgenerate $1,680 in revenue.

3. Find the price that would produce the maximum possible revenue. Explain yourreasoning.

Student Response

1. $30

2. $12

3. $20. Moving the same distance away from each horizontal intercept results in equal revenue.The farther away we move from one horizontal intercept in the direction of the otherintercept, the greater the revenue. An optimal ticket price of $20, producing $2,000 in revenue,is found in the middle point between the two intercepts, at .

Unit 7 Lesson 2 49

Activity Synthesis

Invite students to share their responses and strategies. Make sure students see that the firstquestion can be represented by solving the equation , and that the secondquestion can be represented by solving either or .

Although students are not yet formally introduced to the zero product property, they do haveexperience with finding the zeros of a quadratic function when given an expression in factoredform. This prior knowledge enables them to reason about the solutions to the equation

. For instance, noticing the factor , students are likely to say that one zero of thefunction is 0.

Ask students,

“How can you show, without graphing, that will produce no revenue, or that it is asolution to the equation ?” (When , the factor is 0 and therefore theentire expression equals 0.)

“How can you show that will also produce no revenue, or that it is also a solution to thesame equation?” (When , the factor is 0, and likewise, the entire expressionis 0.)

“Can we use the same reasoning to find the solutions to or? Why or why not?” (No, not easily. Neither show the zeros of a function.

There are many pairs of factors that have 500 as a product.)

Make sure students see that it is fairly straightforward to find the solutions to equations such as, but the same cannot be said about equations such as or

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Highlight that all the equations in this activity are quadratic equations. Explain that a quadraticequation is one that can be written in the form of , and where is not 0.

If time permits, ask students to show how all of the equations seen here can be written in this form.


Representation: Internalize Comprehension. Use color and annotations to illustrate studentthinking. As students share their reasoning about finding the solutions for which the ticketsales would equal zero, scribe their thinking on a visible display. Consider highlighting eachfactor separately and annotating to show that when one factor equals zero, the entireexpression equals zero regardless of the value of the other factor.Supports accessibility for: Visual-spatial processing; Conceptual processing

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Lesson SynthesisHelp students to reflect on the key ideas of the lesson by discussing, as a class or with a partner,questions such as:

“What are some limitations of solving by guessing and checking? Whatabout by graphing?”

“Which equation do you think is easier to solve: or ?Why?”

“Which is easier to solve: or ? Why?”

2.4 The Movie TheatreCool Down: 5 minutesAddressing

HSA-REI.B.4

Building Towards

HSA-CED.A.1

Launch

No graphing technology should be used in this cool-down.


A movie theatre models the revenue from ticket sales in one day as afunction of the ticket price, . Here are two expressions defining thesame revenue function.

1. According to this model, how high would the ticket price have to be for the theater tomake $0 in revenue? Explain your reasoning.

2. What equation can you write to find out what ticket price(s) would allow the theaterto make $600 in revenue?

Student Response

1. $30. Sample reasoning: If , then either or . If the latter is atrue equation, must be 30.

2. or

Student Lesson SummaryThe height of a potato that is launched from a mechanical device can be modeled by afunction, . Here are two expressions that are equivalent and both define function .

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Unit 7 Lesson 2 51

Notice that one expression is in standard form and the other is in factored form.

Suppose we wish to know, without graphing the function, the time when the potato will hitthe ground. We know that the value of the function at that time is 0, so we can write:

Let's try solving , using some familiar moves. For example:

Subtract 96 from each side:

Apply the distributive property to rewrite the expressionon the left:

Divide both sides by -16:

Apply the distributive property to rewrite the expressionon the left:

These steps don’t seem to get us any closer to a solution. We need some new moves!

What if we use the other equation? Can we find the solutions to ?

Earlier, we learned that the zeros of a quadratic function can be identified when theexpression defining the function is in factored form. The solutions toare the zeros to function , so this form may be more helpful! We can reason that:

If is 6, then the value of is 0, so the entire expression has a value of 0.

If is -1, then the value of is 0, so the entire expression also has a value of 0.

This tells us that 6 and -1 are solutions to the equation, and that the potato hits the groundafter 6 seconds. (A negative value of time is not meaningful, so we can disregard the -1.)

Both equations we see here are quadratic equations. In general, a quadratic equation is anequation that can be expressed as .

In upcoming lessons, we will learn how to rewrite quadratic equations into forms that makethe solutions easy to see.

Glossaryfactored form (of a quadratic expression)

quadratic equation

standard form (of a quadratic expression)

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zero (of a function)


StatementA set of kitchen containers can be stacked to save space. The height of the stack is given bythe expression , where is the number of containers.

a. Find the height of a stack made of 8 containers.

b. A tower made of all the containers is 40.6 cm tall. How many containers are in the set?

c. Noah looks at the equation and says, “7.6 must be the height of a single container.” Doyou agree with Noah? Explain your reasoning.

Solutiona. 19.6 cm

b. 22 containers

c. No. Sample reasoning:The expression is equal to 7.6 when , but a stack of zero containers would have noheight.

When is 1, the height of the stack is , which is 9.1 cm.

Problem 2StatementSelect all values of that are solutions to the equation .

A. -7

B. -5

C. -3

D. 0

E. 3

F. 5

G. 7

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Unit 7 Lesson 2 53

Solution["E", "F"]

Problem 3StatementThe expressions and define the same function, .

a. Which expression makes it easier to find ? Explain your reasoning.

b. Find .

c. Which expression makes it easier to find the values of that make the equationtrue? Explain or show your reasoning.

d. Find the values of that make .

Solutiona. . Sample reasoning: It is easy to evaluate the expression

because the first two terms are equal to zero, leaving only the last term,-60.

b. -60

c. . Sample reasoning: With factored form, we know that if the wholeexpression is equal to zero, then one of the factors must be zero.

d. 4 and

Problem 4StatementA band is traveling to a new city to perform a concert. The revenue from their ticket sales is afunction of the ticket price, , and can be modeled with .

What are the ticket prices at which the band would make no money at all?

Solution$6 and $50


Problem 5StatementTwo students built a small rocket from a kit and attached an altimeter (a device for recordingaltitude or height) to the rocket. They record the height of the rocket over time since it islaunched in the table, based on the data from the altimeter.

time (seconds) 0 1 3 4 7 8

height (meters) 0 110.25 236.25 252 110.25 0

Function gives the height in meters as a function of time in seconds, .

a. What is the value of ?

b. What value of gives ?

c. Explain why .

d. Based on the data, which equation about the function couldtrue: or ? Explain your reasoning.

Solutiona. . The height of the rocket at 3 seconds is 236.25 feet.

b. . The rocket reaches 252 feet at 4 seconds.

c. Sample response: is the height of the rocket before launch and is the height of therocket 8 seconds after launch. They both have the same value of 0 feet because the rocketstarts and ends on the ground.

d. . Sample reasoning: means the height of the rocket 2 seconds afterlaunch is 189 feet. means the height after 189 seconds is 2 feet, which isn't likelybecause the rocket lands at 8 seconds.


Problem 6StatementThe screen of a tablet has dimensions 8 inches by 5 inches. The border around the screenhas thickness .

Unit 7 Lesson 2 55

a. Write an expression for the total area of the tablet,including the frame.

b. Write an equation for which your expression is equal to 50.3125. Explain what a solutionto this equation means in this situation.

c. Try to find the solution to the equation. If you get stuck, try guessing and checking. Itmay help to think about tablets that you have seen.

Solutiona. or equivalent

b. . A solution represents the thickness of the border that would makethe total area of the tablet 50.3125 square inches.

c. 0.75 or inch


Problem 7StatementHere are a few pairs of positive numberswhose sum is 15. The pair of numbers thathave a sum of 15 and will produce the largestpossible product is not shown.

Find this pair of numbers.

firstnumber

secondnumber

product

1 14 14

3 12 36

5 10 50

7 8 56

Solution7.5 and 7.5. The product is 56.25


Problem 8StatementKilometer is a measurement in the metric system, while mile is a measurement in thecustomary system. One kilometer equals approximately 0.621 mile.

a. The number of miles, , is a function of the number of kilometers, . What equationcan be written to represent this function?

b. The number of kilometers, , is a function of the number of miles, . What equationcan be written to represent this function?

c. How are these two functions related? Explain how you know.

Solutiona. (or equivalent)

b. (or equivalent)

c. Sample response: The two functions are inverses because they undo each other: the oppositeof multiplying by 1.61 is dividing by 1.61, .


Unit 7 Lesson 2 57

Lesson 3: Solving Quadratic Equations by Reasoning

GoalsFind the solutions to simple quadratic equations and justify (orally) the reasoning that leads tothe solutions.

Understand that a quadratic equation may have two solutions.

Learning TargetsI can find solutions to quadratic equations by reasoning about the values that make theequation true.

I know that quadratic equations may have two solutions.

Lesson NarrativeIn this lesson, students begin to solve quadratic equations by reasoning about what values wouldmake the equations true and by using structure in the equations. The idea that some quadraticequations have two solutions is also made explicit. Students may begin to record their reasoningprocess as steps for solving, but this is not critical at this point as it will be emphasized in a laterlesson.

As students reason about an equation, they may intuitively perform the same operation on eachside of the equal sign to get closer to the solution(s). When they reach equations of the form

, it is important to refrain from telling students to “take the square root of eachside.” Instead, focus on reasoning about values that would make the equation true (MP2).

For example, to solve , we could divide each side by 4 and get . Encouragestudents to interpret this equation as: “Some number being squared gives 25” and to reason: “Thereare two different values that can be squared to get 25: -5 and 5.”

Reasoning this way helps to curb two common misconceptions:

that the only solution to an equation such as is . Each positive number has two

square roots, one positive and the other negative. By convention, the radical symbol

refers to the positive square root. So the number refers only to the positive square rootof 25 and does not capture the negative square root.

that squaring is invertible. The inverse of an operation undoes that operation. Suppose wemultiply a number by 8. Dividing the product by 8 takes us back to the original number, so wesay that division by 8 is the inverse operation of multiplication by 8, and that multiplication by8 is invertible.

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Suppose we square -3, which gives 9. The operation of taking the square root using radical symbol

takes 9 to , which is positive 3, not the original number. Because there are two possiblenumbers whose square is 9, we don’t consider squaring to be invertible.

When solving an equation such as , these notations are commonly used to express thesolutions:

or

and

The use of “or” is really a shorthand for: "If is a number such that , then or .”The use of “and” is a shorthand for: “Both and are values that make the equation

true." Either notation can be appropriate, depending on how the question is stated.

Technology isn't required for this lesson, but there are opportunities for students to choose to useappropriate technology to solve problems. Consider making technology available.

Solving the problems in the lesson gives students many opportunities to engage in sense making,perseverance, and abstract reasoning (MP1, MP2).

Alignments

Addressing


HSA-REI.B.4.b: Solve quadratic equations by inspection (e.g., for ), taking square roots,completing the square, the quadratic formula and factoring, as appropriate to the initial formof the equation. Recognize when the quadratic formula gives complex solutions and writethem as for real numbers and .


Anticipate, Monitor, Select, Sequence, Connect



Let’s find solutions to quadratic equations.

3.1 How Many Solutions?Warm Up: 10 minutesPreviously, students saw that a function could have two input values that give the same outputvalue (which could be 0). The input values were primarily interpreted in terms of a situation. In thiswarm-up, students begin to think more abstractly about this process—in terms of finding the

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Unit 7 Lesson 3 59

solutions to an equation. They recognize that some quadratic equations have one solution andothers have two.

All of the equations can be solved by reasoning and do not require formal knowledge of algebraicmethods such as rewriting into factored form or completing the square. For example, for

, students can reason that must be 4 because that is the only number that, whensubtracted by 1, gives 3.

Finding the solutions of these equations, especially the last few equations, requires perseverance inmaking sense of problems and of representations (MP1).

Addressing

HSA-REI.A.1

HSA-REI.B.4.b

Launch

Ask students to evaluate and . Make sure they recall that both products are positive16.


When solving , some students may confuse with and conclude that the solutions

are and . Clarify that means 2 times , and that the only thing being squared is the .

If both the 2 and are squared, a pair of parentheses is used to group the 2 and the so that weknow both are being squared.


How many solutions does each equation have? What are the solution(s)? Be prepared toexplain how you know.

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2.

3.

4.

5.

6.

7.

Student Response

1. Two solutions: and

2. One solution:

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5. One solution:



Activity Synthesis

Ask students to share their responses and reasoning. After each student explains, ask the class ifthey agree or disagree and discuss any disagreements.

Make sure students see that in cases such as and , the solutions toeach equation may not necessarily be opposites, as was the case in the preceding equations. Forexample, in the last question, we want to find a number that produces 4 when it is squared. Thatnumber can be 2 or -2. If the number is 2, then is 3. If the number is -2, then is -1.

If time permits, discuss questions such as:

"The equation has only one solution, while has two. Whyis that?" (The former has only one solution because the only number that equals 0 whensquared is 0 itself. The latter has two solutions because there are two numbers that, whensquared, equal 4.)

"In an equation like , how can we tell that there are two solutions?" (There are twofactors here, either of which could make the product 0.)

3.2 Finding Pairs of Solutions25 minutesIn this activity, students encounter quadratic equations that are slightly more elaborate than thosein the warm-up but that can still be solved by reasoning in various ways.

Students’ approaches likely vary in efficiency and effectiveness. Monitor for students who:

Substitute different values for until hitting on the ones that work.

Use technology to make a table and looking for the target value.

Use technology to graph and and find theintersection.

Reason about and make use of the structure in the equations. For example, seeingas “432 is 3 times something squared” will lead to 144 as the “something squared” and 12 and-12 as the “something.” Seeing as “something squared is 100” enables them toarrive at 10 and -10 for the value of , and then reason that must be 15 or -5.

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Unit 7 Lesson 3 61

Solve algebraically, by performing the same operation to each side of the equation, and whenarriving at an equation of the form , reasoning that the solutions (thevalues of ) are the positive and negative square roots of that number.

Identify students who use these strategies and ask them to share during the classroom discussion.Alternatively, consider arranging for students who use the same strategy to discuss and thenprepare to share their approach.

Students who use technology to solve the equations engage in choosing tools strategically (MP5).Those who solve by analyzing and taking advantage of the composition of the equations practicemaking use of structure (MP7).

Addressing

HSA-REI.B.4.b




Launch

Consider arranging students in groups of 2 and asking them to work quietly for a few minutesbefore discussing their thinking with a partner.

Give students access to graphing technology and spreadsheet tool, if requested.


Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students forthe whole-class discussion. At the appropriate time, invite student pairs to create a visualdisplay of their strategies for solving the quadratic equations. Allow students time to quietlycirculate and analyze the strategies in at least two other visual displays in the room. Givestudents quiet think time to consider what is the same and what is different about theirsolution strategies. Next, ask students to return to their partner and discuss what they noticed.Listen for and amplify observations that highlight advantages and disadvantages to eachmethod. This will help students make connections between different strategies for solvingquadratic equations.Design Principle(s): Optimize output; Cultivate conversation

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Engagement: Develop Effort and Persistence. Chunk this task into manageable parts for studentswho benefit from support with organizational skills in problem solving. For example, cutproblems into 4 separate slips and have students obtain the new problem only on finishing thelast. Provide students 4 copies of a graphic organizer with two sections—one for work, andanother for notes/explanations. Invite students to record their work for each problem on itsown paper. When presenting multiple strategies and approaches, encourage students to usethe space for notes/explanations to record explanations and alternate strategies.Supports accessibility for: Memory; Organization


Each of these equations has two solutions. What are they? Explain or show your reasoning.

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2.

3.

4.

Student Response

1. 20 and -20. Sample reasoning: Since is 4 more than 400, and can be either20 or -20.

2. 12 and -12. Sample reasoning: Divide each side of the equation by 3. , so or.

3. 11 and -13. Sample reasoning: Since 12 and -12 are the two numbers we can square to make144 and is the number being squared, we need to be one less than 12 or -12. Somust be -13 or 11.

4. 15 and -5


1. How many solutions does the equation have? What are thesolutions?

2. How many solutions does the equation have? What are thesolutions?

3. Write a new equation that has 10 solutions.

Unit 7 Lesson 3 63

Student Response

1. Three solutions: 3, -1, and -5.

2. Two solutions: 2 and 7.

3. Sample response:

Activity Synthesis

Select previously identified students to present their strategies in order of their efficiency, as listedin the Activity Narrative. Where appropriate, help students to make connections between thedifferent strategies. For example, ask students how graphing the expression on each side of theequation and finding the intersection is similar to substituting values for until they find one thatworks.

If no students reasoned about the solutions algebraically, be sure to demonstrate it (without saying“take the square root of each side”) and to record the reasoning process for all to see. For example:

We can interpret as “something plus 4 is 404.” That “something” must be 400, sowe can write . This equation means “something times itself is 400.” That “something”must be 20 or -20, because they each give 400 when squared. The two solutions are therefore20 and -20.

Point out that the reasoning that took us from to gave the sameequation as subtracting 4 from each side of the original equation.

We can see as “144 is something squared,” so the “something” is either 12 or-12. We can represent this with and . The solutions are 11 and -13.

Lesson SynthesisTo help students generalize their reasoning and attend more closely to the structure of quadraticequations, consider revisiting the equations from the lesson and prompting students to articulatewhat the equations might reveal about the solutions. For example:

Display the equations from the warm-up. Ask students if they can tell (without referring to thesolutions they found earlier and without solving again) if an equation would have 0, 1, or 2solutions.

Display the equations from the first activity. Ask students if they can tell (without referring tothe solutions they found earlier) which equations will have two solutions that are opposites(such as 5 and -5) and which will have two solutions that are not opposites (such as 3 and 7).

3.3 Find Both SolutionsCool Down: 5 minutesAddressing

HSA-REI.B.4.b

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Launch

Provide continued access to the tools made available in the previous activity (graphing technology,spreadsheets), if requested.


Find both solutions to the equation . Explain or show your reasoning.

Student Response

9 and -5. Sample reasoning: 100 plus a squared number is 149. That squared number must be 49and the number must be 7 or -7. If , then is 9. If , then .

Student Lesson SummarySome quadratic equations can be solved by performing the same operation to each side ofthe equal sign and reasoning about values of the variable would make the equation true.

Suppose we wanted to solve . We can proceed like this:

Add 75 to each side:

Divide each side by 3:

What number can be squared to get 25?

There are two numbers that work, 5 and -5: and

If , then .

If , then .

This means that both and make the equation true and are solutions to theequation.


StatementConsider the equation .

a. Show that 3, -3, , and are each a solution to the equation.

b. Show that 9 and are each not a solution to the equation.

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Unit 7 Lesson 3 65

Solutiona.

b.

Problem 2StatementSolve . Explain or show your reasoning.

Solution-3 and 5. Sample reasoning: 16 is and .

If is equal to , then is 4 and is 5.

If is , then is -4 and is -3.

Problem 3StatementHere is one way to solve the equation . Explain what is done in each step.

SolutionStep 1: Multiply both sides of the equation by 9.

Step 2: Divide both sides by 5.

Step 3: Find one or more values that, when squared, give 9.

Problem 4StatementDiego and Jada are working together to solve the quadratic equation .

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Diego solves the equation by dividing each side of the equation by 2and then adding 2 to each side. He writes:

Jada asks Diego why he divides each side by 2 and he says, “I want to find a number thatequals 100 when multiplied by itself. That number is half of 100.”

a. What mistake is Diego making?

b. If you were Jada, what could you say to Diego to help him realize his mistake?

Solutiona. Diego wants the square root of 100, not half of 100. Instead of 50 in his equation he should

have 10.

b. Sample responses:“If you substitute 52 back into the original equation, would the two sides be equal? Does

equal 100?”

“What number or numbers, when squared, gives 100?”

Problem 5StatementAs part of a publicity stunt (an event designed to draw attention), a TV host drops awatermelon from the top of a tall building. The height of the watermelon seconds after it isdropped is given by the function , where is in feet.

a. Find . Explain what this value means in this situation.

b. Find . What does this value tell us about the situation?

c. Is the watermelon still in the air 8 seconds after it is dropped? Explain how you know.

Solutiona. 594 feet. This means that 4 seconds after it is dropped, the watermelon is 594 feet off the

ground.

b. 850 feet. Since the watermelon was 850 feet off the ground at the time it was dropped, thismeans that the height of the building is (about) 850 feet tall.

c. No. . The watermelon hits the ground when . Since the height cannot beless than 0 feet, the watermelon will have hit the ground by then.


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Unit 7 Lesson 3 67

Problem 6StatementA zoo offers unlimited drink refills to visitors who purchase its souvenir cup. The cup and thefirst fill cost $10, and refills after that are $2 each. The expression represents thetotal cost of the cup and refills.

a. A family visited the zoo several times over a summer. That summer, they paid $30 forone cup and multiple refills. How many refills did they buy?

b. A visitor has $18 to spend on drinks at the zoo today and buys a souvenir cup. Howmany refills can they afford during the visit?

c. Another visitor spent $10 on this deal. Did they buy any refills? Explain how you know.

Solutiona. 10

b. 4

c. No. Sample reasoning: The first cup of drink is already $10. If that’s the entire amount spenton drinks, no other refills were bought.


Problem 7StatementHere are a few pairs of positive numberswhose sum is 15. The pair of numbers thathave a sum of 15 and will produce the largestpossible product is not shown.

Find this pair of numbers.

firstnumber

secondnumber

product

1 14 14

3 12 36

5 10 50

7 8 56

Solution7.5 and 7.5. The product is 56.25



Problem 8StatementClare is 5 years older than her sister.

a. Write an equation that defines her sister's age, , as a function of Clare’s age, .

b. Write an equation that defines Clare’s age, , as a function of her sister's age, .

c. Graph each function. Be sure to label the axes.

d. Describe how the two graphs compare.

Solutiona.

b.

c. See graphs.

d. Sample response: Thevariables have switchedplaces. The input of onefunction is the output of theother function.


Problem 9StatementThe graph shows the weight of snow as it melts. The weight decreases exponentially.

Unit 7 Lesson 3 69

a. By what factor does the weight of the snowdecrease each hour? Explain how you know.

b. Does the graph predict that the weight of the snow will reach 0? Explain your reasoning.

c. Will the weight of the actual snow, represented by the graph, reach 0? Explain how youknow.

Solutiona. Between 0 and 1 hours, the weight of the snow decreases from 15 kg to 6 kg. The hourly factor

of decrease is .

b. No, the amount of snow predicted by the model will not reach 0. Sample reasoning: Everyhour, the remaining snow is multiplied by . No matter how many times we multiply by , we

always have a positive quantity.

c. Sample response: Yes, the actual snow will completely melt. While the weight predicted by themodel will continue to get smaller and smaller, reaching an amount too small to notice, theactual weight of the snow will become 0. Other factors, like temperature, will impact thepredicted value of the graph.



Lesson 4: Solving Quadratic Equations with the ZeroProduct Property

GoalsGiven quadratic equations where one side is a product of factors and the other is zero, findthe solution(s) and explain (orally and in writing) why the solutions make the equation true.

Understand that the “zero product property” (in written and spoken language) means that ifthe product of two numbers is 0 then one of the factors must also be 0.

Learning TargetsI can explain the meaning of the “zero product property.”

I can find solutions to quadratic equations when one side is a product of factors and the otherside is zero.

Lesson NarrativeIn this lesson, students learn about the zero product property. They use it to reason about thesolutions to quadratic equations that each have a quadratic expression in factored form on oneside and 0 on the other side. They see that when an expression is a product of two or more factorsand that product is 0, one of the factors must be 0. This fact enables us to find unknown values inthe factored expression.

Students also continue to make connections to their earlier work on quadratic functions. They haveseen that sometimes we want to find the input values of a function when the output is zero. Theyalso learned that the factored form can help us identify the zeros of a quadratic function and the

-intercepts of its graph. They have not investigated how or why this form enables us to do so,however. Here, students make use of the structure of a quadratic expression in factored form andthe zero product property to understand the connections between the numbers in the form andthe -intercepts of its graph (MP7).



HSA-REI.B.3: Solve linear equations and inequalities in one variable, including equations withcoefficients represented by letters.

Addressing


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Unit 7 Lesson 4 71

Building Towards



HSA-SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explainproperties of the quantity represented by the expression.


Math Talk


Think Pair Share


Let’s find solutions to equations that contain products that equal zero.

4.1 Math Talk: Solve These EquationsWarm Up: 10 minutesThis Math Talk introduces students to the zero product property and prepares them to use it tosolve quadratic equations. It reminds students that if two numbers are multiplied and the result is0, then one of the numbers has to be 0. Answering the questions mentally prompts students tonotice and make use of structure (MP7).

Building On

HSA-REI.A.1

Building Towards

HSA-REI.B.4.b


Math Talk


Launch

Display one problem at a time and ask students to respond without writing anything down. Givestudents quiet think time for each problem and ask them to give a signal when they have an answer

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and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-classdiscussion.


Representation: Internalize Comprehension. To support working memory, provide students withsticky notes or mini whiteboards.Supports accessibility for: Memory; Organization


What values of the variables make each equation true?

Student Response

-3

0

5

either or

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for allto see. To involve more students in the conversation, consider asking:

“Who can restate ’s reasoning in a different way?”

“Did anyone have the same strategy but would explain it differently?”

“Did anyone solve the problem in a different way?”

“Does anyone want to add on to ’s strategy?”

“Do you agree or disagree? Why?”

Highlight explanations that state that any number multiplied by 0 is 0. Then, introduce the zeroproduct property, which states that if the product of two numbers is 0, then at least one of thenumbers is 0.

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Unit 7 Lesson 4 73


Speaking: MLR8 Discussion Supports. Display sentence frames to support students when theyexplain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .” Somestudents may benefit from the opportunity to rehearse what they will say with a partner beforethey share with the whole class.Design Principle(s): Optimize output (for explanation)

4.2 Take the Zero Product Property Out for a Spin15 minutesIn this activity, students solve equations of increasing complexity and do so by reasoning. Theybegin with linear equations, move toward a series of quadratic expressions in factored form, andend with a cubic expression in factored form. The progression prompts students to reason aboutthe parts and structure of the expressions (MP7), rather than to memorize steps for solving withoutunderstanding, and to notice regularity through repeated reasoning (MP8).

As students discuss their reasoning with their partner, listen for those who invoke the zero productproperty to explain how the last four equations could be solved, and those who notice a pattern inhow the equations could be solved. (Though the last question involves a cubic equation, solving itinvolves the same reasoning as solving quadratic expressions.)

Building On

HSA-REI.A.1

HSA-REI.B.3

Addressing

HSA-REI.B.4


Think Pair Share

Launch

Arrange students in groups of 2. Tell students to work quietly and answer at least half of thequestions before discussing their thinking with a partner.

If needed, remind students that some equations have more than one solution. Because we wantstudents to use reasoning and the structure of equations to develop their solutions, discourage useof graphing technology or spreadsheets in this activity.

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Action and Expression: Internalize Executive Functions. To support development of organizationalskills, check in with students within the first 2–3 minutes of work time, especially whenstudents arrive at problems with two solutions. Look for students who quickly recognize thatthey will need to solve two equations and are organizing their work to account for theappropriate number of solutions for each problem. To support students in recognizingexamples with two solutions, demonstrate annotating the problem by writing two equationsseparately, or drawing arrows from each factor to indicate two solutions. Encourage studentsto prepare their explanations by recording how they recognized the number of solutions asthey work.Supports accessibility for: Memory; Organization


Students may incorrectly think that can represent a different value in each factor in an equation.For example, upon finding -11 and 3 as solutions to , they think that one solutionis for the in and the other for the in .

Remind students that solving the equation is like finding the zeros of thefunction defined by . Although there may be two values of that lead to 0 for thevalue of , only one input can be entered into the function at a time. Ask students tosubstitute the solutions into the equations and check if the expression is equal to 0 each time.

When , the value of the expression is or , which is 0.

When , the value of the expression is or , which is 0.


For each equation, find its solution or solutions. Be prepared to explain your reasoning.

1.

2.

3.

4.

5.

6.

7.

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Unit 7 Lesson 4 75

Student Response

1. 3

2. -11

3.

4. 0 and

5. 3 and -11

6. 3 and

7. 0, -3, and


1. Use factors of 48 to find as many solutions as you can to the equation.

2. Once you think you have all the solutions, explain why these must be the only solutions.

Student Response

1. 7 and -9

2. Sample response: The numbers expressed by and are 8 units apart. The only factorpairs of 48 that are 8 units apart are 4 and 12, as well as -4 and -12. If , then

. If , then .

Activity Synthesis

Invite students to share their strategies for solving the non-linear equations. As they explain, recordand organize each step of their reasoning process and display for all to see.

For example, the equation tells us that, if the product of andis 0, then either is equal to 0, or is equal to 0. We can then organize the rest of thesolving process as:

If is equal to 0, then is 3. If is equal to 0, then

The equation is true when and when .

Emphasize that because at least one of the factors must be 0 for the product to be 0, we can writeeach expression that is a factor to equal to 0 and solve each of these equations separately.


Remind students that we can check our solutions by substituting each one back into the equationand see if the equation remains true. Although the two factors, and , won’t be 0simultaneously when 3 or is substituted for , the expression on the left side of the equation

will have a value of 0 because one of the factors is 0.

When is 3, the expression is or , which is 0.

When is , the expression is or , which is 0.

4.3 Revisiting a Projectile10 minutesThis activity enables students to apply the zero product property to solve a contextual problem andreinforces the idea of solving quadratic equations as a way to reason about quadratic functions.

Previously, students have encountered two equivalent quadratic expressions that define the samequadratic function. Here, they work to show that two quadratic expressions—one in standard formand the other in factored form—really do define the same function. There are several ways to dothis, but an efficient and definitive way to show equivalence would be to use the distributiveproperty to expand quadratic expressions in factored form.

Next, they consider which of the two forms helps them find the zeros of the function and then use itto find the zeros without graphing. The work here reiterates the connections between finding thezeros of a quadratic function and solving a quadratic equation where a quadratic expression thatdefines a function has a value of 0.

Addressing

HSA-REI.B.4

Building Towards

HSA-CED.A.1

HSA-SSE.B.3



Launch

Keep students in groups of 2. Prepare access to graphing technology and spreadsheet tool, in caserequested.

Display the two equations that define for all to see. Tell students that the two equations definethe same function. Ask students how they could show that the two equations indeed define thesame function.

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Unit 7 Lesson 4 77

Give students a moment of quiet time to think of a strategy and test it, and then time to discusswith a partner, if possible. Then, discuss their responses. Some likely strategies:

Graph both equations on the same coordinate plane and show that they coincide.

Inspect a table of values of both equations and show that the same output results for anyinput.

Use the distributive property to multiply the expression in factored form to show that. (Only this reasoning is really a “proof,” but the other

methods supply a lot of evidence that they are the same function.)

Once students see some evidence, ask students to proceed to the activity.


Representation: Internalize Comprehension. Represent the same information through differentmodalities. Display a sketch of a graph of a projectile, and label the -intercept, vertex, andpositive -intercepts. Keep the display visible for the duration of the activity and refer to itwhen students discuss why the negative solution is not viable.Supports accessibility for: Conceptual processing; Language


We have seen quadratic functions modeling the height of a projectile as a function of time.

Here are two ways to define the same function that approximates the height of a projectile inmeters, seconds after launch:

1. Which way of defining the function allows us to use the zero product property to findout when the height of the object is 0 meters?

2. Without graphing, determine at what time the height of the object is 0 meters. Showyour reasoning.

Student Response

1. the one with a quadratic expression in factored form

2. The object has a height of 0 meters after 6 seconds. Sample reasoning: Applying the zeroproduct property to solve gives and . Because the object was

launched at , the negative solution doesn’t make sense in this context.

Activity Synthesis

Ask students to share their responses and reasoning. Discuss questions such as:

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“Why is the factored form more helpful for finding the time when the object has a height of 0meters?” (To find the input values when the output has a value of 0 is to solve the equation

. When the expression is in factored form, we can use the zeroproduct property to find the unknown inputs.)

“What if we tried to solve the equation in standard form by performing the same operation toeach side?” (We would get stuck. For instance, we could add or subtract terms from each side,but then there are no like terms to combine on either side, so we are no closer to isolating thevariable.)

If no students related solving equations in factored form to using the factored form to find thehorizontal intercepts of a graph of a quadratic function, discuss that connection.

“In an earlier unit, we saw that the factored form of a quadratic expression such asallows us to see the -intercepts of its graph, but we didn’t look into why the

graph crosses the -axis at those points. Can you explain why it does now?” (The -interceptshave a -value of 0, which means the quadratic function is 0 at those -values:

. If multiplying two numbers gives 0, one of them must be 0. So eitheror . If , then is 5. If , then is -9.)


Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-classdiscussion. After each student shares, provide the class with the following sentence frames tohelp them respond: “I agree because . . .” or “I disagree because . . . .” If necessary, revoicestudent ideas to demonstrate mathematical language use by restating a statement as aquestion in order to clarify, apply appropriate language, and involve more students. Forexample, a statement such as, “The first one is more helpful” can be restated as a questionsuch as, “Do you agree that the factored form is more helpful for finding when the object has aheight of 0 meters?”Design Principle(s): Support sense-making

Lesson SynthesisTo help students consolidate the ideas in the lesson, discuss questions such as:

“How does the zero product property help us find the solutions to ?” (It tellsus that either or ”, and each of these equations can be solved easily.)

“Can you explain why the solutions to are not 3 and -4?” (The zero productproperty only works when the product of the factors is zero. When the product is any othernumber, we can’t conclude that each factor is that number.)

“The expression is equivalent to . Can we apply the zero productproperty to solve ?” (Only if we rewrite the expression on the left in factored

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Unit 7 Lesson 4 79

form first. We can’t use the zero product property when the expression is not a product offactors.)

“Can we solve by performing the same operation to each side of theequation?” (No, doing that doesn’t help us isolate the variable.)

4.4 Solve This Equation!Cool Down: 5 minutesAddressing

HSA-REI.B.4

Launch

No technology should be used for this cool-down.


Find all solutions to . Explain or show your reasoning.

Student Response

-5 and . Sample reasoning: By the zero product property, either or , so

or .

Student Lesson SummaryThe zero product property says that if the product of two numbers is 0, then one of thenumbers must be 0. In other words, if then either or . This property ishandy when an equation we want to solve states that the product of two factors is 0.

Suppose we want to solve . This equation says that the product of andis 0. For this to be true, either or , so both 0 and -9 are solutions.

Here is another equation: . The equation says the product ofand is 0, so we can use the zero product property to help us find the

values of . For the equation to be true, one of the factors must be 0.

For to be true, would have to be 2.345.

For or to be true, would have to be or .

The solutions are 2.345 and .

In general, when a quadratic expression in factored form is on one side of an equation and 0is on the other side, we can use the zero product property to find its solutions.

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Glossaryzero product property


StatementIf the equation is true, which statement is also true according to the zeroproduct property?

A. only

B. either or

C. either or

D. only

SolutionB

Problem 2StatementWhat are the solutions to the equation ?

A. -10 and 3

B. -10 and 9

C. 10 and 3

D. 10 and 9

SolutionC

Problem 3StatementSolve each equation.

a.

b.

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Unit 7 Lesson 4 81

c.

Solutiona. and

b. and

c. and

Problem 4StatementConsider the expressions and .

Show that the two expressions define the same function.

SolutionStudents may multiply the factors in the first expression to show that it is algebraically equivalent tothe second, or they may graph both and show they produce the same graph. (Only the formerproves that the two expressions are equivalent, but the latter is sufficient for informallyshowing that they are the same function.)

Problem 5StatementKiran saw that if the equation is true, then, by the zero product property,either is 0 or is 0. He then reasoned that, if is true, then either

is equal to 72 or is equal to 72.

Explain why Kiran’s conclusion is incorrect.

SolutionSample response: The zero product property only applies to products that are equal to zero, notany product. Many pairs of factors can be multiplied to be 72. We can’t be sure that one of thefactors is equal to 72.

Problem 6StatementAndre wants to solve the equation . He uses a graphing calculator tograph and and finds that the graphs cross at the pointsand .


a. Substitute each -value Andre found into the expression . Then evaluatethe expression.

b. Why did neither solution make equal exactly 20?

Solutiona. Both -values result in 20.1205, not 20, for the value of the expression.

b. Graphing software gives solutions that have been rounded. To know the exact solutions, weneed to solve the equation without graphing.


Problem 7StatementSelect all the solutions to the equation .

A. 49

B.

C. 7

D. -7

E.

F.

G.

Solution["C", "D", "E", "G"](From Unit 7, Lesson 3.)

Problem 8StatementHere are two graphs that correspond to two patients, A and B. Each graph shows the amountof insulin, in micrograms (mcg) in a patient' body hours after receiving an injection. Theamount of insulin in each patient decreases exponentially.

Unit 7 Lesson 4 83

Patient A Patient B

Select all statements that are true about the insulin level of the two patients.

A. After the injection, the patients have the same amount of insulin in their bodies.

B. An equation for the micrograms of insulin, , in Patient A's body hours after the

injection is .

C. The insulin in Patient A is decaying at a faster rate than in Patient B.

D. After 3 hours, Patient A has more insulin in their body than Patient B.

E. At some time between 2 and 3 hours, the patients have the same insulin level.

Solution["B", "D", "E"](From Unit 5, Lesson 6.)

Problem 9StatementHan says this pattern of dots can be represented by a quadratic relationship because thedots are arranged in a rectangle in each step.

Do you agree? Explain your reasoning.

SolutionI disagree. This is a linear relationship. The total number of dots is growing by 4 for each new step.



Lesson 5: How Many Solutions?

GoalsCoordinate (orally) graphs with no horizontal intercepts, quadratic functions with no (real)zeros, and quadratic equations with no (real) solutions.

Describe (orally and in writing) the relationship between the solutions to quadratic equationsof the form and the horizontal intercepts of the graph of the related function.

Explain (orally and in writing) why dividing each side of a quadratic equation by a variable isnot a reliable way to solve the equation.

Learning TargetsI can explain why dividing by a variable to solve a quadratic equation is not a good strategy.

I know that quadratic equations can have no solutions and can explain why there are none.

Lesson NarrativeThe work in this lesson builds on the idea that both graphing and rewriting quadratic equations inthe form of are useful strategies for solving equations. It also reinforces the tiesbetween the zeros of a function and the horizontal intercepts of its graph, which students beganexploring in an earlier unit.

Previously, to solve an equation such as by graphing, students would graphand and inspect where the parabolic graph and the horizontal line

intersect.

Here, students learn another way to use graphs to solve equations and to anticipate the number ofsolutions. Instead of graphing two separate equations—one quadratic and one linear, studentslearn that they can solve by rearranging the equation into the form , graphing theequation , and finding the horizontal intercepts. Why does this make sense?

The -coordinate of those intercepts produces a -coordinate of 0, so they are the solutions tothe equation .

The number of horizontal intercepts tells us the instances when the -coordinate is 0, whichtells us the number of solutions to the equation.

Later in the lesson, students think about why a quadratic equation that has an expression infactored form on one side of the equal sign but does not have 0 on the other side cannot be solvedthe same way as when the equation is . They also notice that dividing each side of aquadratic equation by a variable is not reliable because it eliminates one of the solutions. As theyexplain why certain maneuvers are acceptable and others are not, students practice constructinglogical arguments (MP3).

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Unit 7 Lesson 5: How Many Solutions? 85

Alignments

Building On

6.EE.B.5: Understand solving an equation or inequality as a process of answering a question:which values from a specified set, if any, make the equation or inequality true? Usesubstitution to determine whether a given number in a specified set makes an equation orinequality true.

HSF-IF.C.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

Addressing




HSA-REI.D: Represent and solve equations and inequalities graphically.

HSA-REI.D.10: Understand that the graph of an equation in two variables is the set of all itssolutions plotted in the coordinate plane, often forming a curve (which could be a line).

Building Towards



Graph It

Math Talk

MLR1: Stronger and Clearer Each Time


Think Pair Share

Required Materials

Graphing technologyExamples of graphing technology are:a handheld graphing calculator, a computer witha graphing calculator application installed, andan internet-enabled device with access to a sitelike desmos.com/calculator or geogebra.org/

graphing. For students using the digitalmaterials, a separate graphing calculator toolisn't necessary; interactive applets areembedded throughout, and a graphingcalculator tool is accessible on the studentdigital toolkit page.

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Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal ifeach student has their own device. (Desmos is available under Math Tools.)


Let’s use graphs to investigate quadratic equations that have two solutions, onesolution, or no solutions.

5.1 Math Talk: Four EquationsWarm Up: 10 minutesThis warm-up reminds students of two facts, that in order to use the zero product property, theproduct of the factors must be 0, and that there is no number that can be squared to get a negativenumber. At this point, students don’t yet know about complex numbers or that squaring a complexnumber produces a negative number. Consequently, for now we can just say that there is nonumber (with a silent “that you know about, yet”) that can be squared to get a negative number.

In explaining why they think each statement is true or false, students practice constructing logicalarguments (MP3).

Building On

6.EE.B.5

Building Towards

HSA-REI.B.4


Math Talk


Launch

Display one problem at a time. Give students quiet think time for each problem and ask them togive a signal when they have an answer and an explanation. Keep all problems displayedthroughout the talk. Follow with a whole-class discussion.



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Decide whether each statement is true or false.

3 is the only solution to .

A solution to is -5.

has two solutions.

5 and -7 are the solutions to .

Student Response

False. There are two solutions because there are two numbers that can be squared to get 9.

False. There is no solution to because no number can be squared to get -25.

True. There are two factors, 0 and 7, that could make the product 0.

False. The product of the two factors is not 0, so we cannot solve it using the zero productproperty (which gives 5 and -7 as solutions). If we evaluate the expression atthose two values, the result will be 0, not 12.

Activity Synthesis

Ask students to share their response and explanation for each problem. Record and display theirresponses for all to see. After each explanation, give the class a chance to agree or disagree. Toinvolve more students in the conversation, consider asking:






Make sure students understand the rationale that makes each statement true or false, as shown inthe student response.



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5.2 Solving by Graphing15 minutesBy now, students recognize that when a quadratic equation is in the form of and theexpression is in factored form, the equation can be solved using the zero product property. In thisactivity, they encounter equations in which one side of the equal sign is not 0. To make one sideequal 0 requires rearrangement. For example, to solve , the equation needs to berearranged to . Yet because the expression on the left is no longer in factored form,the zero product property won’t help after all and another strategy is needed.

Students recall that to solve a quadratic equation in the form of is essentially to findthe zeros of a quadratic function defined by that expression, and that the zeros of a functioncorrespond to the horizontal intercepts of its graph. In the case of , the functionwhose zeros we want to find is defined by . Graphing and examiningthe -intercepts of the graph allow us to see the number of solutions and what they are.

Building On

HSF-IF.C.7.a

Addressing

HSA-REI.D

HSA-REI.D.10


Graph It

Think Pair Share

Launch

Arrange students in groups of 2 and provide access to graphing technology. Give students amoment to think quietly about the first question and then ask them to briefly discuss theirresponse with their partner before continuing with the rest of the activity.


Action and Expression: Provide Access for Physical Action. Support effective and efficient use oftools and assistive technologies. To use graphing technology, some students may benefit froma demonstration or access to step-by-step instructions.Supports accessibility for: Organization; Memory; Attention

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If students enter the equation into their graphing technology, they may see anerror message, or they may see vertical lines. The lines will intersect the -axis at the solutions, butthey are clearly not graphs of a quadratic function. Emphasize that we want to graph the functiondefined by and use its -intercepts to find the solution to the related equation. Allthe points on the two vertical lines do represent solutions to the equation (because the points alongeach vertical line satisfy the equation regardless of the value chosen for ), but understanding this isbeyond the expectations for students in this course.


Han is solving three equations by graphing.

1. To solve the first equation, , he graphed and thenlooked for the -intercepts of the graph.

a. Explain why the -intercepts can be used to solve .

b. What are the solutions?

2. To solve the second equation, Han rewrote it as . He thengraphed .

Use graphing technology to graph . Then, use the graph to solvethe equation. Be prepared to explain how you use the graph for solving.

3. Solve the third equation using Han’s strategy.

4. Think about the strategy you used and the solutions you found.

a. Why might it be helpful to rearrange each equation to equal 0 on one side andthen graph the expression on the non-zero side?

b. How many solutions does each of the three equations have?

Student Response

1. Sample response:

a. The solutions to are -values that make the expression have a value of0. At the -intercepts, the -value of the expression is 0, so the -interceptsgive the solutions.

b. 5 and 3

2. Sample graphs and responses:


The solution is 4 for . The solution is the -coordinate of the -intercept.

3. There are no solutions for . The graph has no -intercept.

4. Sample response:a. When a quadratic expression is equal to 0 and we want to find the unknown values, we

can think of it as finding the zeros of a quadratic function, or finding the -intercepts ofthe graph of that function.

b. There are two solutions for the first equation, one solution for the second equation, andno solutions for the last equation.


The equations , , and all havewhole-number solutions.

1. Use graphing technology to graph each of the following pairs of equations on the samecoordinate plane. Analyze the graphs and explain how each pair helps to solve therelated equation.

and

and

and

2. Use the graphs to help you find a few other equations of the formthat have whole-number solutions.

3. Find a pattern in the values of that give whole-number solutions.

4. Without solving, determine if and havewhole-number solutions. Explain your reasoning.

Student Response

1. Sample response: The solutions to each equation are the intersection points of the twographs.

◦◦◦


2. Sample equations: , ,

3. Sample responses:Starting at , add 1, then 3, then 5, and so on, to find the next -value that yieldswhole-number solutions.

Because the expressions and represent numbers whose difference is 2, thevalues of have the form , , , , and so on.

4. Both have whole-number solutions. The values of that produce whole-number solutions areall 1 less than a square number.

Activity Synthesis

Invite students to share their responses, graphs, and explanations on how they used the graphs tosolve the equations. Discuss questions such as:

“Are the original equation and the rewritten oneequivalent?” (Yes, each pair of equations are equivalent. In that example, 1 is added to bothsides of the original equation.)

"Why might it be helpful to rearrange the equation so that one side is an expression and theother side is 0?" (It allows us to find the zeros of the function defined by that expression. Thezeros correspond to the -intercepts of the graph.)

“What equation would you graph to solve this equation: ?”( ) “What about ?" ( )

Make sure students understand that some quadratic functions have two zeros, some have onezero, and some have no zeros, so their respective graphs will have two, one, or no horizontalintercepts, respectively.

Likewise, some quadratic equations have two solutions, some have one solution, and some have noreal solutions. (Because students won’t know about numbers that aren’t real until a future course,for now it is sufficient to say “no solutions.”)

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5.3 Finding All the SolutionsOptional: 15 minutesThis optional activity gives students an opportunity to practice solving quadratic equations anddeciding on an effective strategy. Some equations can be easily solved by reasoning. Others wouldrequire solving by graphing, because students have not yet learned the strategies to solvealgebraically. Students who use graphing technology only when needed practice choosing toolsstrategically (MP5).

Addressing

HSA-REI.B.4.b

HSA-REI.D


Graph It

Launch

Give students continued access to technology.


Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge.Invite students to select 4 of the 6 equations to solve. Chunking this task into moremanageable parts may also support students who benefit from additional processing time.Supports accessibility for: Organization; Attention; Social-emotional skills


Solve each equation. Be prepared to explain or show your reasoning.

1.

2.

3.

4.

5.

6.

Student Response

1. 11 and -11

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2. 6 and -6

3. 4

4. -4 and 2

5. no solutions

6. 34 and -29

Activity Synthesis

Select students to share their solutions and strategies. If not mentioned by students’ explanations,highlight that:

The first three equations, as well as the equation , can be solved by reasoningand that graphing is not necessary.

The last three equations can be solved by graphing. There are two ways to do so, as shown ina previous activity.

One way is graph each side of the equation separately: and.

Another way is to rearrange the equation such that it is in the form of ,graph , and then find the -intercepts.

The equation states that some number squared is -4. Because no number can besquared to get a negative number, we can reason that there are no solutions. If this equationis solved by graphing , the graph would show no -intercepts. This also tells usthat there are no solutions.

5.4 Analyzing Errors in Equation Solving10 minutesThis activity aims to uncover some common misconceptions in solving quadratic equations and toreinforce that certain familiar moves for solving equations are not effective. Students critiqueseveral arguments on how to solve quadratic equations. In articulating why certain lines ofreasoning are correct or incorrect, they practice constructing logical arguments (MP3).

As students work, look for students who:

explain both the error in Priya’s argument and the validity of Mai’s argument in terms of thezero product property

notice that Diego’s method disregards the second solution of the equation

create and use a graph to verify their critique of Priya, Mai, or Diego's work (for example,graphing to show that has two -intercepts)

•

•

◦

◦

•

•

••


Multiplying or dividing both sides of an equation by a variable expression can change the solutionset of an equation, either by eliminating a solution (as shown in Diego's method) or introducing anew solution (for example, starting with the equation and multiplying both sides by to get

gives an equation that now has 2 solutions). Students will learn more about such moves ina later course.

Addressing

HSA-REI.A.1

HSA-REI.B.4.b



Launch

Keep students in groups of 2 and ask them to work quietly on both questions before discussingtheir responses with a partner.


Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help studentsimprove their written responses by providing them with multiple opportunities to clarify theirexplanations through conversation. At the appropriate time, give students time to meet with2–3 partners to share their response to the first question. Students should first check to see ifthey agree with each other about Priya’s reasoning. Provide listeners with prompts forfeedback that will help their partner add detail to strengthen and clarify their ideas. Forexample, "Your explanation tells me . . .", "Can you say more about why you . . . ?", and "A detail(or word) you could add is _____, because . . . ." Give students with 3–4 minutes to revise theirinitial draft based on feedback from their peers. This will help students evaluate the writtenmathematical arguments of others and improve their own written responses about solvingquadratic equations.Design Principle(s): Optimize output (for justification); Cultivate conversation


1. Consider . Priya reasons that if this is true, then either or. So, the solutions to the original equation are 12 and 6.

Do you agree? If not, where was the mistake in Priya’s reasoning?

2. Consider . Diego says to solve we can just divide each side by to get, so the solution is 10. Mai says, “I wrote the expression on the left in factored

form, which gives , and ended up with two solutions: 0 and 10.”

Do you agree with either strategy? Explain your reasoning.

••

•


Student Response

1. Disagree. Sample explanation: Priya solved the equation using the reasoning we would usewith the zero product property, but the zero product property only works if the product of twofactors is 0. We can tell that 12 isn’t a solution because is 91, not 7.

2. Agree with Mai’s strategy. By substituting 0 for and then 10 for , you can see that they areboth solutions to the original equation. Diego’s strategy eliminates one of the solutions,leaving only one solution.

Activity Synthesis

Select previously identified students to share their responses and reasoning. Here are some keyobservations to highlight:

For the first question, make sure students understand that the zero product property onlyworks when the product of the factors is 0. While substituting 6 for in the expression doesproduce 7, substituting 12 does not give the same result.

For the second question, consider graphing the function so students can see thatthe graph intersects the -axis at two points, which means that there are two -values that givea zero output: 0 and 10. Rewriting into the factored form, writing it to equal 0, andsolving allow us to see that this is indeed the case.

Dividing each side by a variable (as what Diego did) seems to enable us to isolate theremaining variable, but only one solution remains. Dividing each side of an equation by isnot a valid move because when is 0, the expressions on each side become undefined.

Lesson SynthesisTo synthesize the work in this lesson and connect it to prior work, discuss questions such as:

“How would you solve the equation ?” (The simplest way would be to use thezero product property, which tells us that one of the factors must be 0, so and .)

“If you choose to solve by graphing, is it necessary to rearrange and rewrite the equation first?What equation would you graph and how would you use the graph to solve the equation?”(No. We can just graph and look for the -intercepts of the graph.)

“Can we use the zero product property to solve ? Why or why not?” (No,because the expression on the left does not equal 0.)

“How would you solve ?” (We can rewrite the equation as, graph , and see what the -intercepts are.)

“How can the graph tell us how many solutions there are?” (The number of -intercepts revealsthe number of solutions.)

•

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•

•

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•

•

•


Remind students that examining a graph is not a reliable way to get exact solutions to an equation.For example, the -intercepts of the graph for are andand those -coordinates are likely rounded results.

To solve equations exactly, we need to use algebraic means. In upcoming lessons, we’ll learn morestrategies for doing so.

5.5 Two, One, or None?Cool Down: 5 minutesAddressing

HSA-REI.B.4

HSA-REI.D

Launch

Allow students to continue using graphing technology if they choose to.


For each quadratic equation, decide whether it has two solutions, one solution, or nosolutions. Explain how you know.

1.

2.

3.

Student Response

1. No solutions. Sample reasoning:There are no numbers we can square to get a negative product.

The graph of has no -intercepts.

2. Two solutions. Sample reasoning:Both 0 and -2 make the equation true.

The graph of has two -intercepts.

3. One solution. Sample reasoning:Only 3 makes the equation true.

The graph of has one -intercept.

Student Lesson SummaryQuadratic equations can have two, one, or no solutions.

••

◦◦

◦◦

◦◦


We can find out how many solutions a quadratic equation has and what the solutions are byrearranging the equation into the form of , graphing the function that theexpression defines, and determining its zeros. Here are some examples.

Let's first subtract from each side and rewrite the equation as . We canthink of solving this equation as finding the zeros of a function defined by .

If the output of this function is , we can graph and identify where the graphintersects the -axis, where the -coordinate is 0.

From the graph, we can see that the-intercepts are and , so

equals 0 when is 0 and when is 5.

The graph readily shows that there are twosolutions to the equation.

Note that the equation can be solved without graphing, but we need to be careful notto divide both sides by . Doing so will give us but will show no trace of the othersolution, !

Even though dividing both sides by the same value is usually acceptable for solvingequations, we avoid dividing by the same variable because it may eliminate a solution.

Let’s rewrite the equation as , and consider it to represent afunction defined by and whose output, , is 0.

Let's graph and identify the -intercepts.

The graph shows one -intercept at .This tells us that the function definedby has only one zero.

It also means that the equationis true only when

. The value 5 is the only solution to theequation.

•

•


Rearranging the equation gives .

Let’s graph and find the -intercepts.

The graph does not intersect the -axis, sothere are no -intercepts.

This means there are no -values that canmake the expressionequal 0, so the function defined by

has no zeros.

The equation has nosolutions.

We can see that this is the case even without graphing. is. Because no number can be squared to get a negative value, the equation

has no solutions.

Earlier you learned that graphing is not always reliable for showing precise solutions. This isstill true here. The -intercepts of a graph are not always whole-number values. While theycan give us an idea of how many solutions there are and what the values may be (at leastapproximately), for exact solutions we still need to rely on algebraic ways of solving.


StatementRewrite each equation so that the expression on one side could be graphed and the

-intercepts of the graph would show the solutions to the equation.

a.

b.

c.

d.

Solutiona.

b.

c.

•


d.

Problem 2Statement

a. Here are equations that define quadratic functions , and . Sketch a graph, by handor using technology, that represents each equation.

b. Determine how many solutions each , and has. Explain howyou know.

Solutiona. i. Graph of shows an upward opening parabola with -intercept at .

ii. Graph of shows an upward opening parabola with -intercepts at and .

iii. Graph of shows an upward opening parabola with an -intercept at .

b. i. has 0 solutions. Sample reasoning: The graph doesn’t intersect the -axis, so thefunction has no zeros.

ii. has 2 solutions. Sample reasoning: The graph crosses the -axis twice. Functionhas 2 zeros.

iii. has 1 solution. Sample reasoning: The graph touches the -axis once. Functionhas 1 zero.

Problem 3StatementMai is solving the equation . She writes that the solutions are and .Han looks at her work and disagrees. He says that only is a solution. Who do you agreewith? Explain your reasoning.


SolutionSample response: Agree with Han. Sample reasoning:

The only number that, when squared, is equal to 0 is 0.

Mai might have thought that the square meant she needed to have a positive solution and anegative one, but the only value that makes the equation true is .

Problem 4StatementThe graph shows the number of square meters, , covered byalgae in a lake weeks after it was first measured.

In a second lake, the number of square meters, , covered byalgae is defined by the equation , where is

the number of weeks since it was first measured.

For which algae population is the area decreasing more rapidly? Explain how you know.

SolutionThe algae population represented by area decreases more rapidly. Sample explanation: Thedecay factor of the second population is , while the decay factor for the population represented

by area is .


Problem 5StatementIf the equation is true, which is also true according to the zero productproperty?

A. only

B. only

C. or

D. or

◦◦


SolutionC(From Unit 7, Lesson 4.)

Problem 6Statement

a. Solve the equation .

b. Show that your solution or solutions are correct.

Solutiona. and

b. Sample response: , and


Problem 7StatementTo solve the quadratic equation , Andre and Clare wrote the following:

Andre Clare

a. Identify the mistake each student made.

b. Solve the equation and show your reasoning.

Solutiona. Sample response:

Andre distributed the exponent to each term in the parentheses, which cannot be done.

There are two numbers (3 and -3) that, when squared, give 9. Clare left out the -3, so onesolution is missing.

b. Correction:

▪▪



Problem 8StatementDecide if each equation has 0, 1, or 2 solutions and explain how you know.

a.

b.

c.

d.

e.

Solutiona. 2 solutions. 12 and -12 both make the equation true.

b. 0 solutions. There is no number (that the students know of) that can be squared and result ina negative number.

c. 2 solutions. By the zero product property, either is 0 or is 0, so both andare solutions.

d. 1 solution. The zero product property gives the same value of 8 as a solution in either case.

e. 2 solutions. By the zero product property, either is 0 or is 0, so both orare solutions.


Lesson 6: Rewriting Quadratic Expressions inFactored Form (Part 1)

GoalsApply the distributive property to multiply two sums or two differences, using a rectangulardiagram to illustrate the distribution as needed.

Generalize the relationship between equivalent quadratic expressions in standard form andfactored form, and use the generalization to transform expressions from one form to theother.

Use a diagram to represent quadratic expressions in different forms and explain (orally and inwriting) how the numbers in the factors relate to the numbers in the product.

Learning TargetsI can explain how the numbers in a quadratic expression in factored form relate to thenumbers in an equivalent expression in standard form.

When given quadratic expressions in factored form, I can rewrite them in standard form.

When given quadratic expressions in the form of , I can rewrite them in factoredform.

Lesson NarrativePreviously, students learned that a quadratic expression in factored form can be quite handy inrevealing the zeros of a function and the -intercepts of its graph. They also observed that thefactored form can also help us solve quadratic equations algebraically. In this lesson, studentsbegin to rewrite quadratic expressions from standard to factored form.

In an earlier unit, students learned to expand quadratic expressions in factored form and rewritethem in standard form. They did so by applying the distributive property to multiply out thefactors—first by using diagrams for support, and then by relying on structure they observed in theprocess. The attention to structure continues in this lesson. Students relate the numbers in thefactored form to the coefficients of the terms in standard form, looking for structure that can beused to go in reverse—from standard form to factored form (MP7).

This lesson only looks at expressions of the form and where andare positive. This is so that arithmetic doesn’t get in the way of noticing the relationships betweenthe numbers in factored form and the numbers in standard form. In the next lesson, students willencounter expressions of the form and .

Note that this course includes only four lessons on transforming quadratic expressions fromstandard form to factored form. This is intentional. The goal is to help students see factored formconceptually, understand what it can tell us about the function, and use that knowledge inmodeling problems, including problems where the zeros are not rational and algebraic factoring

•

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•

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wouldn't help. Too much attention to the algebraic skill of factoring can obscure these underlyingconcepts.

Sometimes—including in later courses and beyond—expressions do need to be written in factoredform. With the universal availability of computer algebra systems, however, there is less need tospend lots of time learning how to factor by hand.

Alignments

Building On

6.G.A.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons bycomposing into rectangles or decomposing into triangles and other shapes; apply thesetechniques in the context of solving real-world and mathematical problems.

Addressing

HSA-SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, seeas , thus recognizing it as a difference of squares that can be factored as

.

Building Towards



.

HSA-SSE.B.3.a: Factor a quadratic expression to reveal the zeros of the function it defines.



Think Pair Share


Let’s write expressions in factored form.

6.1 Puzzles of RectanglesWarm Up: 10 minutesWhen they write expressions in factored form later in the lesson, students will need to reasonabout factors that yield certain products. This warm-up prompts students to find unknown factorsin the context of area puzzles. Solving the puzzles involves reasoning about the measurements in

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Unit 7 Lesson 6 105

multiple steps. Explaining these steps is an opportunity to practice constructing logical arguments(MP3).

Building On

6.G.A.1

Building Towards

HSA-REI.B.4.b


Think Pair Share

Launch

Arrange students in groups of 2. Give students a few minutes of quiet think time and then time toshare their thinking with their partner. Follow with a whole-class discussion.


Here are two puzzles that involve side lengths and areas of rectangles. Can you find themissing area in Figure A and the missing length in Figure B? Be prepared to explain yourreasoning.

Figure A Figure B

Student Response

Figure A: 20 sq in

Figure B: 9 in

Activity Synthesis

Display the images for all to see. Invite students to share their responses and how they reasonedabout the missing values, using the diagrams to illustrate their thinking.

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After the solution to the second puzzle is presented, draw students’ attention to the rectangle witharea 36 sq in. Point out that, without reasoning about other parts of the puzzle, we cannot knowwhich two numbers are multiplied to get 36 sq in. (The numbers may not be whole numbers.) Butby reasoning about other parts, we can conclude what the missing length must be.

Explain that in this lesson, they will also need to find two factors that yield a certain product and toreason logically about which numbers the factors can or must be.

6.2 Using Diagrams to Understand EquivalentExpressions15 minutesThis activity prompts students to notice the structure that relates quadratic expressions in factoredform and their equivalent counterparts in standard form. Students began this work earlier in thecourse. They applied the distributive property to expand an expression such as intostandard form, using rectangular diagrams to help organize and keep track of the partial products.Here, the focus is on using the structure to go in reverse: rewriting into factored form an expressiongiven in standard form (MP7).

The expressions students encounter here are two sums or two differences, in the form ofor . When working with two differences, it is helpful to think of

subtracting and as adding and and labeling the diagrams accordingly. For example, forthe last expression, , one side of the diagram should be labeled with and -1, and theother labeled with and -7.

Building Towards

HSA-REI.B.4.b

HSA-SSE.A.2

HSA-SSE.B.3.a

Launch

Display this diagram for all to see, and remind students that they haveseen diagrams such as this one in a previous unit on quadratic functions.

Ask students what expression it represents (both and).

•••

Unit 7 Lesson 6 107

Next, ask students what this diagram represents.

Students might say or . Whether theymention both or not, write both expressions for all to see and emphasizethat they are equivalent.

When we want to represent , it is convenient to think of it as and labelthe diagram thusly, so that we keep track in the diagram of what is positive and negative.

Then, ask students what expression or number goes in each blankrectangle. Make sure students see that the rectangles are used to organizethe partial products of , the sum of which is .


Representation: Internalize Comprehension. Use color coding and annotations to highlightconnections between representations. Invite students to use color to identify the like terms intheir diagrams and expressions. Demonstrate how students can use different colors to keeptrack of like terms as they use diagrams to show that each pair of expressions is equivalent.Supports accessibility for: Visual-spatial processing


1. Use a diagram to show that each pair of expressions is equivalent.

and and

and and

and and

2. Observe the pairs of expressions that involve the product of two sums or twodifferences. How is each expression in factored form related to the equivalentexpression in standard form?

Student Response

1. Sample tables:


2. Sample response: If the two numbers that appear in factored form are added, the result is thecoefficient of in standard form. If those same two numbers are multiplied, the result is theconstant term in standard form. (Note that this explanation relies on recasting everysubtraction expression as adding the opposite.)

Activity Synthesis

Invite students to share their diagrams and observations. Make sure students notice that the linearterm in the expression in standard form is the sum of the two numbers in the expression infactored form, and the constant term is the product of the two numbers in the expression infactored form. (For this explanation to be succinct, it requires rewriting any subtractions as addingthe opposite.)

6.3 Let’s Rewrite Some Expressions!10 minutesThis activity allows students to practice rewriting quadratic expressions in standard form by usingthe structure they observed in the earlier activity.

As students work, look for those who approach the work systematically: by looking for two factorsof the constant term and that add up to the coefficient of the linear term of the expression instandard form, listing the possible pairs of factors, and checking their chosen pair. Invite them toshare their strategy during discussion.

Unit 7 Lesson 6 109

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.b

HSA-SSE.B.3.a



Launch

Consider arranging students in groups of 2 and asking them to think quietly about the equivalentexpressions before conferring with their partner. Leave a few minutes for a whole-class discussion.

Ask students to complete as many equivalent expressions as time permits while aiming to completeat least the first seven rows in the table. Then, ask them to generalize their observations in the lastrow.


Some students may struggle to remember how each term in standard form relates to the numbersin the equivalent expression in factored form. Encourage them to use a diagram (as in the earlieractivity) to go from factored form to standard form, and then work backwards.

•

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•



Each row in the table contains a pair ofequivalent expressions.

Complete the table with the missingexpressions. If you get stuck, consider drawinga diagram.

factored form standard form

Student Response

factored form standard form factored form standard form

Unit 7 Lesson 6 111


A mathematician threw a party. She told her guests, “I have a riddle for you. I have threedaughters. The product of their ages is 72. The sum of their ages is the same as my housenumber. How old are my daughters?”

The guests went outside to look at the house number. They thought for a few minutes, andthen said, “This riddle can’t be solved!”

The mathematician said, “Oh yes, I forgot to tell you the last clue. My youngest daughterprefers strawberry ice cream.”

With this last clue, the guests could solve the riddle. How old are the mathematician’sdaughters?

Student Response

2, 6, and 6.

Sample response: if the product of their ages is 72, there are twelve possibilities:

1, 1, 72

1, 2, 36

1, 3, 24

1, 4, 18

1, 6, 12

1, 8, 9

2, 2, 18

2, 3, 12

2, 4, 9

2, 6 ,6

3, 3, 8

3, 4, 6

Because the riddle can’t be solved by knowing the house number (the sum of the three numbers), itmust be the case that there are triplets that have the same sum. The only pair of triplets with thesame sum is 3, 3, 8 and 2, 6, 6, which each add up to 14. From the last clue, we know that there is ayoungest daughter. The only triplet that has a “youngest” age is 2, 6, 6, so these are the ages of thethree daughters.

Activity Synthesis

Focus first on the first three rows. Ask one or more students to share their equivalent expressionsand any diagrams that they drew. Point out that this is a fairly straightforward application of thedistributive property, which students first learned about in grade 6.

Select students to share how they transformed the remaining expressions from standard form tofactored form, using specific examples in their explanations. For the example of ,highlight that:

We are looking for two factors of 12.

We are looking for two numbers with a sum of 13.

One strategy is to list out all the factor pairs of 12.

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••••

••••

•••


Another strategy is to list out all pairs of numbers that add up to 13, but usually the list offactors is shorter.


Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. At theappropriate time, give students, or pairs of students, 2–3 minutes to plan what they will saywhen they present how they transformed their expressions from standard form to factoredform, using specific examples in their explanations. Encourage students to consider whatdetails are important to share and to think about how they will explain their reasoning usingmathematical language.Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson SynthesisTo help students summarize and generalize the reasoning involved in rewriting quadraticexpressions from standard form to factored form, display a few expressions such as these:

For each one, ask students to explain the process of transforming it into factored form bycompleting (in writing or by talking a partner) sentence starters such as these:

To find the factors, I first try to find . . .

Next, I think about .. . .

The equivalent expression in factored form is . . .

To check that the factors are correct, I can . . .

Make sure students see that for , a helpful process goes something like this:

First, we find all pairs of factors of .

Next, find a pair of factors of that add up to equal . (If the factors are and , then we want.)

The factors will be .

We can check by expanding the factored form (by applying the distributive property) and see ifwe get the original expression as a result.

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••••

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Unit 7 Lesson 6 113

6.4 The Missing NumbersCool Down: 5 minutesAddressing

HSA-SSE.A.2


Here are pairs of equivalent expressions—one in standard form and the other in factoredform. Find the missing numbers.

1. and

2. and

3. and

4. and

Student Response

1. 18 and 17. The completed expression is .

2. 7. The completed expression is .

3. 5. The completed expression is .

4. 9 and 20. The completed expression is .

Student Lesson SummaryPreviously, you learned how to expand a quadratic expression in factored form and write it instandard form by applying the distributive property.

For example, to expand , we apply the distributive property to multiply byand 4 by . Then, we apply the property again to multiply by and by 5, and

multiply 4 by and 4 by 5.

To keep track of all the products, we couldmake a diagram like this:

Next, we could write the products of each pairinside the spaces:

The diagram helps us see that is equivalent to , or instandard form, .

•


The linear term, , has a coefficient of 9, which is the sum of 5 and 4.

The constant term, 20, is the product of 5 and 4.

We can use these observations to reason in the other direction: to start with an expression instandard form and write it in factored form.

For example, suppose we wish to write in factored form.

Let’s start by creating a diagram and writing inthe terms and 24.

We need to think of two numbers thatmultiply to make 24 and add up to -11.

After some thinking, we see that -8 and -3meet these conditions.

The product of -8 and -3 is 24. The sum of -8and -3 is -11.

So, written in factored form is .

Glossarycoefficient

constant term

linear term


StatementFind two numbers that satisfy the requirements. If you get stuck, try listing all the factors ofthe first number.

a. Find two numbers that multiply to 17 and add to 18.

b. Find two numbers that multiply to 20 and add to 9.

c. Find two numbers that multiply to 11 and add to -12.

d. Find two numbers that multiply to 36 and add to -20.

Solutiona. 1 and 17

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Unit 7 Lesson 6 115

b. 4 and 5

c. -1 and -11

d. -2 and -18

Problem 2StatementUse the diagram to show that:

is equivalent to is equivalent to.

Solution

Problem 3StatementSelect all expressions that are equivalent to .


A.

B.

C.

D.

E.

F.

G.

Solution["A", "C", "F"]

Problem 4StatementHere are pairs of equivalent expressions—one in standard form and the other in factoredform. Find the missing numbers.

a. and

b. and

c. and

d. and

Solutiona. -12, 27

b. 8

c. -7

d. -5

Problem 5StatementFind all the values for the variable that make each equation true.

a.

Unit 7 Lesson 6 117

b.

c.

d.

Solutiona. or

b.

c. or

d. or


Problem 6StatementLin charges $5.50 per hour to babysit. The amount of money earned, in dollars, is a functionof the number of hours that she babysits.

Which of the following inputs is impossible for this function?

A. -1

B. 2

C. 5

D. 8

SolutionA(From Unit 4, Lesson 10.)

Problem 7StatementConsider the function .

a. Evaluate , writing out every step.

b. Evaluate , writing out every step. You will run into some trouble. Describe it.

c. What is a possible domain for ?


Solutiona. Since , and , therefore, (or equivalent).

b. Since , , and . The trouble is that is undefined.

c. One possible domain is all real numbers except . Any subset of this domain is alsopossible, for example, all negative numbers or all positive numbers except .


Problem 8StatementTechnology required. When solving the equation , Priyagraphs and then looks to find where the graph crosses the -axis.

Tyler looks at her work and says that graphing is unnecessary and Priya can set up theequations and , so the solutions are or .

a. Do you agree with Tyler? If not, where is the mistake in his reasoning?

b. How many solutions does the equation have? Find out by graphing Priya’s equation.

Solutiona. No. Tyler’s mistake was to apply the zero product property when the product was not equal to

zero.

b. None. This equation has 0 solutions because the graph doesn’t intersect the -axis.


Unit 7 Lesson 6 119


GoalsApply the distributive property to multiply a sum and a difference, using a diagram to illustratethe distribution as needed.

Given a quadratic expression of the form , where is negative, write an equivalentexpression in factored form.

When multiplying a sum and a difference, explain (orally and in writing) how the numbers andsigns of the factors relate to the numbers in the product.

Learning TargetsI can explain how the numbers and signs in a quadratic expression in factored form relate tothe numbers and signs in an equivalent expression in standard form.

When given a quadratic expression given in standard form with a negative constant term, I canwrite an equivalent expression in factored form.

Lesson NarrativeIn an earlier lesson, students transformed quadratic expressions from standard form into factoredform. There, the factored expressions are products of two sums, , or two differences,

. Students continue that work in this lesson, extending it to include expressions thatcan be rewritten as products of a sum and a difference, .

Through repeated reasoning, students notice that when we apply the distributive property tomultiply out a sum and a difference, the product has a negative constant term, but the linear termcan be negative or positive (MP8). Students make use of structure as they take this insight totransform quadratic expressions into factored form (MP7). They see that if a quadratic expressionin standard form (with coefficient 1 for ) has a negative constant term, one of its factors musthave a negative constant term and the other must have a positive constant term.

Alignments

Building On

7.NS.A.1: Apply and extend previous understandings of addition and subtraction to add andsubtract rational numbers; represent addition and subtraction on a horizontal or verticalnumber line diagram.

7.NS.A.2: Apply and extend previous understandings of multiplication and division and offractions to multiply and divide rational numbers.

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•

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•


Addressing


.

Building Towards



.




Notice and Wonder

Think Pair Share


Let’s write some more expressions in factored form.

7.1 Sums and ProductsWarm Up: 5 minutesThis warm-up serves two purposes. The first is to recall that if the product of two numbers isnegative, then the two numbers must have opposite signs. The second is to review how to add twonumbers with opposite signs.

Building On

7.NS.A.1

7.NS.A.2

Building Towards

HSA-SSE.A.2


1. The product of the integers 2 and -6 is -12. List all the other pairs of integers whoseproduct is -12.

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Unit 7 Lesson 7 121

2. Of the pairs of factors you found, list all pairs that have a positive sum. Explain why theyall have a positive sum.

3. Of the pairs of factors you found, list all pairs that have a negative sum. Explain whythey all have a negative sum.

Student Response

1. 1 and -12, -1 and 12, 2 and -6, -2 and 6, 3 and -4, -3 and 4

2. -1 and 12, 6 and -2, 4 and -3. The positive number has a larger absolute value than thenegative number (or equivalent).

3. 1 and -12, 2 and -6, 3 and -4. The negative number has a larger absolute value than thepositive number (or equivalent).

Activity Synthesis

Ask students to share their list of factors. Once all the pairs are listed, highlight that each pair has apositive number and a negative number because the product we are after is a negative number,and the product of a positive and a negative number is negative.

Then, invite students to share their responses for the next two questions. Consider displaying anumber line for all to see and using arrows to visualize the additions of factors. Make sure studentsunderstand that when adding a positive number and a negative number, the result is the differenceof the absolute values of the numbers, and that sum takes the sign of the number that is fartherfrom zero.

7.2 Negative Constant Terms15 minutesIn this activity, students encounter quadratic expressions that are in standard form and that have anegative constant term. They notice that, when such expressions are rewritten in factored form,one of the factors is a sum and the other is a difference. They connect this observation to the factthat the product of a positive number and a negative number is a negative number.

Students also recognize that the sum of the two factors of the constant term may be positive ornegative, depending on which factor has a greater absolute value. This means that the sign of thecoefficient of the linear term (which is the sum of the two factors) can reveal the signs of thefactors.

Students use their observations about the structure of these expressions and of operations to helptransform expressions in standard form into factored form (MP7).

Addressing

HSA-SSE.A.2•


Building Towards

HSA-REI.B.4.b

HSA-SSE.B.3.a



Notice and Wonder

Think Pair Share

Launch

Arrange students in groups of 2. Give students a few minutes of quiet time to attempt the firstquestion. Pause for a class discussion before students complete the second question. Make surestudents recall that when transforming an expression in standard form into factored form, they arelooking for two numbers whose sum is the coefficient of the linear term and whose product is theconstant term.

Next, display a completed table for the first question and the incomplete table for the secondquestion for all to see. Ask students to talk to their partner about at least one thing they notice andone thing they wonder about the expressions in the table.

Students may notice that:

The constant terms in the right column of the first table are 30 and 18. They are both positive.

The constant terms in the right column of the second table are all -36.

The linear terms in the right column of the first table are positive and negative. That is also thecase for the linear terms in the second table.

The pairs of factors in the first table are both sums or both differences, while the factors in thesecond table are a sum and a difference.

Students may wonder:

why all the expressions in standard form in the second table have the same squared term andconstant term but different linear terms

whether the missing expression in the first row of the second table will also have the samesquared term and constant term as the rest of the expressions in standard form

whether the factors will be sums, differences, or one of each

Next, ask students to work quietly on the second question before conferring with their partner.

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Unit 7 Lesson 7 123


Conversing: MLR8 Discussion Supports. In groups of two, ask students to take turns describing thedifferences they notice between the expressions in each table. Display the following sentenceframes for all to see: “_____ and _____ are different because . . .”, “One thing that is different is . ..” and “I noticed _____, so . . . .” Encourage students to challenge each other when they disagree.This will help students clarify their reasoning about the relationship between quadraticexpressions with a negative constant term written in factored and standard form.Design Principle(s): Support sense-making; Maximize meta-awareness


Representation: Develop Language and Symbols. Create a display of important terms andvocabulary. During the launch, take time to review terms students will need to access for thisactivity. Invite students to suggest language or diagrams to include that will support theirunderstanding of factored form, standard form, linear term, constant term, coefficient,squared term, expression, and factors.Supports accessibility for: Conceptual processing; Language


1. These expressions are like the ones we have seen before.


Each row has a pair of equivalentexpressions.

Complete the table. If you get stuck,consider drawing a diagram.

2. These expressions are in some ways unlike the ones we have seen before.



Each row has a pair of equivalentexpressions.

Complete the table. If you get stuck,consider drawing a diagram.

3. Name some ways that the expressions in the second table are different from those inthe first table (aside from the fact that the expressions use different numbers).

Student Response

1.factored form standard form


3. Sample responses:In the second table, the expressions in standard form all have a negative constant term.In the first table, they are all positive and the numbers are smaller.

The factored expressions in the first table are either both sums or both differences. Inthe second table, they all consist of a sum and a difference.

Activity Synthesis

Display the incomplete second table for all to see. Invite some students to complete the missingexpressions and explain their reasoning. Discuss questions such as:

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Unit 7 Lesson 7 125

“How did you know what signs the numbers in the factored expressions would take?” (The twonumbers must multiply to -36, which is a negative number, so one of the factors must bepositive and the other must be negative.)

“How do you know which factor should be positive and which one negative?” (If the coefficientof the linear term in standard form is positive, the factor of 36 with the greater absolute valueis positive. If the coefficient of the linear term is negative, the factor of 36 with the greaterabsolute value is negative. This is because the sum of a positive and a negative number takesthe sign of the number with the greater absolute value.)

If not mentioned in students’ explanations, point out that all the factored expressions in the secondtable contain a sum and a difference. This can be attributed to the negative constant term in theequivalent standard form expression.

7.3 Factors of 100 and -10015 minutesThis activity aims to solidify students’ observations about the structure connecting the standardform and factored form. Students find all pairs of factors of a number that would lead to a positivesum, a negative sum, and a zero sum. They then look for patterns in the numbers and draw somegeneral conclusions about what must be true about the numbers to produce a certain kind of sum.Along the way, students practice looking for regularity through repeated reasoning (MP8).

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.b

HSA-SSE.B.3.a

Launch

Give students time to complete the first two questions and then pause for a class discussion. If timeis limited, consider arranging students in groups of 2 and asking one partner to answer the firstquestion and the other to answer the second question. Alternatively, consider offering one pair offactors as an example in each table.

Before students answer the last question, consider displaying the completed tables for all to seeand inviting students to observe any patterns or structure in them. Discuss questions such as:

“How are the pairs of factors of 100 like or unlike those of -100?” (The numbers are the samebut the signs are different.)

“In the first two tables, what do you notice about factor pairs that give positive values?” (Theyare both positive.) “What about factor pairs that give negative values?” (They are negative.)

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“In the next two tables, what do you notice about factor pairs that give positive values?” (Onefactor is positive and the other is negative. The positive number has a greater absolute value.)

“What about factor pairs that give negative values?” (One factor is positive and the other isnegative. The positive number has a greater absolute value.)

“What do you notice about the pair of factors that give a value of 0?” (They are opposites.)

Encourage students to use these insights to answer the last question.


Action and Expression: Provide Access for Physical Action. Support effective and efficient use oftools and assistive technologies. To use graphing technology, some students may benefit froma demonstration or access to step-by-step instructions for a method to find pairs of factors fora given number.Supports accessibility for: Organization; Memory; Attention


When completing the tables to find , some students may multiply the factors rather than addthem. Remind them that what we are looking for is the coefficient of the linear term.

Consider completing one row of the table and displaying a rectangle diagram to remind studentshow the value of is obtained when we rewrite an expression such as in standardform. Applying the distributive property gives or . Point out thatin standard form, the product of the factors,100, is the constant term. If the coefficient of the linearterm is what we are after, we need to find the sum of the factors.


1. Consider the expression .

Complete the first table with all pairs of factors of 100 that would give positive values of, and the second table with factors that would give negative values of .

For each pair, state the value they produce. (Use as many rows as needed.)

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Unit 7 Lesson 7 127

positive value of

factor 1 factor 2 (positive)

negative value of

factor 1 factor 2 (negative)

2. Consider the expression .

Complete the first table with all pairs of factors of -100 that would result in positivevalues of , the second table with factors that would result in negative values of , andthe third table with factors that would result in a zero value of .

For each pair of factors, state the value they produce. (Use as many rows as there arepairs of factors. You may not need all the rows.)

positive value of


negative value of



zero value of

factor 1 factor 2 (zero)

3. Write each expression in factored form:

a.

b.

c.

d.

Student Response

1. positive value of


10 10 20

20 5 25

25 4 29

50 2 52

100 1 101

negative value of


-10 -10 -20

-20 -5 -25

-25 -4 -29

-50 -2 -52

-100 -1 -101

2. positive value of

factor1

factor2 (positive)

25 -4 21

20 -5 15

50 -2 48

100 -1 99

negative value of

factor1

factor2 (negative)

-25 4 -21

-20 5 -15

-50 2 -48

-100 1 -99

zero value of

factor 1 factor 2 (zero)

-10 10 0

Unit 7 Lesson 7 129

3. a.

b.

c.

d.


How many different integers can you find so that the expression can bewritten in factored form?

Student Response

Infinitely many. Sample response: The number needs to have two factors which add up to 10.Here are some examples:

(

There are infinitely many examples because there are infinitely many pairs of integers whose sum is10.

Activity Synthesis

Ask students to share their responses to the last question. Discuss how the work in the first twoquestions helped them rewrite the quadratic expressions in factored form.

Highlight that the sign of the constant term can help us anticipate the signs of the numbers in thefactors, making it a helpful first step in rewriting quadratic expressions in factored form. If theconstant term is positive, the factors will have two negative numbers or two positive numbers. If theconstant term is negative, the factors will have one positive number and one negative number.From there, we can determine which two factors give the specified value of in .

Lesson SynthesisTo help students consolidate the observations and insights from this lesson, consider asking themto describe to a partner or write down their responses to prompts such as:

“How would you explain to a classmate who is absent today how to rewrite infactored form?”

“How would you explain how to rewrite in factored form?”

“Suppose you are rewriting the quadratic expression in factored form. How will the factors be different when the is positive versus when is

negative?”

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7.4 The Missing SymbolsCool Down: 5 minutesAddressing

HSA-SSE.A.2


Here are pairs of equivalent expressions in standard form and factored form. Find themissing symbols and numbers.

1. and

2. and

3. and

4. and

Student Response

1. and . The completed expression is .




Student Lesson SummaryWhen we rewrite expressions in factored form, it is helpful to remember that:

Multiplying two positive numbers or two negative numbers results in a positive product.

Multiplying a positive number and a negative number results in a negative product.

This means that if we want to find two factors whose product is 10, the factors must be bothpositive or both negative. If we want to find two factors whose product is -10, one of thefactors must be positive and the other negative.

Suppose we wanted to rewrite in factored form. Recall that subtracting a numbercan be thought of as adding the opposite of that number, so that expression can also bewritten as . We are looking for two numbers that:

Have a product of 7. The candidates are 7 and 1, and -7 and -1.

Have a sum of -8. Only -7 and -1 from the list of candidates meet this condition.

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Unit 7 Lesson 7 131

The factored form of is therefore or, written another way,.

To write in factored form, we would need two numbers that:

Multiply to make -7. The candidates are 7 and -1, and -7 and 1.

Add up to 6. Only 7 and -1 from the list of candidates add up to 6.

The factored form of is .


StatementFind two numbers that...

a. multiply to -40 and add to -6.

b. multiply to -40 and add to 6.

c. multiply to -36 and add to 9.

d. multiply to -36 and add to -5.

If you get stuck, try listing all the factors of the first number.

Solutiona. 4 and -10

b. -4 and 10

c. 12 and -3

d. 4 and -9

Problem 2StatementCreate a diagram to show that is equivalent to .

Solution

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Problem 3StatementWrite a or a sign in each box so the expressions on each side of the equal sign areequivalent.

a.

b.

c.

d.

Solutiona. ,

b. ,

c. ,

d. ,

Problem 4StatementMatch each quadratic expression in standard form with its equivalent expression in factoredform.

A.

B.

C.

D.

1.

2.

3.

4.

Solution

A: 3

B: 1

C: 4

D: 2

Unit 7 Lesson 7 133

Problem 5StatementRewrite each expression in factored form. If you get stuck, try drawing a diagram.

a.

b.

c.

d.

Solutiona.

b.

c.

d.

Problem 6StatementWhich equation has exactly one solution?

A.

B.

C.

D.

SolutionB(From Unit 7, Lesson 5.)

Problem 7StatementThe graph represents the height of a passenger car on a ferris wheel, in feet, as a function oftime, in seconds.


Use the graph to help you:

a. Find .

b. Does have a solution? Explainhow you know.

c. Describe the domain of the function.

d. Describe the range of the function.

Solutiona. 5 feet

b. No, the ferris wheel does not touch the ground. It is 5 feet off the ground.

c. The domain of the graph is between 0 and 60. Students could describe the domain as thelength of the ride.

d. The range of the function is from 5 to 55 feet.


Problem 8StatementElena solves the equation by dividing both sides by to get . She says thesolution is 7.

Lin solves the equation by rewriting the equation to get . When shegraphs the equation , the -intercepts are and . She says the solutionsare 0 and 7.

Do you agree with either of them? Explain or show how you know.

SolutionSample response: Agree with Lin. Substituting 0 for gives , which is true, andsubstituting 7 for gives , which is also true. Elena is not correct. When she divided by ,the solution was lost.


Unit 7 Lesson 7 135

Problem 9StatementA bacteria population, , can be represented by the equation , where is

the number of days since it was measured.

a. What was the population 3 days before it was measured? Explain how you know.

b. What is the last day when the population was more than 1,000,000? Explain how youknow.

Solutiona. 6,400,000 because

b. Two days before the population was measured it was 1,600,000 and 1 day before it was400,000, so the last day when it was more than 1,000,000 was 2 days before it was measured.




GoalsUnderstand that multiplying a sum and a difference, , results in a quadratic withno linear term and explain (orally) why this is the case.

When given quadratic expressions with no linear term, write equivalent expressions infactored form.

Learning TargetsI can explain why multiplying a sum and a difference, , results in a quadraticexpression with no linear term.

When given quadratic expressions in the form of , I can rewrite them in factoredform.

Lesson NarrativeSo far, the quadratic expressions that students have transformed from standard form to factoredform have at least a squared term and a linear term. In this lesson, students encounter quadraticexpressions without a linear term and consider how to write them in factored form.

Students begin by studying numerical examples and noticing that expressions such asand (which is a difference of two squares) are equivalent. Through

repeated reasoning, students are able to generalize the equivalence of these two forms as(MP8). Then, they make use of the structure relating the two expressions

to rewrite expressions (MP7) from one form to the other.

Along the way, they encounter a variety of quadratic expressions that can be seen as differences oftwo squares, including those in which the squared term has a coefficient other than 1, orexpressions that involve fractions.

Students also consider why a difference of two squares (such as ), can be written in factoredform, but a sum of two squares (such as ) cannot be, even though both are quadraticexpressions with no linear term.

After this lesson, students will have the tools they need to solve factorable quadratic equationsgiven in standard form by first rewriting them in factored form. That work begins in the next lesson.

Alignments

Building On

6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example,apply the distributive property to the expression to produce the equivalentexpression ; apply the distributive property to the expression to produce the

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Unit 7 Lesson 8 137

equivalent expression ; apply properties of operations to to produce theequivalent expression .

Addressing


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Building Towards



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Math Talk

MLR3: Clarify, Critique, Correct


Required Materials

Scientific calculators


Let’s look closely at some special kinds of factors.

8.1 Math Talk: Products of Large-ish NumbersWarm Up: 10 minutesThis Math Talk prompts students to recall strategies for multiplying mentally, which encouragesthem to look for and use structure in the expressions (MP7).

Each expression can be evaluated in different ways. For example, can be viewed as, as , or as , among other ways. Reasoning flexibly

about the structure of numerical expressions encourages students to do the same when rewritingquadratic expressions in this lesson and beyond.

Building On

6.EE.A.3

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Math Talk


Launch

Display one problem at a time. Give students quiet think time for each problem and ask them togive a signal when they have an answer and a strategy. Keep all problems displayed throughout thetalk. Follow with a whole-class discussion.




Find each product mentally.

Student Response

99

399

9,999

11,009

Activity Synthesis






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Unit 7 Lesson 8 139




8.2 Can Products Be Written as Differences?15 minutesIn this activity, students multiply expressions of the form and . Through repeatedreasoning, they discover that the expanded product of such factors can be expressed as adifference of two square numbers: (MP8). Students use diagrams and the distributiveproperty to make sense of their observations.

In the last question, they have an opportunity to notice that an expression in the form isnot equivalent to , to discourage overgeneralizing. Later in the lesson, they will use theirunderstanding of the structure relating the equivalent expressions to transform quadraticexpressions in standard form into factored form and vice versa (MP7).

To answer the first two questions, some students may simply evaluate the expressions rather thanreasoning about their structure. For example, to see if is equivalent to ,they may calculate and and see that both are 91. Ask students to see if they couldshow that this is not a coincidence. Could they show, for example, that would alsobe equivalent to ?

As students work, look for those who expand the factored expression using the distributiveproperty and leave the partial products unevaluated in order to show the equivalence of the twoexpressions.

Building Towards

HSA-SSE.A.2



Launch

Provide access to calculators, in case requested.

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Representation: Internalize Comprehension. Use color coding and annotations to highlightconnections between representations in a problem. For example, use color coding to highlightthe opposites that occur in the linear term after the products are distributed. Then, useannotations to illustrate the sum is zero, which results in no linear term. Encourage students toannotate their work while working independently.Supports accessibility for: Visual-spatial processing


Some students may struggle to generalize the pattern after just a few examples. Before starting theactivity synthesis, provide additional factored expressions for students to expand, forinstance: , , , and .


1. Clare claims that is equivalent to and isequivalent to . Do you agree? Show your reasoning.

2. a. Use your observations from the first question and evaluate .Show your reasoning.

b. Check your answer by computing .

3. Is equivalent to ? Support your answer:

With a diagram: Without a diagram:

4. Is equivalent to ? Support your answer, either with or without adiagram.

Student Response

1. Agree. Sample reasoning:

Unit 7 Lesson 8 141

a.

b.

2. a. .

b.

3. Yes. Sample reasoning:With a diagram:

or

Without a diagram:

4. No. Sample reasoning:


1. Explain how your work in the previous questions can help you mentally evaluateand .

2. Here is a shortcut that can be used to mentally square any two-digit number. Let’s take, for example.

83 is .

Compute and , which give 6,400 and 9. Add these values to get 6,409.

Compute , which is 240. Double it to get 480.

Add 6,409 and 480 to get 6,889.

Try using this method to find the squares of some other two-digit numbers. (With somepractice, it is possible to get really fast at this!) Then, explain why this method works.

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Student Response

1. can be written as , which is equivalent to , which is or396.

can be written as , which is equivalent to , which isor 1,575.

2. Sample response: Any two digit number can be written as , where is 10 times the firstdigit and is the second digit. We are interested in squaring :

We get the from squaring and and adding them, and the from multiplying thetwo values and doubling them.

Activity Synthesis

Invite previously identified students to share their responses and reasoning.

To help students generalize their observations, display the expression and a blankdiagram that can be used to visualize the expansion of the factors. Ask students what expressionsgo in each rectangle.

Illustrate that when the terms in each factor are multiplied out, the resulting expression has twosquares, one with a positive coefficient and the other with a negative coefficient ( and ) andtwo linear terms that are opposites ( and ). Because the sum of and is 0, whatremains is the difference of and , or . There is now no linear term.

or

Emphasize that knowing this structure allows us to rewrite into factored form any quadraticexpression that has no linear term and that is a difference of a squared variable and a squaredconstant. For example, we can write as because we know that when the latteris expanded, the result is .

Unit 7 Lesson 8 143

Use the last question to point out the importance of paying attention to the particulars of thestructure of these expressions (the subtraction in the first expression, the presence of both additionand subtraction in the second). For example, we can't use any patterns observed in this activity torewrite in factored form.


Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share theirexplanations for the last question, present an incorrect answer and explanation. For example,“ is equivalent to because when you square an expression you square eachterm.” Ask students to identify the error, critique the reasoning, and write a correctexplanation. As students discuss with a partner, listen for students who use diagrams orexamples to illustrate the error in the author’s reasoning. Invite students to share theircritiques and corrected explanations with the class. Listen for and amplify the languagestudents use to explain how to use the distributive property to square an expression. Thishelps students evaluate, and improve upon, the written mathematical arguments of others.Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

8.3 What If There is No Linear Term?10 minutesIn this activity, students use the insights from the previous activity to write equivalent quadraticexpressions in standard form and factored forms. They see that when a quadratic expression instandard form is a difference of two squares (a squared variable, , and a squared constant, )and has no linear term, the factored form is . Even if the constant term is not aperfect square, we can still find the factored form, but the numbers in the factors would not be

rational numbers. For example, the expression can be written as ,

because . If we expand , there will be no linear term

because .

Students also notice that when a quadratic expression is a sum (instead of a difference) of asquared variable and a squared constant, it cannot be written in factored form.

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.b

HSA-SSE.B.3.a

•

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Launch

Consider arranging students in groups of 2 and asking them to think quietly about the problemsbefore discussing their responses with their partner.

Ask students to write as many equivalent expressions as time permits while aiming to complete atleast the first six rows and the last row of the table.


Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge.Invite students to work in pairs and take turns selecting rows to complete. Chunking this taskinto more manageable parts may also support students who benefit from additionalprocessing time.Supports accessibility for: Organization; Attention; Social-emotional skills


Some students may struggle to see the numbers in the expressions in standard form as perfectsquares. Prompt them to create a list or table of square numbers ( , , , and soon) to have as a handy reference. Others may benefit by rewriting both terms as squares beforewriting the factored form. Demonstrate how to rewrite as and as

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Unit 7 Lesson 8 145


Each row has a pair of equivalent expressions.

Complete the table.

If you get stuck, consider drawing a diagram.(Heads up: one of them is impossible.)


Student Response

factored form standard form factored form standard form

not possible


Activity Synthesis

Consider displaying the incomplete table for all to see and asking students to record theirresponses. Give the class time to examine the responses and to bring up any disagreements orquestions. Discuss with students:

“How can we check if the expression in factored form is indeed equivalent to the givenexpression in standard form?” (We can expand the factored expression by applying thedistributive property and see if it gives the expression in standard form.)

“Some of the expressions show a squared variable subtracted from a number, instead of theother way around. Can we still write an equivalent expression in factored form?” (Yes. As longas the expression in standard form can be written as a difference of two squares, it can bewritten in factored form.)

“What if the number is not a perfect square, for example: ?” (We can still write it in

factored form, by thinking about what number can be squared to get 5. Both and canbe squared to get 5. Regardless of which number we use, the factored form is

.)

“Why can be written in factored form but cannot?” (One possible approachis to rewrite the former as . We learned previously that, to write this expressionin factored form, we would need to look for two numbers whose product is -100 and whosesum is 0. The numbers 10 and -10 meet this requirement. For , however, weneed two numbers whose product is 100 and whose sum is 0. No such numbers exist. To havea sum of 0, one number has to be positive and the other negative, so their product can’t bepositive 100.)


Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-classdiscussion. After each student shares, provide the class with the following sentence frames tohelp them respond: “I agree because . . .” or “I disagree because . . . .” If necessary, revoicestudent ideas to demonstrate mathematical language use by restating a statement as aquestion in order to clarify, apply appropriate language, and involve more students.Design Principle(s): Support sense-making

Lesson SynthesisTo help students consolidate and articulate their understanding of the relationship between

and , ask them to reflect, in writing or by talking to a partner, on questionssuch as:

“The expression has a sum and a difference, and so does . Whenexpanded into standard form, why does one have a linear term but not the other?”

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Unit 7 Lesson 8 147

(Multiplying out gives two linear terms that are opposites, and , which addup to 0, so the linear term disappears. Multiplying out also gives two linearterms, and . Because they are not opposites, their sum is not 0, so the linear termremains.)

“Can be seen as a difference of two squares? Can it be written in factored form? If

so, what would it be?” (Yes. squared is and is . The factored form is

.

“Think of another example of a quadratic expression in factored form that, when rewritten instandard form, is a difference of two squares and does not have a linear term. What is theexpression in standard form?”

8.4 Can These Be Rewritten in Factored Form?Cool Down: 5 minutesAddressing

HSA-SSE.A.2


Write each expression in factored form. If it is not possible, write “not possible.”

1.

2.

3.

4.

Student Response

1.

2.

3. Not possible

4.

Student Lesson SummarySometimes expressions in standard form don’t have a linear term. Can they still be written infactored form?

Let’s take as an example. To help us write it in factored form, we can think of it ashaving a linear term with a coefficient of 0: . (The expression isequivalent to because 0 times any number is 0, so is 0.)

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We know that we need to find two numbers that multiply to make -9 and add up to 0. Thenumbers 3 and -3 meet both requirements, so the factored form is .

To check that this expression is indeed equivalent to , we can expand the factoredexpression by applying the distributive property: . Adding

and gives 0, so the expanded expression is .

In general, a quadratic expression that is a difference of two squares and has the form:

can be rewritten as:

Here is a more complicated example: . This expression can be written ,so an equivalent expression in factored form is .

What about ? Can it be written in factored form?

Let’s think about this expression as . Can we find two numbers that multiply tomake 9 but add up to 0? Here are factors of 9 and their sums:

9 and 1, sum: 10

-9 and -1, sum: -10

3 and 3, sum: 6

-3 and -3, sum: -6

For two numbers to add up to 0, they need to be opposites (a negative and a positive), but apair of opposites cannot multiply to make positive 9, because multiplying a negative numberand a positive number always gives a negative product.

Because there are no numbers that multiply to make 9 and also add up to 0, it is not possibleto write in factored form using the kinds of numbers that we know about.


StatementMatch each quadratic expression given in factored form with an equivalent expression instandard form. One expression in standard form has no match.

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Unit 7 Lesson 8 149

A.

B.

C.

D.

1.

2.

3.

4.

5.

Solution

A: 3

B: 1

C: 5

D: 4

Problem 2StatementBoth and contain a sum and a difference and have only 3 andin each factor.

If each expression is rewritten in standard form, will the two expressions be the same?Explain or show your reasoning.

SolutionSample response: No. , but .

Problem 3Statement

a. Show that the expressions and are equivalent.

b. The expressions and are equivalent and can help us find theproduct of two numbers. Which two numbers are they?

c. Write as a product of a sum and a difference, and then as a difference of twosquares. What is the value of ?

Solutiona. Sample response: and


b. 28 and 32

c. and . The value is 9,964.

Problem 4StatementWrite each expression in factored form. If not possible, write “not possible.”

a.

b.

c.

d.

e.

Solutiona.

b. not possible

c.

d.

e. not possible


A. and

B. and

C. and

D. and


Unit 7 Lesson 8 151

Problem 6StatementCreate a diagram to show that is equivalent to .

Solution


Problem 7StatementSelect all the expressions that are equivalent to .

A.

B.

C.

D.

E.

F.

G.

Solution["B", "C", "G"](From Unit 7, Lesson 6.)

Problem 8StatementMai fills a tall cup with hot cocoa, 12 centimeters in height. She waits 5 minutes for it to cool.Then, she starts drinking in sips, at an average rate of 2 centimeters of height every 2minutes, until the cup is empty.

The function gives the height of hot cocoa in Mai’s cup, in centimeters, as a function oftime, in minutes.


a. Sketch a possible graph of . Be sure toinclude a label and a scale for each axis.

b. What quantities do the domain andrange represent in this situation?

c. Describe the domain and range of .

Solutiona. See sample graph.

b. The domain represents time, in minutes, since Mai fills hercup with hot cocoa. The range represents the height of theliquid in centimeters.

c. Sample response: The domain includes all values from 0 to17. The range includes all heights from 0 to 12.


Problem 9StatementOne bacteria population, , is modeled by the equation , where is the

number of days since it was first measured.

A second bacteria population, , is modeled by the equation , where is

the number of days since it was first measured.

Which statement is true about the two populations?

A. The second population will always be larger than the first.

B. Both populations are increasing.

C. The second bacteria population decreases more rapidly than the first.

D. When initially measured, the first population is larger than the second.


Unit 7 Lesson 8 153

Lesson 9: Solving Quadratic Equations by UsingFactored Form

GoalsRecognize that the number of solutions to a quadratic equation can be revealed when theequation is written as .

Use factored form and the zero product property to solve quadratic equations.

Learning TargetsI can rearrange a quadratic equation to be written as and findthe solutions.

I can recognize quadratic equations that have 0, 1, or 2 solutions when they are written infactored form.

Lesson NarrativeIn this lesson, students apply what they learned about transforming expressions into factored formto make sense of quadratic equations and persevere in solving them (MP1). They see thatrearranging equations so that one side of the equal sign is 0, rewriting the expression in factoredform, and then using the zero product property make it possible to solve equations that theypreviously could only solve by graphing. These steps also allow them to easily see—withoutgraphing and without necessarily completing the solving process—the number of solutions that theequations have.

Alignments

Building On


Addressing



Building Towards


•

•

•

•

•

•

•

•



Graph It



Notice and Wonder

Think Pair Share

Required Materials






Let’s solve some quadratic equations that before now we could only solve by graphing.

9.1 Why Would You Do That?Warm Up: 10 minutesIn this activity, students find at least one solution of by substituting differentvalues of , evaluating the expression, and checking if it has a value of 0. Experiencing thisinefficient method puts students in a better position to appreciate why it may be desirable to write

in factored form and use the zero product property.

Building Towards

HSA-REI.B.4

Launch

Once students have had a chance to evaluate the expression using their chosennumber for , ask if anyone found a value that made the expression equal 0. (It's 7.) Give students acouple of minutes to look for the other value that makes the expression equal 0.


Let's try to find at least one solution to !

1. Choose a whole number between 0 and 10.

•••••

•

•

Unit 7 Lesson 9 155

2. Evaluate the expression , using your number for .

3. If your number doesn't give a value of 0, look for someone in your class who may havechosen a number that does make the expression equal 0. Which number is it?

4. There is another number that would make the expression equal 0. Can youfind it?

Student Response

1. Answers vary.

2. Here are all the possibilities:

0 1 2 3 4 5 6 7 8 9 10

-35 -36 -35 -32 -27 -20 -11 0 13 28 45

3. 7

4. It's -5, but it is possible that no students will consider negative numbers.

Activity Synthesis

If a student found that -5 makes the expression equal 0, ask them to demonstrate thatequals 0.

Discuss with students:

"Can be written in factored form? What are the factors?" (Yes. and )

"If can be written as , can we solve instead?"(Yes) "Will the solutions change if we use this equation?" (No. The equations are equivalent, sothey have the same solutions.)

"Why might someone choose to rewrite and solveinstead? (Because the expression is equal to 0, rewriting it in factored form allows us to usethe zero product property to find both solutions. It may be more efficient than substitutingand evaluating many values for . It also makes it possible to see how many solutions thereare, which is not always easy to tell when the quadratic expression is in standard form.)

9.2 Let’s Solve Some Equations!15 minutesIn this activity, students solve a variety of quadratic equations by integrating what they learnedabout rewriting quadratic expressions in factored form and their understanding of the zero productproperty. They begin by analyzing and explaining the steps in a solution strategy, and then applyingtheir observations to solve other equations, both of which require sense making and perseverance

••

•


(MP1). Students practice attending to precision (MP6) as they study solution steps andcommunicate what each step does or means.

As students work, notice the equations many students find challenging and those on which errorsare commonly made. Discuss these challenges and errors during the activity synthesis.

Addressing

HSA-REI.B.4.b



Notice and Wonder

Think Pair Share

Launch

Arrange students in groups of 2. Display the worked example in the activity statement for all to see.

Give students 1 minute of quiet time to study the example. Ask them to be prepared to share atleast one thing they notice and one thing they wonder. Give students another minute to discusstheir observations and questions with a partner.

Students may notice that:

The right side is equal to 0 after the first step.

There are two equations starting in step 3.

The product of 11 and -9 is the constant term in step 2.


why there are two equations in the last two steps

where the went when the expression on the left side was written in factored form

whether Tyler's work is correct

Then, give students a moment to work quietly on the first question. Encourage students to userelevant mathematical vocabulary in their explanations (such as constant term, squared term,factored form, and zero product property).

Before students proceed to the second set of questions, pause for a class discussion. Ask studentsto explain each step of Tyler’s solving process and record their explanations for all to see.Encourage students to use reasoning language as opposed to position language: for instance, say"he subtracted 99 from both sides" rather than "he moved the 99 over to the other side."

•

•••

•••

•••

Unit 7 Lesson 9 157

Make sure students recognize that going from the second line to the third line in Tyler’s workinvolves finding two factors of -99 that add up to -2. Also discuss why there are two equations at theend. Students should recall that if two numbers multiply to equal 0, then one of the factors must be0.

Ask students to solve as many equations in the second question as time permits.


Representation: Internalize Comprehension. Activate or supply background knowledge.Demonstrate how students can continue to use diagrams to rewrite equations in factored formby first rewriting each equation so that one side is equal to .Supports accessibility for: Visual-spatial processing; Organization


Some students may struggle with because the coefficient of the linear term, 1, isn'twritten (as is customary). Tell students that means the same thing as , so this equation can bewritten . Encourage them to proceed with the term written this way.


1. To solve the equation , Tyler wrote out the following steps. Analyze Tyler’swork. Write down what Tyler did in each step.

2. Solve each equation by rewriting it in factored form and using the zero productproperty. Show your reasoning.

a.

b.

c.

d.


e.

Student Response

1. Sample response:Step 1: Subtract 99 from each side.

Step 2: Rewrite the left side in factored form.

Step 3: Apply the zero product property, which leads to two separate equations.

Step 4: Solve each equation by performing the same operation on each side.

2. Solutions:a. -3 and -5

b. 7 and 1

c. 11 and -1

d. 7 and -7

e. -10 and 1


Solve this equation and explain or show your reasoning.

Student Response

, , or . Sample reasoning: Rewriting each quadratic expression in factored form

gives . Because the factors andappear on each side of the equation, 5 and -4 are values of that make each side equal to

zero. If those factors are not equal to zero, then we can divide each side of the equation byand are left with , or .

Subtracting the terms from each side and then solving the linear equation gives .

Activity Synthesis

Consider displaying the solutions for all to see and discussing only the equations that studentsfound challenging and any common errors.

The last equation is unlike most equations students have seen. Invite students to share how theysolved that equation. Discuss questions such as:

“Can we use the zero product property to write and ? Why or why not?”(No. The zero product property can be applied only to products that equal 0. The expression

has a product for one of its terms, but the expression itself is a difference.)

◦◦◦◦

•

Unit 7 Lesson 9 159

“How can it be solved, other than by graphing?” (We can expand the factored expressionsusing the distributive property and write an equivalent equation: or

. This last equation can then be written as , which allows itto be solved.)


Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For eachsolution strategy that is shared about the last equation, ask students to restate what theyheard using precise mathematical language. Consider providing students time to restate whatthey hear to a partner before selecting one or two students to share with the class. Ask theoriginal speaker if their peer was accurately able to restate their thinking. Call students’attention to any words or phrases that helped to clarify the original statement. This providesmore students with an opportunity to produce language as they interpret the reasoning ofothers.Design Principle(s): Support sense-making

9.3 Revisiting Quadratic Equations with Only OneSolution10 minutesThis activity reinforces what students learned earlier about the connections between the solutionsof a quadratic equation and the zeros of a quadratic function. Previously, students were givenequations and asked to graph them to determine the number of solutions and their values. Here,they are prompted to work the other way around: to write an equation to represent a quadraticfunction with only one solution. To do so, students need to make use of the structure of thefactored form and their knowledge of the zero product property (MP7).

Building On

HSA-REI.B.4.b

Addressing

HSA-SSE.B.3.a


Graph It


Launch

Give students access to graphing technology.

•

•

•

••



Action and Expression: Provide Access for Physical Action. Support effective and efficient use oftools and assistive technologies. To use graphing technology, some students may benefit froma demonstration or access to step-by-step instructions.Supports accessibility for: Organization; Memory; Attention


1. The other day, we saw that a quadratic equation can have 0, 1, or 2 solutions. Sketchgraphs that represent three quadratic functions: one that has no zeros, one with 1 zero,and one with 2 zeros.

2. Use graphing technology to graph the function defined by . What doyou notice about the -intercepts of the graph? What do the -intercepts reveal aboutthe function?

3. Solve by using the factored form and zero product property. Show yourreasoning. What solutions do you get?

4. Write an equation to represent another quadratic function that you think will only haveone zero. Graph it to check your prediction.

Student Response1. No zeros One zero Two zeros

2. The graph only intersects the -axis at one point: . The function only has one zero.

3. Rewrite the standard form to . The solutions are 1 and 1. Because the twovalues are the same, we say that this equation has one solution.

4. Sample response: . When graphed, this function only has one zero, at -4.

Activity Synthesis

Invite students to share how they solved the equation algebraically. Next, invite students to sharethe equations they generated. Record and display them for all to see.

Unit 7 Lesson 9 161

Students most likely have written equations in the form of . Ask students whythe factored form, rather than the standard form, might have been preferred. Highlight that byusing the same expression for the two factors, we know that the solution to willbe a single number.

Ask students to describe the graph of a quadratic function with one solution. Point out that thismeans that the function will have only one zero, and the graph of the function will have a singlehorizontal intercept.


Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students forthe whole-class discussion. At the appropriate time, invite student pairs to create a visualdisplay of their equation and graph of a quadratic function with only one zero. Allow studentstime to quietly circulate and analyze at least two other visual displays in the room. Givestudents quiet think time to consider how the zero is represented in the equation and graph ofthe quadratic function. Next, ask students to return to their partner and discuss what theynoticed. Listen for and amplify observations that connect the zero of the function with the

-intercept of the graph. This will help students make connections between the algebraic andgraphical representation of quadratic functions.Design Principle(s): Optimize output; Cultivate conversation

Lesson SynthesisDisplay a series of equations that, prior to this lesson, students could only solve by graphing. Forinstance:

Ask students to choose an equation that they think they could solve without graphing. Then, askthem to explain to a partner why they believe they could solve the equation.

Consider displaying for all to see and using it as an example: “I think I can solvebecause I know the expression on the left is equivalent to and I can

rewrite the equation as , which I can solve by rearranging the terms.”


Then, ask if any of the equations appear to be unsolvable other than by graphing and why. Of theequations shown here, is the only one that students aren’t yet equipped to solvebecause the coefficient of is not 1. Students will begin looking at such equations in an upcominglesson.

9.4 Conquering More EquationsCool Down: 5 minutesAddressing

HSA-REI.B.4.b


Solve each equation by rewriting it in factored form and using the zero product property.Show your reasoning.

1.

2.

3.

Student Response

1. -1 and -11. The equation can be written as , whichmeans or .

2. -2 and 2. The equation can be written as , which is ,which means or .

3. 3. The equation can be written as , which is, which means .

Student Lesson SummaryRecently, we learned strategies for transforming expressions from standard form to factoredform. In earlier lessons, we have also seen that when a quadratic expression is in factoredform, it is pretty easy to find values of the variable that make the expression equal zero.Suppose we are solving the equation , which says that the product of andis 0. By the zero product property, we know this means that either or , whichthen tells us that 0 and -4 are solutions.

Together, these two skills—writing quadratic expressions in factored form and using the zeroproduct property when a factored expression equals 0—allow us to solve quadraticequations given in other forms. Here is an example:

•

Unit 7 Lesson 9 163

When a quadratic equation is written as , we can also see thenumber of solutions the equation has.

In the example earlier, it was not obvious how many solutions there would be when theequation was . When the equation was rewritten as ,we could see that there were two numbers that could make the expression equal 0: 14 and-10.

How many solutions does the equation have?

Let’s rewrite it in factored form: . The two factors are identical, whichmeans that there is only one value of that makes the expression equal 0.The equation has only one solution: 10.


StatementFind all the solutions to each equation.

a.

b.

c.

d.

e.

Solutiona. and

b.

c. and

d.

e. and


Problem 2StatementRewrite each equation in factored form and solve using the zero product property.

a.

b.

c.

d.

Solutiona. and

b.

c. and

d.

Problem 3StatementHere is how Elena solves the quadratic equation .

Is her work correct? If you think there is an error, explain theerror and correct it.

Otherwise, check her solutions by substituting them into theoriginal equation and showing that the equation remains true.

SolutionSample response: Elena incorrectly rewrote the expression in factored form. It should be

for the linear terms to add up to . This would give and assolutions.

Problem 4StatementJada is working on solving a quadratic equation, as shown here.

Unit 7 Lesson 9 165

She thinks that her solution is correct because substituting 5for in the original expression gives , whichis or 0.

Explain the mistake that Jada made and show the correct solutions.

SolutionSample response: Jada correctly rewrote the expression in factored form, but instead of using thezero product property, she divided both sides by , so one of the solutions is missing. If

, then or , which means the solutions are and .

Problem 5StatementChoose a statement to correctly describe the zero product property.

If and are numbers, and , then:

A. Both and must equal 0.

B. Neither nor can equal 0.

C. Either or .

D. must equal 0.


Problem 6StatementWhich expression is equivalent to ?

A.

B.

C.

D.



Problem 7StatementThese quadratic expressions are given in standard form. Rewrite each expression in factoredform. If you get stuck, try drawing a diagram.

a.

b.

c.

d.

Solutiona.

b.

c.

d.


Problem 8StatementSelect all the functions whose output values will eventually overtake the output values offunction defined by .

A.

B.

C.

D.

E.

F.

Unit 7 Lesson 9 167

Solution["A", "B", "D", "E"](From Unit 6, Lesson 4.)

Problem 9Statement

A piecewise function, , is defined by this rule:

Find the value of at each given input.

a.

b.

c.

d.

Solutiona.

b.

c.

d.




GoalsGiven a quadratic expression of the form , where is not 1, write an equivalentexpression in factored form.

Write a quadratic equation that represents a context, consider different methods for solving it,and describe (orally) the limitations of each method.

Learning TargetsI can use the factored form of a quadratic expression or a graph of a quadratic function toanswer questions about a situation.

When given quadratic expressions of the form and is not 1, I can writeequivalent expressions in factored form.

Lesson NarrativeUp to this point, most quadratic expressions that students have transformed from standard form tofactored form had a leading coefficient of 1, that is, they were in the form of becausethe squared term had a coefficient of 1. There were a few instances in which students rewroteexpressions in standard form with a leading coefficient other than 1. Those expressions weredifferences of two squares, where there were no linear terms (for instance, or ).Students learned to rewrite these as or , respectively.

In this lesson, students consider how to rewrite expressions in standard form where the leadingcoefficient is not 1 and the expression is not a difference of two squares. They notice that the samestructure used to rewrite as can be used to rewrite expressions such as

, but the process is now a little more involved because the coefficient of has to betaken into account when finding the right pair of factors. The work here gives students manyopportunities to look for and make use of structure (MP7).

This lesson aims to give students a flavor of rewriting more complicated expressions in factoredform, and to suggest that it is not always practical or possible. This experience motivates the needfor other strategies for solving equations and prepares students to complete the square in a seriesof upcoming lessons.



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Unit 7 Lesson 10 169

Addressing



HSA-SSE.A: Interpret the structure of expressions.


.

HSF-IF.B.4: For a function that models a relationship between two quantities, interpret keyfeatures of graphs and tables in terms of the quantities, and sketch graphs showing keyfeatures given a verbal description of the relationship. Key features include: intercepts;intervals where the function is increasing, decreasing, positive, or negative; relative maximumsand minimums; symmetries; end behavior; and periodicity.

Building Towards




Graph It

MLR6: Three Reads


Which One Doesn’t Belong?

Required Materials



•

•••

•

•

•

••••





Let’s transform more-complicated quadratic expressions into the factored form.

10.1 Which One Doesn’t Belong: QuadraticExpressionsWarm Up: 5 minutesThis warm-up prompts students to carefully analyze and compare quadratic expressions. In makingcomparisons, students need to look for common structure and have a reason to use languageprecisely (MP7, MP6). The activity also enables the teacher to hear the terminology students knowand how they talk about characteristics of the different forms of expressions.

As students discuss in groups, listen for rationales that are based on the structure of theexpressions. Select those students or groups to share their thinking during class discussion.

Addressing

HSA-SSE.A


Which One Doesn’t Belong?

Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute ofquiet think time and then time to share their thinking with their small group. In their small groups,ask each student to share their reasoning about why a particular item does not belong, andtogether find at least one reason each item doesn't belong.


Which one doesn’t belong?

A.

B.

C.

D.

Student Response

Sample responses:

•

•

•


A: The only one that is not in standard form. When written in standard form, it is the only one thathas an even number for the constant term.

B: The only expression in standard form that has a leading coefficient other than 1.

C: The only expression in standard form that doesn’t have three terms. It is the only one without alinear term.

D: The only expression in standard form that can’t be written in factored form.

Activity Synthesis

Ask each group to share one reason why a particular expression doesn’t belong. Record and displaythe responses for all to see. After each response, ask the class if they agree or disagree. Since thereis no single correct answer to the question of which one does not belong, attend to students’explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such asstandard form, factored form, linear term, or coefficient. Also, press students on unsubstantiatedclaims. For example, if they claim that B is the only one that cannot be written in factored form, askthem to show how they know.

10.2 A Little More Advanced15 minutesMost of the factored expressions students saw were of the form or . In thisactivity, students work with expressions of the form . They expand expressionssuch as into standard form and look for structure that would allow them to go inreverse (MP7), that is, to transform expressions of the form , where is not 1 (such as

) into factored form.

Going from factored form to standard form is fairly straightforward given students’ experience withthe distributive property. Going in reverse, however, is a bit more challenging when the coefficientof is not 1. With some guessing and checking, students should be able to find the factored formof the expressions in the second question, but they should also notice that this process is notstraightforward.

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.b

HSA-SSE.B.3.a



•

••

•


Launch

Remind students that they have seen quadratic expressions such as and, where the coefficient of the squared term is not 1. Solicit some ideas from students

on how to write the factored form for expressions such as these.

Arrange students in groups of 2–3 and ask them to split up the work for completing the first table,with each group member rewriting one expression into standard form.

Display the incomplete table in the first question for all to see, and then invite students to share theexpanded expressions in standard form. Record the expressions in the right column, and askstudents to make observations about them.

If not mentioned by students, point out that each pair of factors start with and , which multiplyto make . Each pair of factors also has constant terms that multiply to make 4. The resultingexpressions in standard form are all different, however, because using different pairs of factors of 4and arranging them in different orders produce different expanded expressions.

Ask students to keep these observations in mind as they complete the second question.

If time is limited, ask each group member to choose at least two expressions in the second tableand rewrite them into factored form.


Representation: Internalize Comprehension. Activate or supply background knowledge.Demonstrate how students can continue to use diagrams to rewrite expressions in which thecoefficient of the squared term is not 1. Invite students to begin by generating a list of factorsand to test them using the diagram. Encourage students to persist with this method,reiterating the fact that they are not necessarily expected to immediately recognize whichfactors will work without testing them. Allow students to use calculators to ensure inclusiveparticipation.Supports accessibility for: Visual-spatial processing; Organization


Some students may not think to check their answers to the second question and stop as soon asthey think of a pair of factors that give the correct squared term and constant term. Encouragethem to check their answers with a partner by giving time to do so. Consider providing anon-permanent writing surface or extra paper so students could try out their guesses and checktheir work without worrying about having to erase if they make a mistake the first time or two.


Each row in each table has a pair of equivalent expressions. Complete the tables. If you getstuck, try drawing a diagram.




Student Response

1. Sample responses (students may write equivalent expressions):

2. Sample responses (students may write equivalent expressions):

or


Here are three quadratic equations, each with two solutions. Find both solutions to eachequation, using the zero product property somewhere along the way. Show each step in yourreasoning.

Student Response

Sample solution method for each equation is shown.

◦◦◦

◦◦◦


0 and 6. -4 and 1.

If , then .If , then .

and 1.

If , then .If , then .

Activity Synthesis

Display for all to see the incomplete table in the second question. Select students to complete themissing expressions in standard form and to briefly explain their strategy. To rewrite ,students are likely to have tried putting different factors of and of in the factored expressionsuch that when the factors are expanded, they yield a linear term with the coefficient .

Then, help students to reason about the factors more generally. Discuss questions such as:

“To rewrite expressions such as , we looked for two numbers that multiply to makeand add up to . The expressions here are of the form . Are we still looking for

two numbers that multiply to make ? Why or why not?” (Yes. The constant terms in thefactored expression must multiply to make .)

“Do we need to look for factors of ? Why or why not?” (Yes. Those factors will be thecoefficient of in the factored expressions. They must multiply to make .)

“Are we still looking for two factors of that add up to ? Why or why not?” (No. The value ofis no longer just the sum of the two factors of because the two factors of are nowinvolved.) “How does this affect the rewriting process?” (It makes it more complicated, becausenow there are four numbers to contend with, and there are many more possibilities toconsider.)

Tell students that we’ll investigate a bit further quadratic equations in the form ofwhere is not 1, and see if there are manageable ways to rewrite such equations in factored formso that they can be solved.

•

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Speaking, Representing: MLR8 Discussion Supports. Give students time to make sure that everyonein the group can explain or justify each step or part of the problem. Invite groups to rehearsewhat they will say when they share with the whole class. Rehearsing provides students withadditional opportunities to speak and clarify their thinking, and will improve the quality ofexplanations shared during the whole-class discussion. Then make sure to vary who is selectedto represent the work of the group, so that students get accustomed to preparing each otherto fill that role.Design Principle(s): Support sense-making; Cultivate conversation

10.3 Timing A Blob of Water15 minutesEarlier, students were given quadratic expressions of the form where the is not 1.They found that the rewriting process was a bit more involved but was not impossible, at least forthe problems at hand.

In this activity, they encounter a real-world example in which they struggle to find a combination ofrational factors of and that would produce the value of . (Realistic quadratic functions don’talways have rational numbers for their zeros, so the quadratic expressions that define them cannotalways be written in factored form. At this point, students are not yet considering rational andirrational solutions and are not expected to know why some expressions cannot be easily written infactored form.)

By graphing the function, students discover that the horizontal intercepts are decimals (rounded todifferent decimal places, depending on the graphing technology used). The graph allows them toestimate the solution, but that is as far as they could go. The challenges in this activity set the stagefor introducing a more productive technique for solving quadratic equations.

Building On

HSA-REI.B.4.b

Addressing

HSA-REI.D

HSF-IF.B.4


Graph It

MLR6: Three Reads

•

••

••


Launch

Keep students in groups of 2. Let students become briefly frustrated by their unsuccessful attemptsto find factors of the expression in standard form, but move them on to the last question after afew minutes. Provide access to devices that can run Desmos or other graphing technology.


Reading: MLR6 Three Reads. Use this routine to support reading comprehension, withoutsolving, for students. Use the first read to orient students to the situation. Ask students todescribe what the situation is about without using numbers (An engineer is designing a waterfountain). Use the second read to identify quantities and relationships. Ask students what canbe counted or measured without focusing on the values. Listen for, and amplify, the importantquantities that vary in relation to each other in this situation: height of a drop of water, inmeters, and time, in seconds. After the third read, ask students to brainstorm possiblestrategies to answer the questions. This helps students connect the language in the wordproblem and the reasoning needed to solve the problem.Design Principle: Support sense-making


Representation: Internalize Comprehension. Provide appropriate reading accommodations andsupports to ensure student access to written directions, word problems and other text-basedcontent. Clarify any unfamiliar terms or phrases.Supports accessibility for: Language; Conceptual processing


An engineer is designing a fountain that shoots out drops of water. The nozzle from whichthe water is launched is 3 meters above the ground. It shoots out a drop of water at a verticalvelocity of 9 meters per second.

Function models the height in meters, , of a drop of water seconds after it is shot outfrom the nozzle. The function is defined by the equation .

How many seconds until the drop of water hits the ground?

1. Write an equation that we could solve to answer the question.

2. Try to solve the equation by writing the expression in factored form and using the zeroproduct property.

3. Try to solve the equation by graphing the function using graphing technology. Explainhow you found the solution.


Student Response

1. (or equivalent)

2. The expression cannot be written in factored form.

3. About 2 seconds. Sample reasoning: The graph shows two horizontal intercepts, one with apositive -coordinate and one negative -coordinate. The negative one does not apply here.The other horizontal intercept is around .

Activity Synthesis

Ask students to share some challenges they came across when trying to rewrite the expressions infactored form. Solicit some ideas about why this equation presented those challenges. Then,discuss how they found or estimated the solution by graphing.

The approximate solution to the equation, given by the zero of the function and the -intercept ofthe graph, is 2.087 seconds. Some graphing tools would give an approximation with a longerdecimal expansion, giving a clue that it might be trickier to rewrite the equation in factored form.After all, when finding factors, we usually look for integers. (Some quadratic expressions containingnon-integer rational numbers can still be written in factored form. For example, can be

written as .)

Highlight that some equations are difficult or impossible to rewrite in factored form. In fact, whenquadratic models appear in real life, this is usually the case. Graphing is a way to solve theseequations, but there are other techniques, which students will learn over the next several lessons.

10.4 Making It SimplerOptional: 25 minutesThis activity is optional. It allows students to investigate another way (beside guessing and checking)for finding the factors of quadratic expressions in standard form where the leading coefficient isnot 1. The method involves temporarily substituting the squared term with another variable so thatthe leading coefficient is 1 (which makes it easier to spot the factors of the expression), and thensubstituting the original variable back once the expression is in factored form.

Students may find the reasoning and substitution processes challenging. Transitioning from, forexample, to , and then to requires abstractreasoning. When the leading coefficient is not a square number (as in the second half of theactivity), an additional step of multiplying is needed to make the leading coefficient a squarenumber. To keep the value of an expression unchanged, students need to remember that they canonly multiply the expression by 1, but the 1 can be obtained by, say, multiplying by 5 and then by .

Before offering support, allow students to make sense of these processes and to discuss with theirpartner. The work encourages students to persevere in sense making and problem solving (MP1)and to make use of structure (MP7). It also prompts students to attend to precision (MP6). As they


make symbolic substitutions, students need to be clear about what the variables stand for and howthe substitutions transform the expressions.

Later in the unit, students will encounter the use of a placeholder, such as done here, as a way toderive the quadratic formula. If desired, this activity can be done at that point to warm students upto the idea of using a placeholder.

If time is limited and if desired, the activity could be split into two halves and done separately (overtwo class periods).

Addressing

HSA-SSE.A.2



Launch

Display these expressions for all to see and ask students which expressions would be easier towrite in factored form and why.

Students are likely to say that the second and the third expressions are easier because thecoefficient of the squared term in each of those is 1 (or there isn’t another number that needs to befactored aside from the constant term). Tell students that they will study another strategy that cansimplify the process of rewriting quadratic expressions into factored form.

Consider arranging students in groups of 2 and asking them to think quietly about at least the firstcouple of problems before discussing with their partner. After students have had a chance to makesense of the first worked example, pause for a class discussion. Make sure that students can followwhat is happening in the shown steps before trying to apply it with new expressions.

Pause for another class discussion after students have analyzed the second worked example.Before students proceed to the last question, clarify what is happening in each step of the rewritingprocess when the leading coefficient is not a square number.


In the final steps of the last question, students multiply a number by a pair of factors, for example,. Some students may mistakenly apply the distributive property and multiply

to both and . Remind students that the distributive property governs

•

•


multiplication over addition and subtraction, and that and are being multipliedtogether, not added or subtracted.


Here is a clever way to think about quadraticexpressions that would make it easier torewrite them in factored form.

1. Use the distributive property to expand . Show your reasoning andwrite the resulting expression in standard form. Is it equivalent to ?

2. Study the method and make sense of what was done in each step. Make a note of yourthinking and be prepared to explain it.

3. Try the method to write each of these expressions in factored form.

4. You have probably noticed that the coefficient of the squared term in all of the previousexamples is a perfect square. What if that coefficient is not a perfect square?

Here is an example of an expression whosesquared term has a coefficient that is not asquared term.

Use the distributive property to expand . Show your reasoning and writethe resulting expression in standard form. Is it equivalent to ?

5. Study the method and make sense of what was done in each step and why. Make a noteof your thinking and be prepared to explain it.

6. Try the method to write each of these expressions in factored form.


Student Response

1. Yes.

2. Sample reasoning: and have as their common factors. and. If is used to stand in for , the expression becomes simpler. The squared term

has a coefficient of 1, and factors can be found more easily: . Because standsfor , re-substituting the for gives .

3.

4. Yes.

5. Sample reasoning: Multiplying the expression by 5 allows the coefficient of to be a squarenumber. Multiplying by keeps value of the expression unchanged. and have as

a common factor and can be written as and . If we use to stand in for , theexpression becomes simpler. Once the factors are found (using ), the can take the placeof to show the factored form of the original expression.

6.

Activity Synthesis

Invite students to share their attempts to rewrite the expressions using the method they justlearned. Discuss questions such as:

“Do you think this method is simpler or harder than guessing and checking? Is it quicker orslower?”

“In what ways is this method simpler? In what ways is it harder?”


Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-classdiscussion. After each student shares, provide the class with the following sentence frames tohelp them clarify their understanding of the speaker’s ideas and to press for further details:“How did you get . . . ?”, “Why did you . . . ?”, “How do you know . . . ?” If necessary, revoicestudent ideas to demonstrate mathematical language use by restating a statement as aquestion in order to clarify, apply appropriate language, and involve more students.Design Principle(s): Support sense-making

◦◦

◦◦

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Lesson SynthesisEmphasize the point that rewriting quadratic expressions in factored form is not always an efficientor effective way to find solutions to quadratic equations. If time permits, consider generalizing therelationship between and its factored form to help students see why it can be a rathertricky process:

We need to find two factors of . Let’s call them and .

We need to find two factors of . Let’s call them and .

The factored form can be written as .

To see what , , , and should be, we essentially have to apply the distributive property tothe factors to get: or .

We see that must multiply to make and must multiply to make .

The challenge now is to find the right combination of and (factors of ) and and(factors of ) such that is equal to .

Highlight that for some quadratic expressions, the right pairs of factors might be easily spotted, butfor others, the process of guessing and checking can get pretty cumbersome, especially if andhave many pairs of factors. There is also no guarantee that we will find a combination that worksbecause some expressions do not have rational solutions (so we cannot find factors with rationalnumbers). This means that if we rely on putting an expression in factored form to solve anequation, we may get stuck. If we rely on graphing, the solutions may not be exact. We needanother way!

Tell students that in upcoming lessons we will look at other techniques that allow us to solvewithout rewriting expressions in factored form.

10.5 How Would You Solve This Equation?Cool Down: 5 minutesAddressing

HSA-REI.B.4.b

Launch

Provide access to graphing technology if requested.


Solve the equation by any method. Explain your reasoning.

Student Response

(or 2.5) and 1. Sample reasoning: Rewriting the expression in factored form gives

. Using the zero product property: and , so and .

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••

•


Student Lesson SummaryOnly some quadratic equations in the form of can be solved by rewritingthe quadratic expression into factored form and using the zero product property. In somecases, finding the right factors of the quadratic expression is quite difficult.

For example, what is the factored form of ?

We know that it could be , or , but will the secondnumber in each factor be -5 and 7, 5 and -7, 35 and -1, or -35 and 1? And in which order?

We have to do some guessing and checking before finding the equivalent expression thatwould allow us to solve the equation .

Once we find the right factors, we can proceedto solving using the zero product property, asshown here:

What is even trickier is that most quadratic expressions can’t be written in factored form!

Let’s take for example. Can you find two numbers that multiply to make -3 andadd up to -4? Nope! At least not easy-to-find rational numbers.

We can graph the function defined by using technology, which reveals two-intercepts, at around and . These give the approximate zeros of the

function, -0.646 and 4.646, so they are also approximate solutions to .

The fact that the zeros of this function don’tseem to be simple rational numbers is a cluethat it may not be possible to easily rewritethe expression in factored form.

It turns out that rewriting quadraticexpressions in factored form and using thezero product property is a very limited tool forsolving quadratic equations.

In the next several lessons, we will learn some ways to solve quadratic equations that workfor any equation.



StatementTo write in factored form, Diego first listed pairs of factors of -10.

a. Use what Diego started to complete the rewriting.

b. How did you know you’ve found the right pair ofexpressions? What did you look for when trying outdifferent possibilities?

Solutiona.

b. Sample response: I found the right pair of expressions when I multiplied them together andfound . I had to make it so that one number times and the other numbertimes added up to .

Problem 2StatementTo rewrite in factored form, Jada listed some pairs of factors of :

Use what Jada started to rewrite in factoredform.

Solution

Problem 3StatementRewrite each quadratic expression in factored form. Then, use the zero product property tosolve the equation.

a.

b.

c.


Solutiona. or

b. or

c. or

Problem 4StatementHan is solving the equation .

Here is his work:

Describe Han’s mistake. Then, find the correct solutions to the equation.

SolutionSample response: doesn’t make the factor equal zero. The correct solutions are

and .

Problem 5StatementA picture is 10 inches wide by 15 inches long. The area of the picture, including a frame that is

inch thick, can be modeled by the function .

a. Use function notation to write a statement that means: the area of the picture, includinga frame that is 2 inches thick, is 266 square inches.

b. What is the total area if the picture has a frame that is 4 inches thick?

Solutiona.

b. 414 square inches


Problem 6StatementTo solve the equation , Elena uses technology to graph the function

. She finds that the graph crosses the -axis at and .


a. What is the name for the points where the graph of a function crosses the -axis?

b. Use a calculator to compute and .

c. Explain why 1.919 and 5.081 are approximate solutions to the equationand are not exact solutions.

Solutiona. Horizontal intercepts or -intercepts

b. and

c. Sample response: A graph can estimate values but cannot always give exact answers. Thecalculator displays a rounded value of each of those points. The equation would have to besolved to find the exact solutions.


Problem 7StatementWhich equation shows a next step in solving that will lead to the correctsolutions?

A.

B.

C.

D.


Problem 8StatementHere is a description of the temperature at a certain location yesterday.

“It started out cool in the morning, but then the temperature increased until noon. It stayedthe same for a while, until it suddenly dropped quickly! It got colder than it was in themorning, and after that, it was cold for the rest of the day.”

Sketch a graph of the temperature as a function of time.


SolutionThe graph should show an increase until noon,then a constant temperature, then a steep drop,and then flat or nearly flat temperature below theinitial temperature.

Sample graph:


Problem 9StatementTechnology required. The number of people, , who watch a weekly TV show is modeled by theequation , where is the number of weeks since the show first aired.

a. How many people watched the show the first time it aired? Explain how you know.

b. Use technology to graph the equation.

c. On which week does the show first get an audience of more than 500,000 people?

Solutiona. 100,000. When , the value of is 100,000.

b. See graph.

c. On the 17th week since its premier



Lesson 11: What are Perfect Squares?

GoalsComprehend that equations containing a perfect-square expression on both sides of the equalsign can be solved by finding square roots.

Comprehend that perfect squares of the form are equivalent to .

Use the structure of expressions to identify them as perfect squares.

Learning TargetsI can recognize perfect-square expressions written in different forms.

I can recognize quadratic equations that have a perfect-square expression and solve theequations.

Lesson NarrativeThis lesson has two key aims. The first aim is to familiarize students with the structure ofperfect-square expressions. Students analyze various examples of perfect squares. They apply thedistributive property repeatedly to expand perfect-square expressions given in factored form(MP8). The repeated reasoning allows them to generalize expressions of the form asequivalent to .

The second aim is to help students see that perfect squares can be handy for solving equationsbecause we can find their square roots.

Achieving these aims prepares students to solve quadratic equations by completing the square inupcoming lessons. Knowing that quadratic equations can be much more easily solved when oneside is a perfect square and the other side is a number motivates students to transformexpressions into that form. Recognizing the structure of a perfect square equips students to lookfor features that are necessary to complete a square (MP7).

Alignments

Addressing



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••

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Building Towards

HSA-REI.B.4.a: Use the method of completing the square to transform any quadratic equationin into an equation of the form that has the same solutions. Derive thequadratic formula from this form.





Think Pair Share


Let’s see how perfect squares make some equations easier to solve.

11.1 The Thing We Are SquaringWarm Up: 5 minutesIn this warm-up, students reason about equations with quadratic expressions on both sides of theequal sign. They look for and use structure to solve the equations (MP7). Though each equationappears to be more complicated or to have more pieces than the preceding one, the underlyingstructure of all the equations is unchanged: . The last equation is written infactored form, but because the factors are identical, students can see that it can also be written as

. Viewing a complicated expression as a perfect square prepares students toconsider completing the square later.

Building Towards

HSA-REI.B.4.a

HSA-REI.B.4.b

Launch

Give students a moment to observe the list of equations and ask them what they notice andwonder about the equations. Solicit a few observations and questions. Tell students that their job isto think about what should be so that each equation is always true, regardless of what is.


In each equation, what expression could be substituted for so the equation is true for allvalues of ?

1.

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•

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Unit 7 Lesson 11: What are Perfect Squares? 189

2.

3.

4.

5.

6.

7.

Student Response

The opposite of each expression is also a possible response.

1.

2.

3.

4.

5.

6.

Activity Synthesis

Invite students to share their responses and how they viewed the equations. If not mentioned bystudents, point out that the question can be thought of as: “When is squared, it is equal tosomething squared,” so must be equal to that something.

Explain to students that expressions that represent something squared are known as perfectsquares. This term is related to what students learned in earlier grades about the area of a square.Multiplying the side length of a square by itself gives the area of the square, so we can say, forinstance, that 25 is a perfect square because it is the area (in square units) of a square whose sidelength is 5 units. can be thought of as the area of a square with side length .

11.2 Perfect Squares in Different Forms15 minutesThe goal of this activity is to illustrate that a perfect-square expression can take different forms,some of which may not look like or . The work here promptsstudents to recognize structure in perfect-square expressions, particularly when written in standardform, preparing them to complete the square in an upcoming lesson.

Students are given a series of expressions that are clearly perfect squares, for example or, and asked to rewrite them in standard form. As they repeatedly apply the

distributive property to expand these expressions into standard form, students begin to recognize a


pattern in how the two forms of expressions are related (MP8). They see that squaring anexpression such as produces an expression in standard form in which the constant term is

and the linear term is . Then, they use that insight to articulate why certain expressions instandard form (such as ) can be described as perfect squares.

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.a

HSA-REI.B.4.b



Launch


Representation: Internalize Comprehension. Activate or supply background knowledge. Representthe same information through different modalities by using a diagram to write each expressionin expanded form. As students progress through the questions, invite them to notice how eachside of their diagrams will be same since the expressions are perfect squares. If studentsstruggle on the last question, ask them to observe patterns of the entries in their diagrams(such as the lower left and upper right cells are the same) to help them diagram “in reverse,”starting from the standard form.Supports accessibility for: Conceptual processing; Visual-spatial processing


If students have trouble generalizing as from working only with expressions,ask them to draw a rectangular diagram showing and along both sides of the rectangle and seeif they can show on the diagram where the and comes from. (If some scaffolding is needed,consider starting with numbers, for example, , then and then .)

•

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•



1. Each expression is written as the product of factors. Write an equivalent expression instandard form.

a.

b.

c.

d.

e.

f.

2. Why do you think the following expressions can be described as perfect squares?

Student Response

1. a.

b.

c.

d.

e.

f.

2. Sample response: If these expressions are written in factored form, each one is the square ofa linear expression: , , and .


Write each expression in factored form.

1.

2.

3.

Student Response

1.

2.


3.

Activity Synthesis

Invite students to share their responses and reasoning for the last question. Make sure studentssee the structure that relates the expression in standard form and its equivalent expression infactored form. Highlight that:

In each given example, the constant term is a number squared ( ) and the coefficient of thelinear term is twice that number ( ).

This is also true for the expression that contains fractions: is and is .

We call these expressions “perfect squares” because they can be written as somethingsquared: , , and .

In general, when is squared and expanded, we have: .

Tell students that knowing about quadratic expressions that are perfect squares can help us solveall kinds of equations in upcoming lessons.


Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share theirexplanations for the last question, present an incorrect answer and explanation. For example,“Some people may say they’re perfect squares because they have a term that is a square.” Askstudents to identify the error, critique the reasoning, and write a correct explanation. Asstudents discuss with a partner, listen for students who identify and clarify the ambiguouslanguage in the statement. For example, the author probably meant to say that eachexpression can be written as a squared linear expression. This will help students understandthe language of “perfect squares” when referring to quadratic expressions and the relationshipbetween the coefficients in standard and factored form.Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

11.3 Two Methods15 minutesEarlier in the unit, students solved equations with a squared variable on one side and a squarenumber on the other, for example, or . This activity allows students to see thatmore complicated quadratic equations can also be solved relatively easily when both sides of theequal sign are perfect squares. The work here motivates upcoming lessons on completing thesquare.

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•


Addressing

HSA-REI.B.4.b

Building Towards

HSA-REI.B.4.a



Think Pair Share

Launch

Arrange students in groups of 2. Display the two solutions from the task statement for all to see.Give students a moment to study and make sense of the two methods. Then, ask them to talk totheir partner about what Han and Jada did at each step in their solution.

Next, ask students which method they prefer and why. They are likely to prefer Jada’s method sinceit takes far fewer steps, but some might prefer Han’s method because it is more familiar. Tellpartners that they will take turns solving equations with each method and get a better feel for eachmethod.


Conversing: MLR8 Discussion Supports. In their groups of two, display the following sentenceframes for students to use when describing their steps to solve each quadratic equation andquestioning each others’ reasoning: “First, I _____ because . . .”, “I know _____ because . . .”, “Whydid you . . .?”, and “How do you know . . . ?” Encourage students to challenge each other whenthey disagree. This will help students clarify their reasoning when solving quadratic equationsusing multiple methods.Design Principle(s): Support sense-making; Maximize meta-awareness


Han and Jada solved the same equation with different methods. Here they are:

•

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Han’s method: Jada’s method:

Work with a partner to solve these equations. For each equation, one partner solves withHan’s method, and the other partner solves with Jada’s method. Make sure both partners getthe same solutions to the same equation. If not, work together to find your mistakes.

Student Response

and

and

and

and

Activity Synthesis

Select students to display their solutions for all to see. Ask students to reflect on the merits of thesolution methods. Make sure they recognize that when equations are in perfect squares they areeasier to solve because we can find their square roots.

Point out that all the equations in this activity already have perfect squares on both sides, but mostequations that we need to solve do not. Tell students that in upcoming lessons, they will learn howto transform equations so that they have a perfect square and can be more easily solved.


Representation: Internalize Comprehension. Use color coding and annotations to highlightconnections between representations in a problem. For example, use the same color to callstudents' attention to the common elements of each method.Supports accessibility for: Visual-spatial processing; Conceptual processing

••••


Lesson SynthesisReinforce the key points of this lesson by displaying some quadratic expressions such as:

Discuss questions such as:

“Which of these quadratic expressions are perfect squares? Which are not?” (The second andfourth expressions are perfect squares. The other two are not.)

“How can you tell?” (When in standard form, an expression with perfect squares has aconstant term that is a number squared and a linear term that is twice that number. In

, we can see 144 is 12 squared, and 24 is twice 12. In , the term400 is -20 squared and -40 is twice -20.)

“Why isn’t the third equation a perfect square? Isn’t 16 a perfect square?” (16 is 4 squared or -4squared, but for the expression to be a perfect square, the coefficient of the linear termshould be 8 (twice 4) or -8 (twice 4).)

“Suppose we have two equations: and . Which is easierto solve? Why?” (The second one, because the equation has a perfect square on each side. Itcan be rewritten as . There are two numbers that can be squared to make 9,which are 3 and -3, so or .)

“Can’t we simply rewrite the left side of in factored form and solve?”(Rewriting the left side gives , but now we are stuck. We can’t use the zeroproduct property because the expression does not equal 0.)

11.4 A Perfect SquareCool Down: 5 minutesAddressing

HSA-REI.B.4.b

HSA-SSE.A.2


1. Explain why it makes sense to call the expression a “perfect square.”

2. Solve .

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Student Response

1. Sample response: It is equivalent to , so it is a factor squared.

2. -1 and -19

Student Lesson SummaryThese are some examples of perfect squares:

49, because 49 is or .

, because it is equivalent to or .

, because it is equivalent to .

, because it is equivalent to or .

A perfect square is an expression that is something times itself. Usually we are interested insituations in which the something is a rational number or an expression with rationalcoefficients.

When expressions that are perfect squares are written in factored form and standard form,there is a predictable pattern.

is equivalent to .

is equivalent to .

is equivalent to .

In general, is equivalent to .

Quadratic equations that are in the form can be solved ina straightforward manner. Here is an example:

The equation now says: squaring gives 25 as a result. This means must be 5 or-5.

Glossaryperfect square

rational number

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StatementSelect all the expressions that are perfect squares.

A.

B.

C.

D.

E.

F.

G.

Solution["A", "B", "D", "G"]

Problem 2StatementEach diagram represents the square of an expression or a perfect square.

a. Complete the cells in the last table.

b. How are the contents of the three diagrams alike? This diagram represents. Describe your observations about cells 1, 2, 3, and 4.


term_1 term_2

term_1 cell 1 cell 2

term_2 cell 3 cell 4

c. Rewrite the perfect-square expressions , , and in standard

form: .

d. How are the , , and of a perfect square in standard form related to the two termsin ?

Solutiona. See diagram.

b. Cell 1 contains the square of the first term. Cell 2 and cell 3 eachcontain the product of the two terms. Cell 4 contains the square ofthe second term.

c. , , and .

d. The is the square of one term, is the square of the other term,and is twice the product of the two terms.


a.

b.

c.

d.

e.

Solutiona. or

b. or


c.

d. or

e. or

Problem 4StatementExplain or show why the product of a sum and a difference, such as , has nolinear term when written in standard form.

SolutionSample response: When the factors are expanded, two of the partial products—the linearterms—are opposites, so their sum is 0. For , the product will be .The partial products and are opposites, so the product is .


Problem 5StatementTo solve the equation , Han first expanded the squared expression. Here is hisincomplete work:

a. Complete Han’s work and solve the equation.

b. Jada saw the equation and thought, “There are two numbers, 2 and -2, thatequal 4 when squared. This means is either 2 or it is -2. I can find the values offrom there.”

Use Jada’s reasoning to solve the equation.

c. Can Jada use her reasoning to solve ? Explain your reasoning.


Solutiona. Han’s strategy: Jada’s strategy:

b. See Jada's strategy.

c. Sample response: No, because is not a perfect square.

Problem 6StatementA jar full of marbles is displayed. The following table shows the guesses for 10 people. Theactual number of marbles in the jar is 145. Calculate the absolute guessing error for all 10guesses.

guess 190 150 125 133 167 160 148 200 170 115

absolute guessingerror

Solution

guess 190 150 125 133 167 160 148 200 170 115

absolute guessingerror

45 5 20 12 22 15 3 55 25 30



Lesson 12: Completing the Square (Part 1)

GoalsComprehend that to “complete the square” is to determine the value of that will make theexpression a perfect square.

Describe (orally and in writing) how to complete the square.

Solve quadratic equations of the form by rearranging terms and completingthe square.

Learning TargetsI can explain what it means to “complete the square” and describe how to do it.

I can solve quadratic equations by completing the square and finding square roots.

Lesson NarrativePreviously, students saw that a squared expression of the form is equivalent to

. This means that, when written in standard form (where is 1), isequal to and is equal to . Here, students begin to reason the other way around. Theyrecognize that if is a perfect square, then the value being squared to get is half of , or

. Students use this insight to build perfect squares, which they then use to solve quadratic

equations.

Students learn that if we rearrange and rewrite the expression on one side of a quadratic equationto be a perfect square, that is, if we complete the square, we can find the solutions of the equation.

Rearranging parts of an equation strategically so that it can be solved requires students to makeuse of structure (MP7). Maintaining the equality of an equation while transforming it promptsstudents to attend to precision (MP6).

Alignments

Addressing




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.

Building Towards






Think Pair Share


Let’s learn a new method for solving quadratic equations.

12.1 Perfect or Imperfect?Warm Up: 5 minutesThis activity reinforces the meaning of perfect squares and the fact that a perfect square canappear in different forms. To recognize a perfect square, students need to look for and make use ofstructure (MP7).

Addressing

HSA-SSE.A

HSA-SSE.A.2


Select all expressions that are perfect squares. Explain how you know.

1.

2.

3.

4.

5.

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Unit 7 Lesson 12: Completing the Square (Part 1) 203

6.

Student Response

1, 3, and 5, because each of them is the product of an expression and itself. isequivalent to .

Activity Synthesis

Display the expressions for all to see. Invite students to share their responses and record them forall to see. For each expression that they consider a perfect square, ask them to explain how theyknow. For expressions that students believe aren’t perfect squares, ask them to explain why not.

For the last expression, , students may reason that it is not a perfect square because:

The constant term 20 is not a perfect square. (Most students are likely reason this way.)

If 20 is seen as , then the coefficient of the linear term would have to be , not 10.

Though students have been dealing mostly with rational numbers, the second line of reasoning isalso valid and acceptable.

12.2 Building Perfect Squares10 minutesIn this activity, students begin to complete the square. They start by transforming given perfectsquares from standard form to factored form, and vice versa. Then, they are given an incompleteexpression in standard form that contains only the squared term and linear term. Students need todecide what constant term makes the expression a perfect square, and then write the equivalentexpression in factored form. To accomplish these tasks, students must rely on the structure theynoticed in an earlier lesson about the relationship between the standard and factored forms ofperfect squares (MP7).

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.a

HSA-REI.B.4.b



Think Pair Share

••

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••


Launch

Arrange students in groups of 2. Give students a few minutes of quiet think time and then ask themto discuss their responses with their partner. Follow with a whole-class discussion.


Conversing: MLR2 Collect and Display. Listen for and collect vocabulary, gestures, and diagramsstudents use to explain how to rewrite perfect squares from standard form to factored form,and vice versa. Write the students’ words on a visual display and update it throughout theremainder of the lesson. Refer back to the visual display during the activity synthesis toreiterate key ideas. Remind students to borrow language from the display as needed. This willhelp students read and use mathematical language during their partner and whole-classdiscussions.Design Principle(s): Maximize meta-awareness; Support sense-making


If students have trouble determining the constant term in standard form, suggest that they draw arectangular diagram and work backward to determine the factors along the two sides of therectangle. Afterward, they can find the corresponding value of the constant term.

For example, for the expressionwe would write:

Which leads to the the completed diagram:

This diagram represents the expression (in standard form) and the expression(in factored form).


Complete the table so that each row has equivalent expressions that are perfect squares.


standard form factored form

1.

2.

3.

4.

5.

6.

7.

Student Response


1.

2.

3.

4.

5.

6.

7.

Activity Synthesis

Display the incomplete table for all to see. Ask students to complete the missing values orexpressions.

Discuss how students knew what numbers or expressions to write in the last four pairs ofexpressions. Make sure students understand that:

If the constant term in the factored form is , the constant term in standard form is , asquared number.

The coefficient of the linear term in standard form is twice of , or .

•

•


This means that is half of the coefficient of the linear term. If the linear term is , the is10, and the constant term in standard form is or 100.

If the linear term in standard form is (as in the last row of the table), then the constant term

in standard form is , and the constant term in factored form is .

Explain to students that finding the constant term to add in order to create a perfect square iscalled “completing the square.” In the next activity, we will look at completing the square as astrategy to solve quadratic equations.


Representation: Internalize Comprehension. Use color and annotations to illustrate studentthinking. As students share their conclusions about the relationship between the terms indifferent forms of perfect squares, scribe their thinking on a visible display. Choose oneexample from the table. Highlight the coefficient of the linear term in the standard formexpression, labeling it , then highlight the constant term in the standard form expression and

write . Finally, write next to the constant term in the factored form expression and

highlight it in the same color. Encourage students to select and annotate another examplefrom the table on their own to check for understanding.Supports accessibility for: Visual-spatial processing; Conceptual processing

12.3 Dipping Our Toes in Completing the Square20 minutesEarlier, students recognized that an expression in standard form can be written as a perfect squareand learned to write it that way. Here, they learn to use that skill to solve quadratic equations.

Addressing

HSA-REI.B.4.a

HSA-REI.B.4.b



Launch

Consider keeping students in groups of 2.

Display for all to see the two solution methods in the activity statement. Ask students if thequadratic expression in the original equation is a perfect square. Then, ask them to study themethods and make sense of the steps. Afterward, discuss with students:

•

•

••

•


“How are the two solution methods alike?” (They both involve making the expression on theleft a perfect square. They are the steps for solving the same equation, so the solutions are thesame.)

“How they are different?” (In the first solution, Diego first subtracted 9 from each side, andthen added 25 to complete the square. In the second, Mai added 16 to 9 because that is whatis needed to make 25.)

Tell students that either method works, but some people prefer the first approach because movingthe original constant term to the other side of the equal sign (the right side, in this case) allowsthem to see what constant term is needed to make a perfect square on the left side. They also findit to be less prone to errors.


Speaking: MLR8 Discussion Supports. To support students in producing statements about howfeatures of the strategies are alike and how they are different, provide sentence frames forstudents to use when they are comparing and contrasting. For example, “_____ and _____ arealike because . . .”, “_____ and _____ are different because . . .”, “One thing that is the same is . . .”,“One thing that is different is . . . .”Design Principle(s): Support sense-making


Action and Expression: Internalize Executive Functions. Provide students with a two-column graphicorganizer to record their ideas as they compare and contrast the two solution methods. Labelthe left column “alike” and the right column “different.” Encourage students to use theorganizer to take notes and then prepare their ideas to share with the whole class. If studentsare unsure how to start, tell them to number the steps in the examples and compare each stepindividually, placing it in the correct spot on the diagram.Supports accessibility for: Language; Organization


One technique for solving quadratic equations is called completing the square. Here are twoexamples of how Diego and Mai completed the square to solve the same equation.

•

•


Diego: Mai:

Study the worked examples. Then, try solving these equations by completing the square:

1.

2.

3.

4.

5.

Student Response

1. -4 and -2

2. -13 and 1

3. 3 and 7

4. -8 and 10

5. 4 and 10


Here is a diagram made of a square and two congruent rectangles. Itstotal area is square units.

1. What is the length of the unlabeled side of each of the two rectangles?

2. If we add lines to make the figure a square, what is the area of the entire figure?

3. How is the process of finding the area of the entire figure like the process of buildingperfect squares for expressions like ?


Student Response

1. 17.5 units

2.

3. Sample response: First we had to figure out what number was half of 35 (which is 17.5) to findthe side length of the rectangles. Then, we had to square the number 17.5 to find the area ofthe missing piece.

Activity Synthesis

Select students to share their solutions and to display their reasoning for all to see. After eachstudent presents, ask if others found the same solution but completed the square in a differentway. Make sure students see that the steps could vary, but the solutions should be the same ifequality between the two sides of the equal sign is maintained throughout the solving process.

Ask students how they could check their solutions. One way is by substituting the solutions backinto the equation and seeing if the equation is true at those values of the variable. For example, tosee if -4 and -2 are the right solutions to the equation , we can evaluate

and and see if each has a value of 0. Both andgive 0, so the solutions are correct.

Lesson SynthesisTo close the lesson, help students connect the new method to earlier skills. Discuss questions suchas:

“Earlier we solved by completing the square. Could we have solved byrewriting it in factored form and using the zero product property? Why or why not?” (Yes, theequation can be rewritten as and solved easily.)

“Is solving that equation by completing the square a quicker way?” (In this case, no.)

“Can be solved by rewriting it in factored form?” (It can be written infactored form as , but that’s as far as we could go.)

“Can it be solved by completing the square?” (Yes. To make it a perfect square we can add 16to each side, which makes the equation or , which can besolved by finding square roots of 36.)

“When might we prefer to complete the square than to rewrite an equation in factored form?”(Not all equations can be written in factored form so it is not always possible to solve that way.In those cases, we can solve by completing the square.)

Tell students that in upcoming lessons, we will look at other examples of equations that cannot beeasily rewritten in factored form but can be solved by completing the square.

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••

•

•


12.4 Make It a Perfect SquareCool Down: 5 minutesAddressing

HSA-REI.B.4.a

HSA-REI.B.4.b


1. What could be added to each expression to make it a perfect square?

a.

b.

c.

2. Solve the equation by completing the square. Show your reasoning.

Student Response

1. a. 36

b. 8

c. 59

2. 6 and 10. Sample reasoning:

Student Lesson SummaryTurning an expression into a perfect square can be a good way to solve a quadratic equation.Suppose we wanted to solve .

The expression on the left, , is not a perfect square, but is aperfect square. Let’s transform that side of the equation into a perfect square (while keepingthe equality of the two sides).

••


One helpful way to start is by firstmoving the constant that is not a perfectsquare out of the way. Let’s subtract 10from each side:

And then add 49 to each side:

The left side is now a perfect squarebecause it’s equivalent toor . Let’s rewrite it:

If a number squared is 9, the number hasto be 3 or -3. To finish up:

This method of solving quadratic equations is called completing the square. In general,

perfect squares in standard form look like , so to complete the square, take

half of the coefficient of the linear term and square it.

In the example, half of -14 is -7, and is 49. We wanted to make the left sideTo keep the equation true and maintain equality of the two sides of the

equation, we added 49 to each side.

Glossarycompleting the square


StatementAdd the number that would make the expression a perfect square. Next, write an equivalentexpression in factored form.

a.

b.

c.

d.

e.

Solutiona. and

•

•

•

•

•


b. and

c. and

d. and

e. and

Problem 2StatementMai is solving the equation . She writes:

Jada looks at Mai’s work and is confused. Shedoesn’t see how Mai got her answer.

Complete Mai’s missing steps to help Jada seehow Mai solved the equation.

Solution

Problem 3StatementMatch each equation to an equivalent equation with a perfect square on one side.

A.

B.

C.

D.

E.

F.

1.

2.

3.

4.

5.

6.

Solution


A: 4

B: 2

C: 1

D: 5

E: 6

F: 3

Problem 4StatementSolve each equation by completing the square.

SolutionThe solutions to are and .

The solutions to are and .

The solutions to are and .

The solution to is .

Problem 5StatementRewrite each expression in standard form.

a.

b.

c.

d.

Solutiona.

◦◦◦◦


b.

c.

d.


Problem 6StatementTo find the product without a calculator, Priya wrote . Veryquickly, and without writing anything else, she arrived at 39,991. Explain how writing the twofactors as a sum and a difference may have helped Priya.

Solution. Both and are simple to compute. ,

which is equal to 39,991.


Problem 7StatementA basketball is dropped from the roof of a building and itsheight in feet is modeled by the function .

Here is a graph representing .

Select all the true statements about this situation.

A. When the height is 0 feet.

B. The basketball falls at a constant speed.

C. The expression that defines is linear.

D. The expression that defines is quadratic.

E. When the ball is about 50 feet above the ground.

F. The basketball lands on the ground about 1.75 seconds after it is dropped.


Solution["D", "E", "F"](From Unit 6, Lesson 5.)

Problem 8StatementA group of students are guessing the numberof paper clips in a small box.

The guesses and the guessing errors areplotted on a coordinate plane.

What is the actual number of paper clips in thebox?

Solution22




GoalsExpress any quadratic equation in the form and solve the equation by findingsquare roots.

Generalize (orally) a process for completing the square to express any quadratic equation inthe form .

Solve quadratic equations in which the squared term has a coefficient of 1 by completing thesquare.

Learning TargetsWhen given a quadratic equation in which the coefficient of the squared term is 1, I can solve itby completing the square.

Lesson NarrativeIn this lesson, students learn that completing the square can be used to solve any quadraticequation, including equations that involve rational numbers that are not integers. Students noticethat the process of completing the square is the same when the equations involve messiernumbers as when they have simple integers, but the calculations may be more time consuming andprone to error. An error-analysis activity highlights some common errors related to completing thesquare.

Although any equation can be solved by completing the square, equations that are really difficult tosolve by this method are not included here. Students will solve such equations when they haveaccess to the quadratic formula. What is important in this lesson is to recognize that putting aquadratic equation in the form of allows them to solve it, but there are cases in whichdoing so may not always be the most efficient strategy.

Completing the square for quadratic expressions that are more elaborate encourages students tolook for and make use of the same structure that helped them when they were working with lesscomplicated expressions (MP7).

Alignments

Addressing

HSA-REI.A: Understand solving equations as a process of reasoning and explain the reasoning.


•

•

•

•

••



Math Talk



Required Materials



Let’s solve some harder quadratic equations.

13.1 Math Talk: Equations with FractionsWarm Up: 5 minutesThe purpose of this Math Talk is to elicit strategies and understandings students have foroperations on fractions and solving equations. These understandings help students develop fluencyand will be helpful later in this lesson when students will need to use similar computations to solveequations.

Addressing

HSA-REI.A


Math Talk


Launch

Display one problem at a time. Give students quiet think time for each problem and ask them togive a signal when they have an answer and a strategy. Keep all problems displayed throughout thetalk. Follow with a whole-class discussion.




Solve each equation mentally.

•••

•

•

••


Student Response

(or equivalent)

Activity Synthesis








Speaking: MLR8 Discussion Supports. Display sentence frames to support students when theyexplain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ."Some students may benefit from the opportunity to rehearse what they will say with a partner

before they share with the whole class.Design Principle(s): Optimize output (for explanation)

13.2 Solving Some Harder Equations20 minutesIn this activity, students encounter equations that contain rational numbers. They learn that, eventhough the numbers are not as straightforward as in equations they have previously seen, they canstill solve the equations by completing the square. The less-friendly numbers and the moreelaborate equations make the computations more challenging, so students need to attend to

•

•

•

•

•••••


precision (MP6) and mind the steps for rearranging parts of the equations, as well as the signs ofthe numbers, especially negative numbers.

As students work, they may need to be reminded to rewrite the equation in a different form, isolatea term to make it easier to complete the square, apply the zero product property only when aproduct is equal to zero, and so on. Make note of common errors or hurdles and address themduring whole-class discussion.

Addressing

HSA-REI.B.4.b



Launch

Give students a moment to look at the list of equations and notice how they are like or unlike otherequations they have seen and solved before. Invite students to share some observations. They maynotice, for instance:

does not have 0 on one side, so we cannot use the zero product propertyright away.

and involve fractions.

has an odd number for the coefficient of the linear term.

has 0 on one side but the other side is a sum, not a product.

has decimals.

Explain to students that each equation can be solved by completing the square and using the samereasoning as when we solved simpler-looking equations earlier.

Consider arranging students in groups of 2. If there is not enough time for students to answer allthe questions, consider asking students to look at the list of equations and to choose one they thinklooks easier (or more familiar) and one that looks harder (or less familiar).


Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge.Invite students to select at least 3 of the equations to solve. Encourage students to select theproblems ahead, and persist with the problems they have chosen for the duration of the task(not reselecting if a problem becomes challenging for them). Chunking this task into moremanageable parts may also support students who benefit from additional processing time.Supports accessibility for: Organization; Attention; Social-emotional skills

•

•

•

•

•••



Some students may have a difficult time getting started given the increased complexity of theseproblems. Encourage these students to use completing the square to solve a more familiar problemfirst, such as , and then to use the structure of the process they followed (MP7) asa guide to help them solve these more challenging problems.


Solve these equations by completing the square.

1.

2.

3.

4.

5.

Student Response

1. -2 and 4

2. and

3. -2 and -1

4. 4 and 6

5. -0.7 and -0.9


1. Show that the equation is equivalent to .

2. Write an equation that is equivalent to and that includes .

3. Does this method help you find solutions to the equations? Explain your reasoning.

Student Response

1. Expanding the expression gives . Combining like terms thengives .

2. Because is equivalent to , the equation is equivalent to.

3. This method is not helpful in finding the solutions to the equations. Sample reasoning: Tosolve , the next step would be . But because the right sidecontains a variable, the usual method of solving doesn’t work.


Activity Synthesis

Select students to share their solutions and display their work for all to see. If students foundsolutions that differ from each other, as time permits, invite these students to describe theirthinking to the class, and encourage them to try to identify any errors and determine whichsolutions are correct. Discuss any common challenges or errors.

Make sure students understand that the process of solving by completing the square is the samewhether the numbers in the equations are integers or other rational numbers. Elicit from studentsa generalization of the process. For example:

Start with a quadratic expression in standard form on one side of the equal sign and a number(which could be 0) on the other side.

Take half of the coefficient of the linear term and square it. The result is the constant term thatwould make the quadratic expression a perfect square.

Perform the same operations to both sides of the equal sign so the constant term that is onone side is a perfect square that makes the expression a perfect square.

Rewrite the expression (in standard form) as a squared factor.

Find both values of the factor that, when squared, gives the number on the other side of theequal sign.


Conversing: MLR2 Collect and Display. During the whole-class discussion, listen for and collectlanguage students use to generalize the process of solving by completing the square. Writestudents’ words and phrases on a visual display, keeping track of each step from start to finish.If necessary, use a specific example to make connections to the generalized steps byhighlighting, making annotations, or drawing arrows. For example, when describing the“coefficient of the linear term” or the “perfect square” in the generalized process, highlight theactual terms in the example. Remind students to borrow language from the display and use itas a resource to draw language from during small-group and whole-class discussions.Design Principle(s): Maximize meta-awareness; Support sense-making

13.3 Spot Those Errors!10 minutesIn this activity, students analyze worked examples of equations solved by completing the square.The work allows them to further develop their understanding of the method when used withrational numbers and to notice common errors. Identifying errors in worked examples is anopportunity to attend to precision (MP6) and to analyze and critique the reasoning of others (MP3).

•

•

•

••


Addressing

HSA-REI.B.4.b



Launch

Display the four equations in the task statement for all to see:

1.

2.

3.

4.

Arrange students in groups of 3–4. Assign one equation (or more, depending on time constraints)for each group to solve by completing the square. Once group members agree on the solutions, askthem to look for errors in the worked solution for the same equation (as the one they solved).


Engagement: Develop Effort and Persistence. Chunk this task into manageable parts for studentswho benefit from support with organizational skills in problem solving. Check in with studentsafter the first 2–3 minutes of work time. Invite 1–2 students to share how they identified theerror in the first worked solution. Record their thinking on a display for all to see, and keep thework visible as students continue.Supports accessibility for: Organization; Attention


Here are four equations, followed by worked solutions of the equations. Each solution has atleast one error.

Solve one or more of these equations by completing the square.

Then, look at the worked solution of the same equation as the one you solved. Find anddescribe the error or errors in the worked solution.

1.

2.

3.

•

•

••


4.

Worked solutions (with errors):

1. 2.

3. 4.

Student Response

1. or . Error: The number being added, 28, is not a perfect square. It should be 49.

2. or . Error: The second to last step should have 3 and -3 on the right side, not 9 and-9.

3. or . Error: In the last step, only the positive square root is written and thenegative square root is neglected.

4. or . Error: In the second step, different amounts are added to each side. On the

left, is added. On the right, is added.

Activity Synthesis

Display the four solutions in the task statement for all to see. Select students to share the errorsthey spotted and their proposed corrections.

To involve more students in the discussion, after each student presents, consider asking students toclassify each error by type (not limited to one type per error) and explain their classification. Hereare some examples of types of errors:


Careless errors—for example, writing the wrong number or symbol, repeating the same steptwice, leaving out a negative sign, or not following directions.

Computational errors—for example, adding or subtracting incorrectly or putting a decimalpoint in the wrong place.

Gaps in understanding—for example, misunderstanding of the problem or the rules ofalgebra, applying an ineffective strategy, or choosing the wrong operation or step.

Lack of precision or incomplete communication—for example, missing steps or explanations,incorrect notation, or forgetting labels or parentheses.


Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make surethat everyone in their group can explain the errors they identified and proposed corrections.Invite groups to rehearse what they will say when they share with the whole class. Rehearsingprovides students with additional opportunities to speak and clarify their thinking, and willimprove the quality of explanations shared during the whole-class discussion. Make sure tovary who is selected to represent the work of the group, so that students get accustomed topreparing each other to fill that role.Design Principle(s): Support sense-making; Cultivate conversation

Lesson SynthesisTo help students generalize the process of solving quadratic equations by completing the square,ask students,

“How was the process of completing the square in this lesson different than that in an earlierlesson?” (The numbers in the equations are not as simple, so the calculations may be moreinvolved and time consuming. In some cases, more steps were needed to complete thesquare, and the process may be more prone to error.)

“How was the process of completing the square in this lesson like that in the previous lesson?”(The steps are essentially the same: turn one side into a perfect-square expression, write it asa squared factor, and reason about the value of the factor.)

“What are some errors you have seen or have made when the solving process involvesmultiple steps, especially when the numbers are tricky to work with?” (Some common errors:

Only positive solutions are shown. Negative solutions are neglected.

Different amounts are added to the two sides of the equal sign.

The wrong amount is added so the quadratic expression is not a perfect square.

Incorrect calculations or missed steps.)

•

•

•

•

•

•

•

◦◦◦◦


“When might we want to choose to solve quadratic equations by rewriting them in factoredform instead of by completing the square? When might it be preferable to complete thesquare?” (When we could easily see the factors in a quadratic equation, then it makes sense tosolve the equation by putting it in factored. When it is not obvious what the factors are, orwhen the equation involves non-integers, then completing the square may be preferable.)

Emphasize that completing the square is a handy technique that can be used to solve any quadraticequation, but depending on the numbers in the equation, it may or may not be an efficient method.

13.4 How Did We Get Those Solutions?Cool Down: 5 minutesAddressing

HSA-REI.B.4.b


The solutions to this equation are and . Show how to find those solutions by completing

the square.

Student Response

and . Sample reasoning:

Student Lesson SummaryCompleting the square can be a useful method for solving quadratic equations in cases inwhich it is not easy to rewrite an expression in factored form. For example, let’s solve thisequation:

•

•


First, we’ll add to each side to make things easier on ourselves.

To complete the square, take of the coefficient of the linear term 5, which is , and square

it, which is . Add this to each side:

Notice that is equal to 25 and rewrite it:

Since the left side is now a perfect square, let’s rewrite it:

For this equation to be true, one of these equations must true:

To finish up, we can subtract from each side of the equal sign in each equation.

It takes some practice to become proficient at completing the square, but it makes it possibleto solve many more equations than you could by methods you learned previously.



StatementAdd the number that would make the expression a perfect square. Next, write an equivalentexpression in factored form.

a.

b.

c.

d.

e.

Solutiona. and

b. and

c. and

d. and

e. and

Problem 2StatementNoah is solving the equation . He begins by rewriting the expression on theleft in factored form and writes . He does not know what to do next.

Noah knows that the solutions are and , but is not sure how to get to thesevalues from his equation.

Solve the original equation by completing the square.

Solution


Problem 3StatementAn equation and its solutions are given. Explain or show how to solve the equation bycompleting the square.

a. . The solutions are and .

b. . The solutions are and .

c. . The solutions are and .

Solution


a.

b.

c.

Solutiona. and

b. and

c. and


Problem 5StatementMatch each quadratic expression given in factored form with an equivalent expression instandard form. One expression in standard form has no match.

A.

B.

C.

D.

1.

2.

3.

4.

5.

Solution

A: 4

B: 5

C: 1

D: 3


Problem 6StatementFour students solved the equation . Their work is shown here. Only one studentsolved it correctly.

Student A: Student B:

Student C: Student D:


Determine which student solved the equation correctly. For each of the incorrect solutions,explain the mistake.

SolutionStudent A wrote two solutions but there are no numbers (that we know of at the moment) thatcan be squared to make a negative number.

Student B solved it correctly.

Student C rewrote the quadratic expression in factored form but the expression is notequivalent to the original one. The expression cannot be rewritten in factored form.

Student D subtracted 225 from the left side of the equal sign but added 225 on the right side.


◦

◦◦

◦



GoalsGeneralize (orally) a process for completing the square to express any quadratic equation inthe form .

Solve quadratic equations in which the squared term has a coefficient other than 1 bycompleting the square.

Learning TargetsI can complete the square for quadratic expressions of the form when is not 1and explain the process.

I can solve quadratic equations in which the squared term coefficient is not 1 by completingthe square.

Lesson NarrativePrior to this lesson, students have solved quadratic equations by completing the square, but all theequations were monic quadratic equations, in which the squared term has a coefficient of 1. In thislesson, students complete the square to solve non-monic quadratic equations, in which the squaredterm has a coefficient other than 1.

Students begin by noticing that the structure for expanding expressions such as can alsobe used to expand expressions such as . The expanded expression is always

. If the perfect square in standard form is , then is , is , andis . Recognizing this structure allows students to complete the square for expressions

when is not 1, and then to solve equations with such expressions (MP7).

Completing the square when is not 1 can be rather laborious, even when is a perfect square andis an even number. It is even more time consuming and complicated when is not a perfect

square and is not an even number. Students are not expected to master the skill of solvingnon-monic quadratic equations by completing the square. In fact, they should see that this methodhas its limits and seek a more efficient strategy.

This lesson aims only to show that non-monic quadratic equations can be solved by completing thesquare and exposing students to how it can be done. This exposure provides some backgroundknowledge that will be helpful when students derive the quadratic formula later.

Alignments

Addressing


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•

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•




.

Building Towards




.





Take Turns


Let’s complete the square for some more complicated expressions.

14.1 Perfect Squares in Two FormsWarm Up: 5 minutesIn this warm-up, students consider perfect squares in standard form in which the leading coefficientis not 1. They use what they know about expanding to analyze the expansion of anexpression of the form and to identify an error. In explaining and correcting the error,students practice constructing logical arguments (MP3).

Building Towards

HSA-SSE.A.2

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•

•

•

••••

•

•


Launch

Display the entire task for all to see. Give students 2 minutes of quiet think time. Select students toshare their responses and how they reasoned about the error in Elena’s statement.


Elena says, “ can be expanded into . Likewise, can be expandedinto .”

Find an error in Elena’s statement and correct the error. Show your reasoning.

Student Response

is not equivalent to . Sample reasoning: Applying the distributive property togives , which is .

Activity Synthesis

Select 1–2 students to share their responses. Make sure students see that expanding theexpression involves the same structure as expanding . Highlight the structureusing a diagram and the distributive property.

When we expand , the squared term is , the linear term is 2 times , which is , andthe constant term is . When we expand , the squared term is , the linear term is 2times , which is , and the constant term is .

Students will have opportunities to transform more of such expressions later and, with practice, willbetter see the effects of squaring a linear expression like , in which the coefficient of thevariable is not 1.

14.2 Perfect in A Different Way15 minutesIn this activity, students use the structure they saw earlier to rewrite squared expressions of theform , where is not 1, into standard form. Through repeated reasoning, students seehow the coefficient is related to the values of and when the expression is expanded into

, and how the is related to (MP8).

As students work, monitor for the following strategies for rewriting the expressions in the firstquestion (from factored form to standard form):


Drawing rectangular diagrams.

Applying the distributive property, writing out all the expanded terms, and combining liketerms.

Generalizing that when expanding an expression such as :The coefficient 4 in gets squared, so in is .

The coefficient 4 in gets multiplied by the constant term 1 twice, so is .

The constant term 1 in is squared, so is .

Identify students using each strategy and invite them to share during the class discussion.

Addressing

HSA-SSE.A.2

Building Towards

HSA-REI.B.4.a

HSA-REI.B.4.b




Launch

Explain to students that they will now practice squaring expressions such as and ,

where the coefficient of the linear term is not 1, and writing them in standard form.

Consider arranging students in groups of 2 and asking them to think quietly about each questionbefore conferring with their partner.


1. Write each expression in standard form:

a.

b.

c.

d.

e.

••

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•

••

••


2. Decide if each expression is a perfect square. If so, write an equivalent expression of theform . If not, suggest one change to turn it into a perfect square.

a.

b.

Student Response

1. a.

b.

c.

d. or

e.

2. a. Yes, a perfect square. .

b. No, not a perfect square. Sample suggestions for turning it into a perfect square:Changing the coefficient of the linear term to 20 gives , which isequivalent to .

Changing the coefficient of the linear term to -20 gives , which isequivalent to .

Changing the constant term to 4 gives , which is equivalent to.

Activity Synthesis

Select previously identified students to share their responses and strategies for the first question,starting with students who used diagrams and ending with students who generalized the pattern inthe rewriting process. As students explain, display or record their reasoning for all to see.

If not mentioned in students’ explanations, highlight the generalization noted in the activitynarrative.

Then, discuss how students used their insights from the first question to help them identify perfectsquares, or to turn expressions into perfect squares, in the second question. Make sure studentssee the structure behind the values of , , and when the expression is a perfectsquare. This structure will help them complete the square in the next activity.

▪

▪

▪



Representing, Conversing: MLR7 Compare and Connect. As students share their strategies for thefirst question, call students’ attention to the different approaches to rewriting the expressionsin standard form. Take a close look at the last two parts of the first problem and how the valueof is represented in the expression . Wherever possible, amplify studentwords and actions that describe the connections between a specific feature of onemathematical representation and a specific feature of another representation.Design Principle(s): Maximize meta-awareness; Cultivate conversation


Action and Expression: Develop Expression and Communication. To help get students started,display sentence frames that elicit observations about the connection between squared andstandard form for each expression. For example, “I notice that . . . .” and “It looks like . . . .” Afterstudents have discussed these initial observations, display frames that encourage students, intheir partner groups, to create generalizations from their observations that they can share withthe whole class. For example, “_____ will always _____ because . . . .” and “Is it always true that . .. ?”Supports accessibility for: Language; Organization

14.3 When All the Stars Align15 minutesEarlier, students noticed that when an expression is squared and written in standard form

, the value of is and the value of is . In this activity, they use theseobservations to complete expressions to make them perfect squares and then write them assquared factors. Then, they apply this skill to solve quadratic equations.

Addressing

HSA-REI.B.4.b

HSA-SSE.A.2

Launch

To reiterate the connections between and its equivalent expression of the form, display these two expressions for all to see and ask students:

“If the two expressions are perfect squares and are equivalent, how is related to ?” ( is .)

“How is related to ?” ( is .)

••

••


“How is related to and ?” ( is .)

Tell students they will use these insights to complete some squares. Consider keeping students ingroups of 2.


Representation: Internalize Comprehension. Use color and annotations to illustrate studentthinking. As students share their reasoning about the connections between and

, scribe their thinking on a visible display. Invite students to suggest language ordiagrams to include that will support their understanding. Keep the display visible as studentscontinue to work.Supports accessibility for: Visual-spatial processing; Conceptual processing


If students struggle to follow the generalized relationships between and its equivalentexpression of the form (as discussed in the launch), consider revisiting a concreteexample from an earlier activity—for example, . Ask students to relate each term in thesquared factor to the terms of its expression in standard form, .

If further scaffolding is helpful, consider using an example with an additional variable (forinstance, and its counterpart in standard form, ) before generalizingthe equivalence of the two forms in entirely abstract terms.


1. Find the value of to make each expression in the left column a perfect square instandard form. Then, write an equivalent expression in the form of squared factors. Inthe last row, write your own pair of equivalent expressions.

standard form squared factors

2. Solve each equation by completing the square:

•


Student Response

1.standard form squared factors

sample response:

2. The solutions to are and .

The solution to are and .

Activity Synthesis

Ask students to write the standard-form expression they invented on a piece of scrap paper, exceptwithout the constant term. (From the blank row of the table.) Invite them to switch papers with apartner, and complete the square and write each other’s expressions in factored form. If they getstuck, encourage them to talk with their partner to work together and fix any mistakes.

If time permits, invite students to share their responses and strategies. Discuss questions such as:

“How did you know what value to use for in, say, ?” ( , so is either10 or -10. If is 10, then and . We know that , so is or 16. If is-10,then and . Squaring -4 still gives 16.)

“How did you know whether would be positive or negative?” ( is some number squared, so itwill always be positive.)

“In the equation , the expression on the left already has 10 as a constantterm but it is not a perfect square. How do we make it a perfect square?” (25 would make it aperfect square, so we can either add 15 to each side of the equation, or first subtract 10 fromeach side and then add 25.)

As students explain their solution methods on the second question, record and display theirreasoning for all to see, or display a worked solution such as:

◦

◦

•

•

•


14.4 Putting Stars into AlignmentOptional: 30 minutesThis activity is optional. It shows three different methods for solving an equation: by rewriting it infactored form, by transforming it into an expression in which the squared term has a coefficient 1and then rewriting in factored form, and by completing the square. The second method involvestemporarily substituting a part of the expression with another variable so that the squared termhas a coefficient of 1, which makes it easier to rewrite in factored form. Students encountered thisstrategy in an optional activity (“Making It Simpler”) in an earlier lesson. If students did not do thatoptional activity, consider doing it first before using this activity.

The numbers in the equations here do not lend themselves to be easily rewritten in factored form(without guessing and checking or substitution), or to be easily rewritten into perfect squares(because the leading coefficient and the constant terms are not perfect squares), but students seethat these equations can be solved with these strategies. The activity further illustrates that any ofthese methods can be rather tedious, providing further motivation to solve using the quadraticformula, which students will learn in a future lesson.

Addressing

HSA-REI.B.4.a

HSA-REI.B.4.b



Take Turns

Launch

Display the perfect squares students saw in the preceding activity. Ask them what they notice aboutthe coefficient of the square term in each expression.

••

••


Students are likely to notice that the coefficient of every is a perfect square, which made it easierto rewrite the completed square as squared factors. Solicit some ideas about whether we could stillcomplete the square if a quadratic expression does not have a perfect square for the coefficient of

. Then, tell students that they will explore a few methods for solving such equations.

Arrange students in groups of 2. Ask one partner to study the first two methods and the otherpartner to study the third method, and then take turns explaining their understanding to eachother. Discuss the methods, especially the third one, before students begin using them to solveequations.

Make sure students see that, in both the second and third methods, the first step involvesmultiplying both sides of the equation by 3 to make the coefficient of a perfect square. Each termon the left side of the equation changes by a factor of 3, but the right side of the equation remains 0because multiplying 0 by any number results in 0.

If time is limited, consider asking students to solve only three equations, using each method once.


Action and Expression: Internalize Executive Functions. To support development of organizationalskills in problem-solving, chunk this task into more manageable parts. For example, cut thethree methods apart into separate pieces. Begin by presenting one method at a time and invitestudents to annotate the examples as they analyze the steps. After students annotate the firstmethod, invite students who have great clarity in their annotations to share their work, anddisplay for all to see. Encourage students to refer to their annotations as they approach thelast problems themselves.Supports accessibility for: Memory; Organization


Here are three methods for solving.

Try to make sense of each method.

Method 1:


Method 2: Method 3:

Once you understand the methods, use each method at least one time to solve theseequations.

1.

2.

3.

4.

5.

6.

Student Response

1. -3 and

2. and

3. and 4

4. and

5. and

6. and


Find the solutions to . Explain your reasoning.


Student Response

and . Sample reasoning: Using the techniques of this lesson, the first step is to multiply

by 3 to obtain . Another approach is to begin by dividing each side of the

equation by 3 and then completing the square.

Activity Synthesis

Consider asking students who solve the same equation using different methods to compare andcontrast their solution strategies. Then, invite them to reflect on the solving process:

“Did the numbers in certain equations make it easier to solve by one method versus theothers?”

“What are the advantages of each method? What about its drawbacks?”

“Is there a strategy you prefer or find reliable? Which one and why?”

Tell students that in upcoming lessons, they will look at a more efficient method for solvingquadratic equations.


Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-classdiscussion. Before addressing the two questions, “What are the advantages of each method?What about its drawbacks?”, give students 2–3 minutes to plan what they will say when theypresent their responses to the class. Encourage students to consider what details areimportant to share and to think about how they will explain their reasoning usingmathematical language. If needed, revoice student ideas to demonstrate mathematicallanguage use by restating a statement as a question in order to clarify, apply appropriatelanguage, and involve more students.Design Principle(s): Support sense-making

Lesson SynthesisInvite students to reflect on the process of completing the square for various kinds of equations.Display equations such as the following and ask students which ones would be fairly easy to solveby completing the square, which ones would not be, and why.

•

••


Discuss questions such as:

“What number would you add to (the third equation on the list) to make it a perfectsquare? How do you know?” (64. The coefficient 32 is equal to , so is 8. The number tobe added is , which is 64.)

“What about ? How do you know?” (1. The coefficient 6 is equal to , so is 1.The number to be added is , which is 1.)

“Do certain features or numbers in an equation make it easier or harder to solve it bycompleting the square? If so, which features or what kinds of numbers?” (It’s easier when thecoefficient of the squared term is 1 or another perfect square. It’s harder when some of thecoefficients are fractions or the quadratic term’s coefficient is not a perfect square.)

14.5 One More EquationCool Down: 5 minutesAddressing

HSA-REI.B.4.a

HSA-REI.B.4.b


1. Find the value of c to make the expression a perfect square. Then, write an equivalentexpression in the form of squared factors.

standard form squared factors

2. Solve the equation by completing the square. Show your reasoning.

Student Response

1. is 49. Standard form: . Squared factors:

2. and (or equivalent). Sample reasoning:

•

•

•

••


Student Lesson SummaryIn earlier lessons, we worked with perfect squares such as and . Welearned that their equivalent expressions in standard form follow a predictable pattern:

In general, can be written as .

If a quadratic expression is of the form , and the value of is 1, then thevalue of is , and the value of is for some value of .

In this lesson, the variable in the factors being squared had a coefficient other than 1, forexample and . Their equivalent expression in standard form alsofollowed the same pattern we saw earlier.

squared factors standard form

In general, can be written as:

or

If a quadratic expression is of the form , then:

the value of is

the value of is

the value of is

We can use this pattern to help us complete the square and solve equations when thesquared term has a coefficient other than 1—for example: .

What constant term can we add to make the expression on the left of the equal sign aperfect square? And how do we write this expression as squared factors?

16 is , so the squared factors could be .

••

•••

•


40 is equal to , so or . This means that .

If is , then or .

So the expression is a perfect square and is equivalent to .

Let’s solve the equation by completing the square!

.


StatementSelect all expressions that are perfect squares.

A.

B.

C.

D.

E.

F.

Solution["A", "C", "E", "F"]

Problem 2StatementFind the missing number that makes the expression a perfect square. Next, write theexpression in factored form.

a.

•••


b.

c.

d.

e.

Solutiona. 56 and

b. 24 and

c. 20 and

d. 18 and

e. 66 and

Problem 3StatementFind the missing number that makes the expression a perfect square. Next, write theexpression in factored form.

a.

b.

c.

d.

e.

Solutiona. 49 and

b. 4 and

c. 121 and

d. 81 and

e. 36 and


Problem 4Statement

a. Find the value of to make the expression a perfect square. Then, write an equivalentexpression in factored form.


b. Solve each equation by completing the square.

Solution

a.standard form factored form

b. Solutions to : and .

Solutions to : and

Problem 5StatementFor each function , decide if the equation has 0, 1, or 2 solutions. Explain how youknow.

A B C

▪▪


D E F

SolutionA: 2 solutions. The graph has 2 -intercepts. The function has 2 zeros.

B: 1 solution. The graph has 1 -intercept. The function has 1 zero.

C: 0 solutions. The graph does not intersect the -axis. The function has no zeros.

D: 2 solutions. The graph has 2 -intercepts. The function has 2 zeros.

E: 1 solution. The graph has 1 -intercept. The function has 1 zero.

F: 0 solutions. The graph does not intersect the -axis. The function has no zeros.



Solution

or or or


Problem 7StatementWhich function could represent the height in meters of an object thrown upwards from aheight of 25 meters above the ground seconds after being launched?

◦◦◦◦◦◦


A.

B.

C.

D.

SolutionD(From Unit 6, Lesson 6.)

Problem 8StatementA group of children are guessing the number of pebbles in a glass jar. The guesses and theguessing errors are plotted on a coordinate plane.

a. Which guess is furthest away from theactual number?

b. How far is the furthest guess away fromthe actual number?

Solutiona. 28

b. 9



Lesson 15: Quadratic Equations with IrrationalSolutions

GoalsCoordinate and compare (orally and in writing) solutions to quadratic equations obtained bycompleting the square and those obtained by graphing.

Understand that the “plus-minus” symbol is used to represent both square roots of a numberand that the square root notation expresses only the positive square root.

Use radical and “plus-minus” symbols to express solutions to quadratic equations.

Learning TargetsI can use the radical and “plus-minus” symbols to represent solutions to quadratic equations.

I know why the plus-minus symbol is used when solving quadratic equations by finding squareroots.

Lesson NarrativeThis lesson serves two main purposes: to reiterate that some solutions to quadratic equations areirrational, and to give students the tools to express those solutions exactly and succinctly.

Students recall that the radical symbol ( ) can be used to denote the positive square root of anumber. Many quadratic equations have a positive and a negative solution, and up until this point,students have been writing them separately. For example, the solutions of are and

. Here, students are introduced to the plus-minus symbol ( ) as a way to express bothsolutions (for example, ).

Students also briefly recall the meanings of rational and irrational numbers. (They will have a morethorough review later in the unit.) They see that sometimes the solutions are expressions that

involve a rational number and an irrational number—for example, . While this is acompact, exact, and efficient way to express irrational solutions, it is not always easy to intuit thesize of the solutions just by looking at the expressions. Students make sense of these solutions byfinding their decimal approximations and by solving the equations by graphing. The work here givesstudents opportunities to reason quantitatively and abstractly (MP2).

Alignments

Building On

8.EE.A.2: Use square root and cube root symbols to represent solutions to equations of theform and , where is a positive rational number. Evaluate square roots of small

perfect squares and cube roots of small perfect cubes. Know that is irrational.

•

•

•

••

•


Addressing




Building Towards

HSN-RN.B: Use properties of rational and irrational numbers.


Graph It



Required Materials






Let’s find exact solutions to quadratic equations even if the solutions are irrational.

15.1 Roots of SquaresWarm Up: 5 minutes

This warm-up reminds students that we can use the notation to express the side length of a

square with area and that the value of may be not be a whole number or a fraction.

To find the area of a tilted square, students might choose different strategies:

Decompose the square and rearrange the parts into rectangles.

•

•

•

•

•••

•

•


Enclose the tilted square with another square that is on the grid and subtract the areas of theextra triangles.

Use the Pythagorean Theorem to calculate the squared value of the side length.

Building On

8.EE.A.2

Launch

Display the entire task for all to see. Give students 3 minutes of quiet think time. Select students toshare their responses and how they reasoned about the side length and area of each square.


Here are some squares whose vertices are on a grid.

Find the area and the side length each square.

square area (square units) side length (units)

A

B

C

Student Response

A: 9 square units, 3 units

B: 2 square units, units

C: 10 square units, units

Activity Synthesis

Invite students to share their solutions. It is not essential to discuss how students found the area.

What is important is that students recall that we can use the square root notation to refer to

the side length of a square with area 9 square units and that . When the area is 2 or 10

•

•

•

•••


square units, the square root of each number is not a whole number or a fraction, but we can write

and rather than writing their decimal approximations.

Remind students that any positive number has two square roots because there are two numbers(one positive and one negative) that, when squared, give that number. Also remind students that

the symbol refers only to the positive square root of a number. In this context, if is the sidelength of a square and its area, we can describe the relationship between the two quantities as

because only positive side lengths make sense.

15.2 Solutions Written as Square Roots15 minutesThis activity introduces the use of notation as a simple way to express the two square roots of anumber. Students solve several simple equations by finding square roots and express theirsolutions using the notation.

Students also see that sometimes the solutions are not rational numbers and can be expressedexactly using the radical notation rather than using their decimal approximations. For example, the

solutions to can be written as rather than and the solutions to

can be written as instead of .

Addressing

HSA-REI.B.4.b

Building Towards

HSN-RN.B



Launch

Remind students that we have seen that some quadratic equations have two solutions. Take theequation , for example. For the equation to be true, can be 5 and -5 because and

. We have been writing the solutions as: and . Explain that a shorter way toconvey the same information is by writing .


When using the notation for the first time, some students may struggle to remember that itshows a number and its opposite. Encourage those students to continue writing their answers bothways in this lesson to reinforce the meaning of the new notation.

Some students may still struggle to understand the meaning of the square root. Watch for studentswho evaluate a square root by dividing by 2. Ask these students what the square root of 9 is. If this

•

•

•


does not convince students that dividing by two is the incorrect operation, demonstrate that we call3 a square root of 9 because .


Solve each equation. Use the notation when appropriate.

1.

2.

3.

4.

5.

6.

Student Response

1.

2.

3. No solution

4. (or equivalent)

5. (or equivalent)

6. (or equivalent)

Activity Synthesis

Invite students to share their solutions. Record and display the solutions for all to see. Discuss anydisagreements, if there are any.

Draw students’ attention to the last three sets of solutions, which are irrational. Ask students torecall the meaning of rational numbers and irrational numbers. Remind students that a rationalnumber is a fraction or its opposite, for example: 12, -7, , , 90.38, or 0.005. Irrational numbers

are numbers that are not rational, for example: , and . (Students will have moreopportunities to review and classify rational and irrational numbers later in the unit, so for now, itsuffices that they remember that an irrational number is a number that is not rational.)

Highlight that when the solution is irrational, the most concise way to write an exact solution is, for

example, . Writing this in decimal form only allows us to write an approximatesolution. For example, in this case, 3.24 and -5.24 are approximate solutions.


Then, display the variations in writing the solutions to for all to see. Discuss how theyare all different ways of writing the same two solutions.

and

and


Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their solutions,present an incorrect solution and explanation for the equation . For example, “Thesolutions are 9 and -9. I took the square root of each side and divided by 2. Since we are doingsquare roots, I made sure to put a positive and negative solution.” Ask students to identify theerror, critique the reasoning, and write a correct explanation. As students discuss with apartner, monitor for students who clarify the meaning of square root by using a simplerexample. Highlight explanations that verify the solution by multiplying the same number byitself. This helps students evaluate, and improve upon, the written mathematical arguments ofothers, as they solidify their understanding of square root.Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

15.3 Finding Irrational Solutions by Completing theSquare15 minutesThis activity allows students to integrate several skills and ideas they have learned so far: solvingquadratic equations by graphing and by completing the square, using notation, and expressingsolutions both approximately and exactly (using the square root symbol). In solving the equationsalgebraically and using the notations to express and verify solutions, students practice attending toprecision (MP6).

The last equation may be challenging for some students as it involves fractions and messier-lookingsolutions. It is not essential that all students get to this equation. They will have a chance toencounter more complicated equations and solutions in upcoming lessons.

Addressing

HSA-REI.B.4.a

HSA-REI.B.4.b

HSA-REI.D

••••

•••


Building Towards

HSN-RN.B


Graph It


Launch

Display the equation for all to see. Ask students if they could solve it by rewriting itin factored form. (The expression cannot be rewritten in factored form.) Remind students that theyknow how to solve any equation by completing the square, which would give exact solutions. Theyalso know how to solve by graphing, which would give approximate solutions.

Arrange students in groups of two and provide access to graphing technology. One partner shouldfind exact solutions by completing the square and the other should find approximate solutions bygraphing. Partners should confirm that the -intercepts of the graph (or the zeros of the functionrepresented by the graph) approximate the exact solutions obtained algebraically. Ask partners toswitch roles for each equation.


Conversing: MLR8 Discussion Supports. Use this routine to support small-group discussion asstudents take turns solving the quadratic equations using different methods and explainingtheir reasoning. Display the following sentence frames for all to see: “That could (or couldn’t)be true because . . .”, “How do you know . . .?”, and “How did you get . . . ?” Encourage studentsto challenge each other when they disagree. This will help students clarify their reasoningabout solving quadratic expressions by completing the square and by graphing.Design Principle(s): Support sense-making; Maximize meta-awareness


Engagement: Develop Effort and Persistence. Provide a checklist for completing the activity thatsupports increasing the length of on-task orientation in the face of distractions. For example,provide students with a two-column checklist with the headers of “graphing” and “completingthe square” for students to record their verification. Number the list 1–4 for each problem. Asstudents work through each list item, have them initial by each method to show they havecompleted a verification, and that they are switching roles accordingly. To support studentfocus, consider displaying a classroom timer of 3 minutes for each of the 4 verifications.Supports accessibility for: Attention; Social-emotional skills

•

••



Here is an example of an equation being solved by graphing and by completing the square.

Verify: is approximately 1.414.

So and .

For each equation, find the exact solutions by completing the square and the approximatesolutions by graphing. Then, verify that the solutions found using the two methods are close.If you get stuck, study the example.

1.

2.

3.

4.

Student Response

Exact and approximate solutions to three decimal places:

1. , or approximately -3.732 and -0.268

2. , or approximately 2.354 and 7.646




Write a quadratic equation of the form whose solutions are

and .

Student Response

or equivalent. Sample response: Working backward, these are solutions toIn standard form, this is .


Activity Synthesis

Discuss any common struggles or common mistakes made when solving the equations. Then, invitestudents to reflect on the merits and challenges of solving by each method. Ask questions such as:

“What are some benefits of solving by graphing? What are some drawbacks?” (Benefits: It isquick and straightforward. It shows the solutions, or that there are no solutions, even whenthe equations involve fractions or very large numbers. Drawbacks: It does not give exactsolutions. Some tools may require adjusting the graphing window quite a bit to see thesolutions.)

“What are some benefits and drawbacks of solving by completing the square?” (Benefit: It canbe used to find exact solutions to any equation. Drawbacks: It can be pretty time consuming.When the equations have fractions or very large or very small numbers, the calculations getcomplicated and may be prone to error.)

Lesson SynthesisTo highlight and help students connect the key ideas in the lesson, discuss questions such as:

“How is the notation useful when solving quadratic equations? Will all solutions need thisnotation?” (The notation is useful for expressing solutions that are the two square roots of anumber—for example, the solutions to . It is not needed when an equation has nosolutions—for example, , or only one solution—for example, .)

“How is the symbol useful when solving quadratic equations?” (It helps us express thesolutions exactly, especially when the solutions are irrational. We can just say, for example,

that the solutions are , instead of using the decimal approximations.)

“Does the expression mean there are two solutions, one positive and one negative?” (No.

The symbol by convention refers only to the positive square root of a number. To show

the negative square root, we need to write . Or to show both solutions, we need to write

.)

“What are some benefits and drawbacks of expressing the solutions using the square root

symbol and the plus-minus notation—as in —instead of writing them as decimals?”(Benefits: The solutions are exact. The radical and notations are very succinct and savecomputation time. Drawbacks: It is not always easy to tell what the approximate values of thesolutions are. When the solutions are written as -1.586 and -4.414, we have an intuition about

their size or where they are on the number line. When expressed as , it isn’timmediately clear how large they are or if they are positive or negative.)

15.4 Finding Exact SolutionsCool Down: 5 minutes

•

•

•

•

•

•


The approximate solutions are given so students can check (using a calculator) if their exactsolutions match.

Addressing

HSA-REI.B.4.b


For the equation , the approximate solutions are -8.732 and -5.268.

Find the exact solutions by completing the square. Show your reasoning.

Student Response

Student Lesson SummaryWhen solving quadratic equations, it is important to remember that:

Any positive number has two square roots, one positive and one negative, becausethere are two numbers that can be squared to make that number. (For example, and

both equal 36, so 6 and -6 are both square roots of 36.)

The square root symbol ( ) can be used to express the positive square root of a

number. For example, the square root of 36 is 6, but it can also be written as

because .

To express the negative square root of a number, say 36, we can write -6 or .

When a number is not a perfect square—for example, 40—we can express its square

roots by writing and .

How could we write the solutions to an equation like ? This equation is saying,

“something squared is 11.” To make the equation true, that something must be or

. We can write:

•

•

•

••


A more compact way to write the two solutions to the equation is: .

About how large or small are those numbers? Are they positive or negative? We can use acalculator to compute the approximate values of both expressions:

We can also approximate the solutions bygraphing. The equation isequivalent to , so we cangraph the function and findits zeros by locating the -intercepts of thegraph.

Glossaryirrational number


StatementSolve each equation and write the solutions using notation.

a.

b.

c.

d.

e.

f.

Solutiona.

b.

c.

d.

•


e.

f. (or equivalent)

Problem 2StatementMatch each expression to an equivalent expression.

A.

B.

C.

D.

E.

1. -17 and 5

2. and

3. 8 and 12

4. 3 and 5

5. and

Solution

A: 4

B: 3

C: 1

D: 5

E: 2

Problem 3Statement

a. Is a positive or negative number? Explain your reasoning.

b. Is a positive or negative number? Explain your reasoning.

c. Explain the difference between and the solutions to .

Solutiona. Positive. By convention we take to mean the positive square root of 4.

b. Positive. By convention we take to mean the positive square root of 5.


c. is equal to 3, while there are two numbers that make true: 3 and -3

Problem 4StatementTechnology required. For each equation, find the exact solutions by completing the square andthe approximate solutions by graphing. Then, verify that the solutions found using the twomethods are close.

Solution

Exact solutions: .

Approximate solutions:

Using a calculator, and

.

Exact solutions: .

Approximate solutions:

Using a calculator, and

.

Problem 5StatementJada is working on solving a quadratic equation, as shown here.

She thinks that her solution is correct because substituting 5for in the original expression gives , whichis or 0.

Explain the mistake that Jada made and show the correct solutions.


SolutionSample response: Jada correctly rewrote the expression in factored form, but instead of using thezero product property, she divided both sides by , so one of the solutions is missing. If

, then or , which means the solutions are and .

Problem 6StatementWhich expression in factored form is equivalent to ?

A.

B.

C.

D.


Problem 7StatementTwo rocks are launched straight up in the air. The height of Rock A is given by the function ,where . The height of Rock B is given by , where .In both functions, is time measured in seconds and height is measured in feet.

a. Which rock is launched from a higher point?

b. Which rock is launched with a greater velocity?

Solutiona. Rock B

b. Rock A


Problem 8Statement

a. Describe how the graph of has to be shifted to match the given graph.


b. Find an equation for the function represented by the graph.

Solutiona. The graph is shifted 4 units to the right.

b. (or equivalent)



Lesson 16: The Quadratic Formula

GoalsRecognize that the solutions obtained using the quadratic formula are the same as thosefound by using factored form or by completing the square.

Use the quadratic formula to solve quadratic equations of the form .

Learning TargetsI can use the quadratic formula to solve quadratic equations.

I know some methods for solving quadratic equations can be more convenient than others.

Lesson NarrativeIn this lesson, students encounter the quadratic formula and learn that it can be used to solve anyquadratic equation. They use the formula and verify that it produces the same solutions as thosefound using other methods, but can be much more practical for certain equations.

In upcoming lessons, students will continue to develop their understanding of the formula and itsstructure by using it to solve contextual problems and by analyzing its parts. After students havegained some experience working with the formula, they will investigate how it is derived.

Using the quadratic formula to solve equations requires students to attend carefully to theparameters in the given equations (MP6) and to apply different properties of operations flexibly asthey reason symbolically (MP2).

Alignments

Building On



Addressing



Building Towards

HSA-REI.B.4.b: Solve quadratic equations by inspection (e.g., for ), taking square roots,completing the square, the quadratic formula and factoring, as appropriate to the initial form

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of the equation. Recognize when the quadratic formula gives complex solutions and writethem as for real numbers and .



Think Pair Share

Required Materials



Let’s learn a formula for finding solutions to quadratic equations.

16.1 Evaluate ItWarm Up: 5 minutesThis warm-up prompts students to evaluate the kinds of numerical expressions they will see in thelesson. The expressions involve rational square roots, fractions, and the notation.

As students work, notice any common errors or challenges so they can be addressed during theclass discussion.

Building On

8.EE.A.2

Addressing

HSA-SSE.A

Building Towards

HSA-REI.B.4.b

Launch

Tell students to evaluate the expressions without using a calculator.


Students may be unfamiliar with evaluating rational expressions in which the numerator containsmore than one term. To help students see the structure of the expressions, consider decomposingthem into a sum of two fractions. For example, show that can be written as . This

approach can also help to avoid a common error of dividing only the first term by the denominator

( ). Some students may incorrectly write as . Point out that the first expression

is equal to 3 while the other has to be greater than 3 since .

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Unit 7 Lesson 16: The Quadratic Formula 267


Each expression represents two numbers. Evaluate the expressions and find the twonumbers.

1.

2.

3.

4.

Student Response

1. 8 and -6

2. 2 and

3. 3 and -3

4. -2 and -4

Activity Synthesis

Select students to share their responses and reasoning. Address any common errors. As needed,remind students of the properties and order of operations and the meaning and use of thesymbol.

16.2 Pesky Equations10 minutesIn this activity, students encounter equations that are challenging to solve using the methods theyhave learned, motivating students to seek a more efficient method.

Addressing

HSA-REI.B.4.b



Think Pair Share

Launch

Arrange students in groups of 2. Ask partners to choose the same equation. Give students quiettime to solve the equation and then time to discuss their solutions and strategy. If they finish

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solving their chosen equation, ask them to choose another one to solve. Leave a few minutes for awhole-class discussion.

Provide access to calculators for numerical computations.


Speaking, Representing: MLR8 Discussion Supports. Use this routine to support small-groupdiscussion. Provide the class with the following sentence frames to help them discuss theirsolution strategies with their partner: "First I _____ because. . .”, "I tried _____ and whathappened was . . .”, “How did you get . . . ?”, and “Why did you . . . ?” Encourage students tochallenge each other when they disagree. This will help students clarify their reasoning whensolving quadratic equations.Design Principle(s): Support sense-making


Engagement: Develop Effort and Persistence. Connect a new concept to one with which studentshave experienced success. Activate prior knowledge by reminding students that they havealready successfully completed tasks by both factoring and completing the square. Invite themto brainstorm what might be important to consider when selecting a method to use.Supports accessibility for: Social-emotional skills; Conceptual processing


Choose one equation to solve, either by rewriting it in factored form or by completing thesquare. Be prepared to explain your choice of method.

1.

2.

3.

4.

Student Response

1. 2.5 and -0.5 (or equivalent)

2. -4 and 2.2

3. 0.25 and -1.5

4.


Activity Synthesis

Consider arranging students who solved the same equation in groups of 2 to 3 to discuss theirstrategies and then displaying the correct solutions for all to see.

Invite students to share their reflections on the solving process. Discuss questions such as:

“What method did you choose and why?”

“Did you find yourself choosing one method and then switching to another? If so, whatprompted you to do that?”

“Did you run into any challenges when rewriting the equation (or completing the square)?What were some of the challenges?”

Acknowledge that all of these equations are cumbersome to solve by either rewriting in factoredform or completing the square. The last equation cannot be written in factored form (with rationalcoefficients), so completing the square is the only way to go. Tell students they are about to learn aformula that gives the solutions to any quadratic equation.

16.3 Meet the Quadratic Formula20 minutesThis activity introduces the quadratic formula. Students begin by applying the formula and using itto solve various equations, ranging from those that can be easily solved using other methods to thekinds that would be quite tedious to solve without the formula. They then verify that the formulagives the same solutions as those calculated by another method. Students notice that, forequations that cannot be easily rewritten using factored form or solved by completing the square,the formula offers quite an efficient way to find the solutions.

Addressing

HSA-REI.B.4.b



Launch

Display the equation for all to see. Tell students that , , and are numbers andis not 0. Ask students if they could complete the square for this equation without first replacing ,

, and with numbers.

Explain that this can indeed be done! The outcome of completing the square is not going to benumerical solutions (because no numbers are used), but rather a general formula for finding thesolutions of the quadratic equation. While students won’t have to complete the square for thisequation now, they will see the formula and try using it.

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Display the quadratic formula for all to see. Tell students that when an equation is of the form, where , , and are numbers and is not 0, we can find its solutions by using

the formula:

Guide students through the steps of using the formula to find the solutions to .

“First, what do we expect the solutions to be?” (3 and 5, because we can rewrite it as.)

“If is , what are the values of , , and ?” ( .)

Write out the formula as is.

Replace , , and with the corresponding numbers from the equation:

Evaluate one part of the expression at a time, ending with and .


If time is limited, ask students to complete at least 2 equations, including an equation in which theleading coefficient is not 1.


Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge.Invite students to select 4 questions to complete. Chunking this task into more manageableparts may also support students who benefit from additional processing time.Supports accessibility for: Organization; Attention; Social-emotional skills


Here is a formula called the quadratic formula.

The formula can be used to find the solutions to any quadratic equation in the form of, where , , and are numbers and is not 0.

This example shows how it is used to solve , in which , , and.

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Here are some quadratic equations and their solutions. Use the quadratic formula to showthat the solutions are correct.

1. . The solutions are and .


3. . The solutions are .

4. . The solutions are .

5. . The solution is .


Student Response

1. , which means or .

2. , which means or .

3. , which means .

4. , which means .

5. , which means .


6. , which means or

.


1. Use the quadratic formula to solve . Let’s call the resulting equation P.

2. Solve the equation in two ways, showing your reasoning for each:

Without using any formulas. Using equation P.

3. Check that you got the same solutions using each method.

4. Use the quadratic formula to solve . Let’s call the resulting equation Q.

5. Solve the equation in two ways, showing your reasoning for each:

Without using any formulas. Using equation P.

6. Check that you got the same solutions using each method.

Student Response

1. or . Either is an acceptable form of equation P.

2. The solutions are and . Sample reasoning:Without any formulas: The equation is equivalent to , so the solutions are 3 and -3.

Using equation P:

3. No response required.

4. , or simply and . Either is an acceptable form of equation Q.

5. The solutions are and . Sample reasoning:Without any formulas: The equation is equivalent to . Setting each of thefactors equal to zero, or .

Using equation Q: One solution is always zero. The other is . For this

equation, , or -2.5.

6. No response required.

Activity Synthesis

Much of the student discussion will have happened in small groups. Focus the whole-classconversation on whether the quadratic formula works for solving all equations and when it mightbe a preferred method. Ask students,

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“Look at the list of equations and the work you did to solve them. Do certain equations lendthemselves to certain methods of solving? Why?” (We can see the factors of some expressionsright away, so rewriting it in factored form would be the quickest method to solve theequations. Other expressions cannot be easily rewritten in factored form, but can be easilytransformed into perfect squares. For example, when the coefficient of is 1, the coefficientof the linear term is an even number, and the constant term is also a whole number,completing the square is straightforward. When the squared term has a coefficient other than1, the other two methods are less practical, so the quadratic formula may be the quickestapproach.)

“Why do you think the quadratic formula is useful only when the in is not0?” (If is 0, the expression is no longer quadratic because the squared term disappears,leaving only , which is linear.)

Select students who used the quadratic formula to solve the last few equations to explain theirsolutions and display their work for all to see. Discuss any challenges or disagreements in using theformula.

Tell students that they will use the formula to solve other equations and find out more about itsmerits and how it compares to other methods of solving.


Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. At theappropriate time, give students 2–3 minutes to plan what they will say when they share theirresponses to the question, “Look at the list of equations and the work you did to solve them.Do certain equations lend themselves to certain methods of solving? Why?” in the activitysynthesis. Encourage students to consider what details are important to share and to thinkabout how they will explain their reasoning using mathematical language. Invite students torehearse what they will say to the whole class with their partner. Rehearsing provides studentswith additional opportunities to speak and clarify their thinking, and will improve the quality ofexplanations shared during the whole-class discussion.Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson SynthesisDisplay the equation for all to see. Ask students how they prefer to solve it (byrewriting the expression in factored form, completing the square, or using the quadratic formula)and why. Then, ask them to solve the equation using their preferred method.

Possible explanations for the different methods:

Rewriting in factored form: The equation can be rewritten as and solvedusing the zero product property.

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Completing the square: is a perfect square, so we can just add 9 to either sideof the original equation and find the square root of 9.

The quadratic formula, because it always works.

The solutions (by any method) are and .

16.4 Solving and CheckingCool Down: 5 minutesAddressing

HSA-REI.B.4.b


Here is the quadratic formula: .

Use the formula to solve the equation .

Student Response

The solutions are -8 and -1. The quadratic formula applied for

: , which means or .

Student Lesson SummaryWe have learned a couple of methods for solving quadratic equations algebraically:

by rewriting the equation as and using the zero product property

by completing the square

Some equations can be solved quickly with one of these methods, but many cannot. Here isan example: . The expression on the left cannot be rewritten in factoredform with rational coefficients. Because the coefficient of the squared term is not a perfectsquare, and the coefficient of the linear term is an odd number, completing the square wouldbe inconvenient and would result in a perfect square with fractions.

The quadratic formula can be used to find the solutions to any quadratic equation, includingthose that are tricky to solve with other methods.

For an equation of the form , where , , andare numbers and , the solutions are given by:

For the equation , we see that , , and . Let’s solve it!

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A calculator gives approximate solutions of 0.84 and -0.24 for and .

We can also use the formula for simpler equations like , but it may not be themost efficient way. If the quadratic expression can be easily rewritten in factored form ormade into a perfect square, those methods may be preferable. For example, rewriting

as immediately tells us that the solutions are 1 and 8.

Glossaryquadratic formula


StatementFor each equation, identify the values of , , and that you would substitute into thequadratic formula to solve the equation.

a.

b.

c.

d.

e.

Solutiona. , ,

b. , ,

c. , ,

d. , ,

e. , ,

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Problem 2StatementUse the quadratic formula to show that the given solutions are correct.

a. . The solutions are and .

b. . The solutions are and .

c. . The solutions are .

Solution

a. , which is -4 or -5.

b. , which is 3 or 7.

c. , which is .

Problem 3StatementSelect all the equations that are equivalent to

A.

B.

C.

D.

E.

F.

G.

Solution["C", "D"](From Unit 7, Lesson 14.)


Problem 4StatementTechnology required. Two objects are launched upward. Each function gives the distance fromthe ground in meters as a function of time, , in seconds.

Object A: Object B:

Use graphing technology to graph each function.

a. Which object reaches the ground first? Explain how you know.

b. What is the maximum height of each object?

Solutiona. Object B. The graph shows the positive zero of function is 5 and the positive zero of function

is between 3 and 4 seconds.

b. Object A: 45 feet. Object B: 35 feet.


Problem 5StatementIdentify the values of , , and that you would substitute into the quadratic formula to solvethe equation.

a.

b.

c.

d.

e.

f.

Solutiona. , , and

b. , , and

c. , , and

d. , , and


e. , , and

f. , , and

Problem 6StatementOn the same coordinate plane, sketch a graph of eachfunction.

Function , defined by

Function , defined by

Solution


◦◦


Lesson 17: Applying the Quadratic Formula (Part 1)

GoalsInterpret (orally and in writing) the solutions to quadratic equations in context.

Practice using the quadratic formula to solve quadratic equations, rearranging the equationsinto if not already given in this form.

Learning TargetsI can use the quadratic formula to solve an equation and interpret the solutions in terms of asituation.

Lesson NarrativeIn this lesson, students return to some quadratic functions they have seen. They write quadraticequations to represent relationships and use the quadratic formula to solve problems that they didnot previously have the tools to solve (other than by graphing). In some cases, the quadraticformula is the only practical way to find the solutions. In others, students can decide to use othermethods that might be more straightforward (MP5).

The work in this lesson—writing equations, solving them, and interpreting the solutions incontext—encourages students to reason quantitatively and abstractly (MP2).

Alignments

Addressing


HSA-REI.A: Understand solving equations as a process of reasoning and explain the reasoning.


HSF-IF.B.5: Relate the domain of a function to its graph and, where applicable, to thequantitative relationship it describes. For example, if the function gives the number ofperson-hours it takes to assemble engines in a factory, then the positive integers would bean appropriate domain for the function.

Building Towards


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Think Pair Share

Required Materials



Let’s use the quadratic formula to solve some problems.

17.1 No Solutions for You!Warm Up: 5 minutesPreviously, students have seen that some quadratic functions have no zeros and that somequadratic equations have no solutions. In this warm-up, they recall that when a solution is a squareroot of a negative number, the equation has no solutions. Note that students don’t yet know aboutany numbers that aren’t real at this point, so it is unnecessary to specify “no real solutions.” (Tostudents, the word “real” would seem like an extra word added for no reason.)

Addressing

HSA-REI.A

Building Towards

HSA-REI.B.4


Think Pair Share

Launch

Arrange students in groups of 2. Give students quiet think time, and then time to share theirthinking with a partner.


Here is an example of someone solving a quadratic equation that has no solutions:

1. Study the example. At what point did you realize the equation had no solutions?

2. Explain how you know the equation has no solutions.

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Student Response

1. Sample responses:In the second line, when noticing that the squared quantity equals a negative number.

In the third line, when seeing the square root of a negative number.

2. Sample response: Solving this equation involves rearranging the terms, which leads to onone side and a negative number on the other. It is not possible for the square of a number tobe negative.

Activity Synthesis

Select students to share their responses and reasoning.

If not mentioned in students’ explanations, point out that we can look at and reason about thestructure of these equations to tell which ones have no solutions, without taking any steps to solve.For example, the equation means “a square plus 9 equals 0.” Here are two ways toreason about its structure:

For two numbers to add up to 0, they have to be opposites. Since the number 9 is positive, theother number must be negative, which a square cannot be.

A square is always positive, so a square plus a positive number must also be positive and cannever equal zero.

These lines of reasoning also allow us to see that has no solutions.

17.2 The Potato and the Pumpkin15 minutesBy now, students are pretty familiar with quadratic functions that model the height of objects as afunction of time after being launched up or being dropped. They have found the zeros of suchfunctions graphically, by identifying horizontal intercepts, as well as algebraically, by writingequations of the form and applying the zero product property.

Students have also used graphs to estimate input values that yield non-zero output values. Prior tolearning about completing the square or the quadratic formula, however, they did not have thetools to solve for such inputs algebraically. In this activity, students apply the knowledge and skillsthey recently developed to solve contextual problems that they couldn’t previously solve withoutgraphing.

Because algebraic reasoning is the aim of this activity, graphing technology should not be used tosolve the equations. It could be used to verify solutions during the activity synthesis, however.

If time is limited, assign the first question to half of the class and the second question to the otherhalf. Alternatively, ask students to choose one question to answer.

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Addressing

HSA-CED.A.1

HSA-REI.B.4.b

HSF-IF.B.5



Launch

Remind students that in earlier lessons they encountered two functions that modeled the height ofa launched object as a function of time. They solved problems about the functions by graphing. Tellstudents that they now have additional strategies at their disposal and ask them to solve theseproblems without graphing. Provide access to calculators. Tell students to use them only fornumerical computations. If time is limited, consider asking students to answer only the first set ofquestions.


Representation: Develop Language and Symbols. Display or provide charts with symbols andmeanings. Display important phrases from the questions and elicit student responses toconnect them to symbols. For example, ask students what and represent and scribe theirresponses. Then, display the phrase “hits the ground,” and connect it to the output, .Similarly, display “40 feet off the ground,” and show that . When students have writtentheir equations, use the display to scribe and record the steps for correctly entering theexpression into the calculator. Keep the display visible for all to see as students continue towork.Supports accessibility for: Conceptual processing; Memory


Watch for students who incorrectly substitute 40 for and evaluate , or who writeinstead of evaluating . These students may be having trouble distinguishing between the inputand output because the meaning of words such as “the height is a function of time” is still unclear tothem. Remind these students that “height is a function of time” means that the variable in theformula given represents time, and that the formula can be used to find the height. Point out that

represents the output for input .


Answer each question without graphing. Explain or show your reasoning.

1. The equation represented the height, in feet, of a potatoseconds after it has been launched.

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a. Write an equation that can be solved to find when the potato hits the ground.Then solve the equation.

b. Write an equation that can be solved to find when the potato is 40 feet off theground. Then solve the equation.

2. The equation models the height, in meters, of a pumpkinseconds after it has been launched from a catapult.

a. Is the pumpkin still in the air 8 seconds later? Explain or show how you know.

b. At what value of does the pumpkin hit the ground? Show your reasoning.

Student Response

1. a. . The potato hits the ground after about 5.7 seconds. Samplereasoning: Using the quadratic formula with , , and gives two

solutions: , which are approximately -0.702 and 5.702. Only the positive

solution makes sense here.

b. which is equivalent to . The potato is 40 feetoff the ground after about 5.3 seconds. Using the quadratic formula with , ,

and gives two solutions: , which are approximately -0.284 and 5.284.

Only the positive solution makes sense in this context.

2. a. No. Sample reasoning: . If the pumpkin is still in theair, the value of would be positive.

b. Just a little under 5 seconds. Sample reasoning: . Using thequadratic formula with , and gives two solutions:

, or approximately -0.083 and 4.920. Only the positive solution makes

sense.

Activity Synthesis

Make sure students understand what equations to write and what it means to solve each equationin the given contexts. Then, focus the discussion on how the solutions can be found, interpreted,and verified. Ask questions such as:

“Once you have the equations, how did you find the solutions when the equation has a 0 onone side?” (By using the quadratic formula or by completing the square.)

“What about when the equation has a non-zero number on one side?” (Rearrange it so that itis in standard form and has 0 on one side, and then use one of the solving methods.)

“How many solutions did you get?” (Two for each equation.) “Do both solutions make sense inthe given context?” (No. Only positive solutions are meaningful because they represent timeafter the objects were launched.)

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“How can we check if our solutions are accurate?” (Solve with a different method, substitutethem back into the equation and see if the equation is true, or check by graphing.)

Consider displaying the graphs of the two functions to verify students’ solutions.


Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their equationsand strategies to find the time of the potato when it is 40 feet off the ground, present anincorrect application of the quadratic formula that represents a common misconception. For

example, “I used the quadratic formula to get 5.28 seconds since .” Askstudents to identify the error, critique the reasoning, and write a correct explanation. Asstudents discuss with a partner, listen for students who clarify the misapplication of thequadratic formula and the importance of maintaining the signs of the coefficients. Invitestudents to share their critiques, corrected explanations, and solution with the class. Listen forand amplify the language students use to describe what happens when the sign of thecoefficient is not maintained. This will help students understand how to apply the quadraticformula and how to use the context to reason about appropriate solutions.Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

17.3 Back to the Framer15 minutesIn this activity, students return to the framing problem they encountered when starting the unit. Inthat first lesson, they were challenged to use an entire sheet of paper to frame a picture and ensurethat the thickness is uniform all around. At that time, students did not have adequate knowledge tosolve the problem methodically, so they relied mainly on guessing and checking. Since then,students have developed their understandings around writing and solving quadratic equations.They are now ready to formulate the problem effectively and solve it methodically.

Students engage in aspects of mathematical modeling (MP4) as they identify the constraints in asituation, formulate a problem, construct a model, and interpret their solutions in context.

Because algebraic reasoning is the aim of this activity, graphing technology should not be used tosolve the equations. It could be used to verify solutions during the activity synthesis, however.

Addressing

HSA-CED.A.1

HSA-REI.B.4.b



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•


Think Pair Share

Launch

Ask students to recall the framing problem from the beginning of the unit, how they tried to solve it,and what challenges they encountered. Consider preparing a copy of the picture and framingmaterial from that lesson to serve as a visual aid. Tell students that they now have enoughknowledge and skills to solve the problem more effectively and no longer have to rely on guessingand checking.

Arrange students in groups of 2. Give students a moment of quiet think time to interpret theequation in the first question and then discuss it with a partner. Display somesentence stems to help students articulate their interpretation. For example:

The two factors in the equation represent . . .

The number 38 represents . . .

The variable represents . . .

Solving the equation means finding . . . . Pause afterward for a brief discussion. Make surestudents understand the meaning of the equation before they proceed to solve it.

This activity was designed to be completed without graphing, so ask students to put away anygraphing devices. Provide continued access to calculators for numerical computations.

If time is limited, consider asking students to complete only the first question.


Representation: Develop Language and Symbols. Create a display of important terms andvocabulary. During the launch, take time to review terms that students will need to access forthis activity. Invite students to suggest language or diagrams to include that will support theirunderstanding of methods for solving quadratic equations. Invite student input on a list of themethods (graphing, writing in factored form and applying the zero product property,completing the square, and the quadratic formula) and write them in separate areas on thechart. Ask students to suggest conditions under which each method can be applied, andbrainstorm ideas about how to recognize the methods. Encourage students to recall solutionsfrom previous lessons to construct the chart. As students begin to work, check in with them toask which method they have selected. Keeping the chart displayed throughout the activity willhelp students contextualize the application of new knowledge.Supports accessibility for: Conceptual processing; Language

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1. In an earlier lesson, we tried to frame a picture that was 7 inches by 4 inches using anentire sheet of paper that was 4 inches by 2.5 inches. One equation we wrote was

.

a. Explain or show what the equation tells us about thesituation and what it would mean to solve it. Use the diagram, as needed.

b. Solve the equation without graphing. Show your reasoning.

2. Suppose you have another picture that is 10 inches by 5 inches, and are now using afancy paper that is 8.5 inches by 4 inches to frame the picture. Again, the frame is to beuniform in thickness all the way around. No fancy framing paper is to be wasted!

Find out how thick the frame should be.

Student Response

1. a. Sample response: The two factors, and , represent the overall length andthe width of the picture and frame. 38 is the total area, in square inches, of the pictureand the frame when all the framing material is used ( ). representsthe thickness of the frame. Solving the equation means finding the frame thickness thatis uniform all around and uses all the framing material.

b. About 0.42 inch. Sample reasoning: can be rewritten into the form.

Applying the quadratic formula for , , and :


The solutions, to three decimal places, are 0.422 and -5.922. Only the positive solutionmakes sense in this context.

2. 1 inch. Sample reasoning: The equation is , which can be rewritten instandard form as and then in factored form as . Bythe zero product property, or , so or . Only the positive

solution is meaningful here.


Suppose that your border paper is 6 inches by 8 inches. You want to use all the paper tomake a half-inch border around some rectangular picture.

1. Find two possible pairs of length and width of a rectangular picture that could beframed with a half-inch border and no leftover materials.

2. What must be true about the length and width of any rectangular picture that can beframed this way? Explain how you know.

Student Response

1. Sample response: 23 inches by 24 inches, 15 inches by 32 inches.

2. The sum of the length and width must be 47 inches. Sample reasoning: If the side lengths ofthe picture are by , then linear inches of border paper are needed to frame thefour sides and four corners. Because there are 48 square inches of border paper, there shouldbe 96 linear inches of half-inch strips available to make a frame. This means the side lengths ofthe picture plus 4 linear inches must equal 96 linear inches, which can be expressedas , or .

Activity Synthesis

Invite students to share their solution strategies. If not mentioned in students’ explanations, makesure to discuss:

Why negative solutions aren’t meaningful in this context.•


Which strategy they found effective and efficient for solving the equations. (The quadraticformula might not be the speediest way to solve the last equation, for example, as it can besolved by rewriting the equation in factored form and by completing the square.)

If time permits, consider asking students to verify their solution to the first question using thepicture and framing materials from the blackline master from the first lesson of the unit. Askstudents to cut the paper into strips that are as thick as the solution they calculated and arrangethe strips around the picture. (If their solution is correct, there should be no leftover framingmaterial and the frame should be uniform all around the picture.)

Lesson SynthesisTo help students consolidate the insights from this lesson, ask students to reflect on questions suchas:

“Before this lesson, you were able to solve many application problems that involve quadraticfunctions, but you didn’t use the quadratic formula. What does the quadratic formula allowyou to do that you couldn’t do before?”

“Suppose a situation can be modeled with a quadratic function, defined by an expression inthe vertex form, and we want to find its zeros. We can write and solvethe equation. Would you choose the quadratic formula to solve it? Why or why not?” (Probablynot. The equation can be much more quickly solved by using the zero product property.)

“What if we want to find when the function has a value of 75? Would the quadratic formula bea good way to solve ? Why or why not?” (Yes. We cannot use the zeroproduct property and completing the square is likely pretty laborious. We could graph, but thesolutions might not be precise.)

“How do we know what the , and are in the equation ? (We needto first rewrite it so that it is in the form of , and then identify those values.)

17.4 Tennis Ball Up, Tennis Ball DownCool Down: 5 minutesAddressing

HSA-CED.A.1

HSA-REI.B.4.b

Launch

Provide continued access to the quadratic formula and to calculators for numerical computations.Ask students to put away their graphing devices.


Function gives the height of a tennis ball, in feet, seconds after it is tossed straight up inthe air. The equation defines function .

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Write and solve an equation to find when the ball hits the ground. Show your reasoning.

Student Response

The equation can be rewritten as . By the zero productproperty, or , so or . Only the positive solution makes sense

here, so the ball hits the ground 1.25 seconds after being tossed up.

Student Lesson SummaryQuadratic equations that represent situations cannot always be neatly put into factored formor easily solved by finding square roots. Completing the square is a workable strategy, but forsome equations, it may involve many cumbersome steps. Graphing is also a handy way tosolve the equations, but it doesn’t always give us precise solutions.

With the quadratic formula, we can solve these equations more readily and precisely.

Here’s an example: Function models the height of an object, in meters, seconds after it islaunched into the air. It is is defined by .

To know how much time it would take the object to reach 15 meters, we could solve theequation . How should we do it?

Rewriting it in standard form gives . The expression on the left sideof the equation cannot be written in factored form, however.

Completing the square isn't convenient because the coefficient of the squared term isnot a perfect square and the coefficient the linear term is an odd number.

Let’s use the quadratic formula, using !

The expression represents the two

exact solutions of the equation.

We can also get approximate solutions byusing a calculator, or by reasoning that

.

The solutions tell us that there are two times after the launch when the object is at a height of15 meters: at about 0.7 seconds (as the object is going up) and 4.3 seconds (as it comes backdown).

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StatementSelect all the equations that have 2 solutions.

A.

B.

C.

D.

E.

F.

G.

Solution["A", "C", "E", "G"]

Problem 2StatementA frog jumps in the air. The height, in inches, of the frog is modeled by the function

, where is the time after it jumped, measured in seconds.

Solve . What do the solutions tell us about the jumping frog?

SolutionSample response: and . The height of the frog was 0 inches before it jumped. The frog

landed seconds after it jumped and was on the ground again.

Problem 3StatementA tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeledby the equation , where is measured in seconds since the ball wasthrown.

a. Find the solutions to the equation .

b. What do the solutions tell us about the tennis ball?


Solutiona. and . Sample reasoning using factored form: , so either

or . This means or .

b. Sample response: The ball hits the ground 1 second after it is hit. The negative solution doesn’thave meaning because a negative value for time doesn't make sense.

Problem 4StatementRewrite each quadratic expression in standard form.

a.

b.

c.

d.

Solutiona.

b.

c.

d.


Problem 5StatementFind the missing expression in parentheses so that each pair of quadratic expressions isequivalent. Show that your expression meets this requirement.

a. and

b. and

c. and

Solutiona. . Sample reasoning:


b. . Sample reasoning: is and is -4. These are the squared term and theconstant term in standard form. Multiplying by -2 gives . Multiplying 2 by gives .The sum of the two linear terms is or , which is the middle term.

c. . Sample reasoning:


Problem 6StatementThe number of downloads of a song is a function, , of the number of weeks, , since thesong was released. The equation defines this function.

a. What does the number 100,000 tell you about the downloads? What about the ?

b. Is meaningful in this situation? Explain your reasoning.

Solutiona. There were 100,000 downloads of the song the week it was released. Each week after that, the

number of downloads decreased by a factor of .

b. No. A negative value of would mean downloads were made before the song was released,which is not possible.


Problem 7StatementConsider the equation .

a. Identify the values of , , and that you would substitute into the quadratic formula tosolve the equation.

b. Evaluate each expression using the values of , , and .

c. The solutions to the equation are and . Do these match the values of the

last expression you evaluated in the previous question?


Solutiona. , , and

b. is 4. is 16. is -240. is 256.

is 16. is 20or -12.

is 8. is or .

c. The solutions are the same as the values of the last expression evaluated (or equivalent).


Problem 8Statement

a. Describe the graph of . (Does it open upward or downward? Where is its-intercept? What about its -intercepts?)

b. Without graphing, describe how adding to would change each feature of thegraph of . (If you get stuck, consider writing the expression in factored form.)

i. the -intercepts

ii. the vertex

iii. the -intercept

iv. the direction of opening of the U-shape graph

Solutiona. The graph opens downward. The vertex is at and this point is also the -intercept and

-intercept of the graph.

b. Sample response:i. The equation is equivalent to . The factored form reveals

that the -intercepts are at and . This means the graph has shifted to theright and up such that it now crosses the -axis at two points.

ii. The coordinate of the vertex is halfway between 0 and 16, so it is 8. The coordinate isor , which is 64. The vertex is at .

iii. The -intercept is at , because .

iv. The graph still opens downward because the coefficient of the squared term is stillnegative.



Lesson 18: Applying the Quadratic Formula (Part 2)

GoalsAnalyze and critique (orally and in writing) solutions to quadratic equations that are foundusing the quadratic formula.

Determine whether a given value is a solution to a quadratic equation.

Learning TargetsI can identify common errors when using the quadratic formula.

I know some ways to tell if a number is a solution to a quadratic equation.

Lesson NarrativeIn earlier lessons, students saw that the quadratic formula can be used to solve any quadraticequation, but also that it might not be the most practical approach for all equations. Here, studentsdeepen their understanding about the merits and potential drawbacks of using the quadraticformula. They pay close attention to each step in the solving process and analyze errors commonlymade when applying the formula. Students also consider how to verify that the solutions theyobtained with the quadratic formula are correct.

In analyzing the solution process and checking their work and the work of others, students practiceattending to precision and critiquing the reasoning of others (MP6, MP3).

Technology isn’t required for this lesson, but there are opportunities for students to choose to useappropriate technology to solve problems. Consider making technology available.

Alignments

Building On

6.EE.A.2.c: Evaluate expressions at specific values of their variables. Include expressions thatarise from formulas used in real-world problems. Perform arithmetic operations, includingthose involving whole-number exponents, in the conventional order when there are noparentheses to specify a particular order (Order of Operations). For example, use the formulas

and to find the volume and surface area of a cube with sides of length .



Addressing


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HSF-IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpretstatements that use function notation in terms of a context.

Building Towards






Think Pair Share

Required Materials



Be prepared to display a graph for all to see in the activity synthesis of “Sure About That?” There isan image to display if graphing technology is not available.


Let’s use the quadratic formula and solve quadratic equations with care.

18.1 Bits and PiecesWarm Up: 5 minutesIn this warm-up, students evaluate variable expressions that resemble those in the quadraticformula. The aim is to preview the calculations that are the source of some common errors insolving equations using the quadratic formula, which students will analyze in the next activity.

Building On

6.EE.A.2.c

8.EE.A.2

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Building Towards

HSA-REI.B.4.b


Evaluate each expression for , , and

1.

2.

3.

4.

Student Response

1. 5

2. 25

3. 97

4. 8 and 2

Activity Synthesis

Invite students to share their responses and discuss any disagreement. For each expression, ask ifthey can think of an error someone might make when evaluating such an expression. Somepossible errors:

: Forgetting that is really and the product of two negative numbers is positive.

: When is -5, evaluating , instead of .

: Forgetting that subtracting by is equivalent to adding , or neglecting to seethat if is negative, is positive, not negative.

: Neglecting the negative in front of , or neglecting to see that the expression takes

two different values. Tell students that in the next activity, they will spot some errors in solvingquadratic equations.

18.2 Using the Formula with Care15 minutesSolving an equation with the quadratic formula involves multiple calculations, some of which maybe prone to error. This activity acquaints students with some of the commonly made mistakes andencourages them to write and evaluate expressions carefully. As students analyze the progressionof reasoning in worked solutions, they practice reasoning abstractly (MP2) and attending toprecision in communicating a solution (MP3).

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As students work, identify those who can clearly and correctly explain the errors in the workedsolutions. Ask them to share their responses during the class discussion.

Addressing

HSA-REI.B.4.b



Think Pair Share

Launch

Arrange students in groups of 2. Give students a few minutes of quiet work time to solve 1–2equations using the quadratic formula, and then a moment to discuss their response with theirpartner. Then, ask them to study the worked solutions (in the activity statement) for the sameequations that they had solved and identify the errors.

If time permits, ask groups to identify the errors in the remaining equations. If they get stuck,suggest that they try solving the equations first and then look for the errors.



Here are four equations, followed by attempts to solve them using the quadratic formula.Each attempt contains at least one error.

Solve 1–2 equations by using the quadratic formula.

Then, find and describe the error(s) in the worked solutions of the same equations asthe ones you solved.

Equation 1: Equation 2:


Here are the worked solutions with errors:

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Student Response

Equation 1:

The correct solutions are or or equivalent.

The second-to-last step is not evaluated correctly, so the solution is incorrect. is not

.

Equation 2:The correct solutions are and .

The equation was not rewritten as before the , , and are identified.The should be -10, not 10.

Equation 3:

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•◦◦

•


The correct solutions are (or equivalent).

The value of is not multiplied to 2 in the denominator of the formula.

Equation 4:

The correct solutions are (or equivalent).

When is -10, the value of should be 10. The value of should be positive even whenis negative.

Activity Synthesis

Display the worked solutions in the activity statement for all to see. Select previously identifiedstudents to identify and explain the error(s) in each worked solution.

After each student presents, ask the class to classify the error(s) by type and to explain theirclassification. For instance, are the errors careless mistakes or computational mistakes? Do theyshow gaps in understanding, incomplete communication, or a lack of precision?

If not mentioned by students, point out that these errors are easy to make but not always easy tonotice. Sometimes, unless the solutions really make no sense, we won’t know if the solutions areactually correct or completely off unless we check them. Tell students that they will consider waysto check their solutions in the next activity.


Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. Aftereach student shares what they identified and explained as the error in the worked solutions,ask students to restate what they heard using precise mathematical language. Providestudents with time to restate what they hear to a partner before selecting one or two studentsto share with the class. Ask the original speaker if their peer was accurately able to restate theirthinking. Call students’ attention to any words or phrases that helped to clarify the originalstatement. This provides more students with an opportunity to produce language as theyinterpret the reasoning of others.Design Principle(s): Support sense-making

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Representation: Internalize Comprehension. Use color coding and annotations to highlightconnections between representations in a problem. For example, as students share theirthinking, display their work and the worked solutions (with errors) side by side for all to see.When the students identify the error, use a highlighter to indicate the spot where the error wasinitially made, and highlight the corresponding spot in the correct example. Encouragestudents to do the same on their own materials. Before having students classify errors, invitestudents to suggest a list of possible types of errors and display the list. As students connectthe types of errors to the example, annotate the worked examples and label them as such.Supports accessibility for: Visual-spatial processing

18.3 Sure About That?15 minutesEarlier, students practiced identifying likely mistakes in solving quadratic equations using thequadratic formula. This activity prompts students to check the solutions to quadratic equations andto recognize that there is more than one way to do so. Different approaches include substitutingsolutions to see if the equation is true, solving using a different algebraic method, or solving bygraphing.

Monitor for students who verify the solutions to the equation by:

Solving the equation using another algebraic method and seeing if they get the samesolutions. For instance, if they used the quadratic formula initially, they might choose to solveagain by completing the square or by rewriting the equation in factored form.

Substituting the solutions for in the equation and seeing if the equation is true.

Graphing and and seeing if the horizontal coordinates of theintersections match those of the solutions.

Graphing and checking the horizontal intercepts of the graph.

Identify students who use contrasting methods and ask them to share during discussion. Thealgebraic strategies are more precise but could lead to errors if the computations are performedincorrectly. The graphing strategies can be accomplished quickly with the use of technology butmay only give approximate solutions.

Making calculators and graphing technology available gives students an opportunity to chooseappropriate tools strategically (MP5).

Addressing

HSA-CED.A.1

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•

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HSA-REI.B.4.b

HSF-IF.A.2




Launch

Keep students in groups of 2. Ask students to think quietly about each question before conferringwith their partner. Encourage partners to consider different ways of checking their solutions.

If time is limited, consider asking half of the class to answer the first question and the other half toanswer the second question.

Provide access to calculators for numerical computations and to graphing technology, in caserequested for solution checking.


1. The equation represents the height, as a function of time, of apumpkin that was catapulted up in the air. Height is measured in meters and time ismeasured in seconds.

a. The pumpkin reached a maximum height of 47 meters. How many seconds afterlaunch did that happen? Show your reasoning.

b. Suppose someone was unconvinced by your solution. Find another way (besidesthe steps you already took) to show your solution is correct.

2. The equation models the revenue a band expects to collect as afunction of the price of one concert ticket. Ticket prices and revenues are in dollars.

A band member says that a ticket price of either $15.50 or $74.50 would generateapproximately $1,000 in revenue. Do you agree? Show your reasoning.

Student Response

1. a. 3 seconds after launch. Sample reasoning:

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••


b. Sample responses:Substituting 3 for in the original expression and evaluating it gives

or , which is 47.

Graphing and shows an intersection is at .

Graphing shows a zero at .

2. Partially agree. A ticket price of $15.50 will generate about $1,000 in revenue, but a ticket priceof $74.50 will generate much less (about $410). Sample reasoning:

Using the quadratic formula to solve gives and asthe solutions.

Substituting 15.50 into the expression gives 999.75, which is close to 1,000.( ). Substituting 74.50 into the expression gives409.75.

Graphing the equation and shows two intersections at approximatelyand .


Function is defined by the equation . Its graph opens downward.

1. Find the zeros of function without graphing. Show your reasoning.

2. Explain or show how the zeros you found can tell us the vertex of the graph of .

3. Study the expressions that define functions and (which defined the height of thepumpkin). Explain how the maximum of function , once we know it, can tell us themaximum of .

Student Response

1. Function has one zero: 3. Sample reasoning:is equivalent to . Solving using the

quadratic formula gives .

▪

▪▪

◦

◦

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◦


can be rewritten as and then as . Applyingthe zero product property to solve gives .

2. Sample response: Function only has one zero, which means its graph has only onehorizontal intercept, occurring when . Because the point is the only intercept, itmust also be vertex of the graph, otherwise there would have been either two horizontalintercepts or none.

3. Sample response: The value of expression for is 47 less than that of , so if we know themaximum of , we know that the maximum of is 47 less than that. We saw that themaximum of is 47, so the maximum of is 0.

Activity Synthesis

Select previously identified students to present the different ways in which they checked theirsolutions. Sequence their presentations based on the precision of the verification (from moreprecise to less precise), as shown in the Activity Narrative. Highlight as many different strategies astime permits.

If no one mentions solving with a different strategy or graphing as ways to verify solutions, bringthese up. Display a graph such as this one to show that the graph can immediately show that 74.50is not a solution to the equation , but 64.50 is very, very close.

Many students may have chosen to substitute the solutions back into the equation as a way tocheck them. Acknowledge that this is a very useful strategy for checking, but it also has its limits.

Sometimes the solutions are not rational numbers—for example, . Substituting such

an expression back into the equation and evaluating it would mean complicated computations,which may themselves be subject to errors.

We could approximate the values of expressions like and write them as decimals before

checking. This is a workable strategy if a calculator is handy.

◦



Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students forthe whole-class discussion. At the appropriate time, invite students to create a visual displayshowing their two strategies and calculations for the first question involving. Allow studentstime to quietly circulate and analyze the strategies in at least 2 other displays in the room. Givestudents quiet think time to consider what is the same and what is different. Next, ask studentsto find a partner to discuss what they noticed. Listen for and amplify observations thathighlight advantages and disadvantages to each method. This will help students makeconnections between different methods for solving quadratic equations.Design Principle(s): Optimize output; Cultivate conversation

Lesson SynthesisInvite students to name some familiar mistakes that are made when using the quadratic formula,not only those seen in the activities, but also mistakes that the students themselves might havemade.

Then, to encourage students to think about ways of checking solutions, discuss questions such as:

“What might be some good ways to check if and are solutions to

” (Some possibilities:Substitute each solution into the equation and see if the equation is true. For example,

see if equals 0.

Rewrite in factored form: and apply the zeroproduct property to solve.

Graph and find the -intercepts, when the graph has a -coordinate of0.)

“Can completing the square be used to check the solution?” (Yes, but it is not simple becausethe leading coefficient is not a perfect square and the linear coefficient is an odd number.)

“Suppose we want to verify if are correct solutions to . Issubstituting the expressions in the equation an effective way to check?” (In this case,

substituting for makes the checking cumbersome. The expression to be evaluated

would look like: , which is not easy to evaluate.)

“What might be some good ways to check?” (We could:Solve by completing the square. The leading coefficient of is 1 and the linearterm coefficient is an even number, which makes it fairly simple to complete the square.

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Approximate the value of the solutions with a calculator, then graph andsee if the solutions match the -intercepts.)

18.4 Where Did It Go Wrong?Cool Down: 5 minutesAddressing

HSA-REI.B.4.b


Here is someone’s solution to the equation .

Identify as many errors as you can and briefly explain each error.

Student Response

Sample response:

is -4, so the value should be , which is 4.

When the equation is rewritten as , the constant term is -20, not 20, so thelast part of the numerator should be .

should be 16. Squaring a negative number gives a positive number.

If the earlier issues were fixed, then the values under the square root symbol would beor 256.

Student Lesson SummaryThe quadratic formula has many parts in it. A small error in any one part can lead to incorrectsolutions.

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••


Suppose we are solving . To use the formula, let's rewrite it in the form of, which gives: .

Here are some common errors to avoid:

Using the wrong values for , , and in the formula.

Nope! is -11, so is , which is 11, not -11.

That’s better!

Forgetting to multiply 2 by for the denominator in the formula.

Nope! The denominator is , which is or 4. That’s better!

Forgetting that squaring a negative number produces a positive number.

Nope! is 121, not -121. That’s better!

Forgetting that a negative number times a positive number is a negative number.

Nope! and is . That’s better!

Making calculation errors or not following the properties of algebra.

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•

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Nope! Both parts of the numerator, the 11 and the ,

get divided by 4. Also, is not .

That’s better!

Let’s finish by evaluating correctly:

To make sure our solutions are indeed correct, we can substitute the solutions back into theoriginal equations and see whether each solution keeps the equation true.

Checking 6 as a solution: Checking as a solution:

We can also graph the equation and find its-intercepts to see whether our solutions to

are accurate (or close to accurate).



StatementMai and Jada are solving the equation using the quadratic formula but founddifferent solutions.

Mai wrote: Jada wrote:

a. If this equation is written in standard form, , what are the values of ,and ?

b. Do you agree with either of them? Explain your reasoning.

Solutiona. , , and .

b. Sample response: Disagree with both of them. Mai used 7 for instead of -7. For in theformula, Jada computed instead of , so she wrote -49 instead of 49.

Problem 2StatementThe equation represents the height, in feet, of a potato secondsafter it was launched from a mechanical device.

a. Write an equation that would allow us to find the time the potato hits the ground.

b. Solve the equation without graphing. Show your reasoning.

Solutiona.

b. . Sample reasoning: Using the quadratic formula, with , , and :


The potato hits the ground about 5.7 seconds after being launched. (The negative solution hasno meaning in this situation.)

Problem 3StatementPriya found and as solutions to . Is she correct? Show how youknow.

SolutionYes. Sample responses:

Substituting each 3 and -1 for in the original quadratic expression and then evaluating theexpression gives a value of 0.

Graphing the equation shows -intercepts at and .

Substituting 3 for , -6 for , and -9 for in the quadratic formula gives and as thesolutions.

Problem 4StatementLin says she can tell that and are perfect squares becauseeach expression has the following characteristics, which she saw in other perfect squares instandard form:

The first term is a perfect square. The last term is also a perfect square.

If we multiply a square root of the first term and a square root of the last term and thendouble the product, the result is the middle term.

a. Show that each expression has the characteristics Lin described.

b. Write each expression in factored form.

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◦◦

◦◦


Solutiona. :

is or .

16 is or .

is . Doubling it gives .

:

is or .

64 is or .

is or is .Doubling it gives .

b. and or



A.

B.

C.

D.


Problem 6StatementSolve each equation by rewriting the quadratic expression in factored form and using thezero product property, or by completing the square. Then, check if your solutions are correctby using the quadratic formula.

a.

b.

c.

▪▪▪

▪▪▪


Solutiona. and . Sample reasoning: , so either or . Quadratic

formula: which is -8 and -3.

b. . Sample reasoning: , so , which means . Quadratic

formula: which is .

c. . Sample reasoning:

Quadratic formula: which is approximately -0.583 or -8.583. The

approximate values of are the same.


Problem 7StatementHere are the graphs of three equations.

Match each graph with the appropriate equation.

A.

B.

C.

1. X

2. Y

3. Z

Solution

A: 1


B: 3

C: 2


Problem 8StatementThe function is defined by .

a. What are the -intercepts of the graph of ?

b. Find the coordinates of the vertex of the graph of . Show your reasoning.

c. Sketch a graph of .

Solutiona. ,

b. . The vertex, located at thehalfway point between the -intercepts, hasan -coordinate of -3.5, so the -coordinateis or -6.25.

c. See graph.



Lesson 19: Deriving the Quadratic Formula

GoalsExplain (in writing) the steps used to derive the quadratic formula.

Explain (orally and in writing) how the solutions obtained by completing the square areexpressed by the quadratic formula.

Understand that the quadratic formula can be derived by generalizing the process ofcompleting the square.

Learning TargetsI can explain the steps and complete some missing steps for deriving the quadratic formula.

I know how the quadratic formula is related to the process of completing the square for aquadratic equation .

Lesson NarrativeBy now, students are familiar with the quadratic formula and how it can be used to solve quadraticequations. Students started to recognize when the quadratic formula is useful and when anothermethod might be preferred. They also saw some errors that are commonly made when applyingthe formula. Up to this point, however, students simply took the formula for granted and have notconsidered how it produces solutions to any quadratic equations. That work takes place in thislesson.

Students learn that the quadratic formula came from the steps of completing the square. Becausecompleting the square always works for solving any quadratic equation, the steps can begeneralized into a single formula for solving any equation of the form .

To prepare students to complete the square with , , and remaining as letters, students firsttransform perfect squares from factored form into standard form, but without evaluating anything.For example, they rewrite as , and then as

. Doing so reinforces and makes explicit the structural connectionsbetween the two forms, equipping students to reason in reverse as they complete the square for

.

There are different ways to derive the quadratic formula. The path chosen here involvestemporarily replacing the in with a single letter, say , so the expression forwhich we are completing the square is a monic quadratic expression: . An optionalactivity in the last lesson on completing the square includes this strategy. If desired, consider usingit to familiarize students with the idea of using a temporary placeholder to reason with complicatedexpressions.

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In this lesson, students analyze and complete partially worked-out derivations of the quadraticformula, explaining each step along the way. As they do so, students practice constructing logicalarguments (MP3).

Alignments

Addressing



.

Building Towards




Think Pair Share


Let’s find out where the quadratic formula comes from.

19.1 Studying StructureWarm Up: 10 minutesThis warm-up reminds students of the structure that governs the relationship between perfectsquares written as squared factors and their equivalent expression in standard form. The key goalis for students to see that when we expand a squared factor of the form , the equivalentexpression in standard form has this structure: . Seeing the coefficient of thesquared term as and the coefficient of the linear term as 2 times a product of twonumbers ( and ) would enable students to complete the square more easily. This would in turnhelp them make sense of where the quadratic formula comes from, which they will explore in thislesson.

Students practice looking for and making use of structure as they think about the relationshipsbetween equivalent expressions (MP7).

Addressing

HSA-SSE.A.2

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Unit 7 Lesson 19: Deriving the Quadratic Formula 315

Building Towards

HSA-REI.B.4.a


Think Pair Share

Launch

Ask students to expand the squared factor into standard form and show all the steps.Students are likely to show , or

.

Next, arrange students in groups of 2. Tell students to use the relationship between the factoredand standard forms to write some equivalent perfect-square expressions. Ask students to thinkquietly about the expressions in the table before conferring with their partner.


Here are some perfect squares in factored and standard forms, and an expression showinghow the two forms are related.

1. Study the first few examples, and then complete the missing numbers in the rest of thetable.


2. Look at the expression in the last row of the table. If is equivalent to, how are , and related to and ?

Student Response

1.

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2. Sample response: , , and .

Activity Synthesis

Display the table for all to see and ask students for the missing values. Then, discuss the followingquestions to highlight the ways in which the numbers in the middle column are related to the twonumbers in each expression in factored form.

“For the expressions in the middle column, how do you know what to write for the firstsquared term?” (It’s the square of the linear term of the expression in factored form.)

“The coefficient of the linear term is always 2 times the two other numbers. How do you knowwhat those are?” (It’s the two terms from factored form: the linear term and the constantterm.)

“How do you know what to write for the last squared term?” (It’s the square of the constantterm of the expression in factored form.)

“When making a perfect square, how do you know what values to write in themiddle column?” (The first squared term is that equals , so that “something”must be because . The linear term is that equals , sothat other thing must be 3 because . The last squared term is ,so it is .)

Emphasize the general structure of the equivalent expressions, as shown in the last row. Whenis expanded, the standard form always has the structure of . Also,

make sure students understand how the numbers in the factored form are related to ,that is, that they recognize that , , and .

Highlight that when we want to complete the square for and write an equivalentexpression of the form , having a perfect square for makes it much easier to find , andhaving an even number for makes it easier to find (because an even number gives a wholenumber when divided by 2).

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19.2 Complete the Square using a Placeholder10 minutesThis is the first of two activities that help students derive the quadratic formula. Here students try tocomplete the square for a quadratic equation but without evaluating some of the numericalexpressions along the way. Doing so enables them to recognize the parts of the quadratic formula

but in numerical form (for example, instead of seeing , they see ).

The linear coefficient of the given equation is an odd number. Students learn that there arestrategies to transform the quadratic expression such that it has an even number for the linearcoefficient and 1 for the leading coefficient, which makes it much easier to complete the square.These strategies involve multiplying the equation by a helpful factor and temporarily using simplervariables to stand in for complicated parts of an expression.

(The strategy of multiplying an equation by a number to make the a perfect square is illustrated inan optional activity in the last lesson on completing the square. Students who completed thatactivity may recall this approach. Using a simple variable to stand in for a messier expression is alsoexplored in an optional activity in an earlier lesson—the last lesson on rewriting expressions infactored form.)

Addressing

HSA-REI.B.4.a

Launch

The quadratic equation is in the form of . What are the values of ,, and ? Talk to a partner about whether it would be relatively simple to complete the square from

this equation. (It’s doable, but probably pretty messy, as is an odd number.)

Tell students that one way to make it easier to complete the square is to multiply the equation by anumber such that the is an even number and the is a perfect square.

Let’s try multiplying by 2. We get . (The right side remains 0because 0 times any number is 0.) We have 10, an even number for , but the is not a perfectsquare.

Let’s try multiplying it by 4. We get . Now the is a perfect square and theis still an even number. To complete the square, it helps to isolate the existing constant and

rewrite it as: .

It’s still not immediately obvious what constant term to add to make a perfect square. It wouldbe easier if the or the coefficient of the squared term is 1. One way to deal with this is tothink of as . What would the “something” be? ( )

If we use a placeholder to stand for , we can write:

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Ask students to try completing the square for this simpler-looking equation.


1. One way to solve the quadratic equation is by completing the square. Apartially solved equation is shown here. Study the steps.

Then, knowing that is a placeholder for , continue to solve for but withoutevaluating any part of the expression. Be prepared to explain each step.

2. Explain how the solution is related to the quadratic formula.

Student Response

1. Sample response:


2. Sample response: The numbers in the solution are the values of , , and in the originalequation, , which is in the form of . The -5 is . The is .The 12, which is , is the value of . The in the original equation is 1, and is 2,which is the number in the denominator in the solution.

Activity Synthesis

Highlight the connections between the numbers in the solution to the

parameters in the original equation.

The -5 in the expression is .

The is .

The 12 is or .

The 2 in the denominator is or .


Representation: Internalize Comprehension. Use color and annotations to support informationprocessing. Invite students to use color to identify and keep track of terms from one step tothe next.Supports accessibility for: Visual-spatial processing; Organization

19.3 Decoding the Quadratic Formula15 minutesIn the previous activity, students solved a quadratic equation by completing the square, but theydid so without evaluating any of the numerical expressions. Students explained each step along thesolving process and saw how it produced a solution that looks almost exactly like the quadraticformula, except that it contains numbers instead of and . That activity gave students concreteinsights into the process of deriving the quadratic formula, preparing them to do the same in moreabstract terms.

••••


In this activity, students study a series of steps taken to solve by completing thesquare and make sense of how it leads to the quadratic formula. Along the way, they see that thesolving process involves similar maneuvers as those seen earlier, for example, multiplying theequation by a factor to make the leading coefficient a perfect square, writing the linear coefficientas 2 times two numbers, and adding the square of one of the numbers to complete the square.They see that the result of solving the equation by completing the square is the quadratic formula.

Addressing

HSA-REI.B.4.a



Launch

Tell students that they will now study a worked-out solution to . There are nonumbers in this equation, but the process of solving should be familiar. Ask students to analyze thesolution and record an explanation for each step. (They should explain why each step is taken andnot only what happens in each step.)


Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to helpstudents improve their written explanations for each step in the derivation of the quadraticformula. Give students time to meet with 2–3 partners to share and get feedback on theirresponses. Display feedback prompts that will help students strengthen their ideas and clarifytheir language. For example, “How do you know . . . ?”, “Why do you think . . . ?”, “Whathappened between _____ and _____?”, and “Is it always true that . . . ?” Invite students to go backand revise or refine their written explanation based on the feedback from peers. This will helpstudents understand how completing the square leads to the quadratic formula throughcommunicating their reasoning with a partner.Design Principle(s): Support sense-making; Cultivate conversation


Action and Expression: Internalize Executive Functions. Chunk this task into manageable parts forstudents who benefit from support with organizational skills in problem solving. Invitestudents to number each line to facilitate identification of each step. Check in with studentsafter the first 2–3 minutes of work time. Invite 1–2 students to share what they noticed for thefirst few steps. Record their thinking on a display for all to see, and keep the work visible asstudents continue.Supports accessibility for: Organization; Attention

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Some students may think that we multiply just by 4 and wonder where the comes from. Point outthat we need the first term to be a perfect square. In the previous activity, the value was 1 (whichwas already a perfect square). We can’t be certain that is a perfect square, so we multiply by 4 and

to make the first term , which is , a perfect square.

Some may wonder where the comes from. Consider displaying a completed solution from theprevious activity that parallels this work. Remind students of the placeholder used there. Invitethem to compare other steps in the previous activity to those in this activity to help them explainthe derivation.


Here is one way to make sense of how the quadratic formula came about. Study thederivation until you can explain what happened in each step. Record your explanation next toeach step.

Student Response

Sample response:



Here is another way to derive the quadratic formula by completing the square.

First, divide each side of the equation by to get .

Then, complete the square for .

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1. The beginning steps of this approach are shown here. Briefly explain what happens ineach step.

2. Continue the solving process until you have the equation .

Student Response

1. [1] Subtract from each side.

[2] Notice that is half of . Complete the square by adding to each side.

[3] Rewrite the left side of the equation in factored form; square the expression .

[4] To find a common denominator, multiply by .

[5] Add the two fractions, writing the term before the term in the numerator.

[6] Since , must equal either of the square roots of .

[7] Use the property that .

2. Sample response:


Activity Synthesis

Invite students to share their explanations for each step. Highlight a few key maneuvers:

Multiplying the equation by makes the coefficient of a perfect square and the coefficientof the linear term an even number, both of which make completing the square much easier.

Some students may wonder why is chosen to complete the square when solving. Why not , , or ? Any of these would work. The equation could also

have been divided by to make the first term , which is a perfect square. (If time permits,consider asking students to try completing the square using one of these alternatives.) Whilethese alternatives result in a perfect square for the first term, they may not give an evennumber for the coefficient of the second term or they may produce an equation with largernumbers, making it a bit trickier to manipulate.

We can tell what constant term is needed to complete the square by dividing the coefficient ofthe linear term by the positive square root of the coefficient of , taking half of it, andsquaring that number. In this case:

The coefficient of is .

The coefficient of is , so the square root is .

Dividing by gives . Half of is .

The constant term that completes the square is .

Emphasize that the quadratic formula essentially captures the steps for completing the square inone expression. Every time we solve a quadratic equation by completing the square, we areessentially using the quadratic formula, but in a less condensed way.

Lesson SynthesisStudents are not expected to internalize how to derive the quadratic formula by the end of thelesson. They should, however, understand that each part of the complicated expression can betraced back to a step in completing the square. In other words, they should recognize that theformula is not an isolated, opaque method that mysteriously produces solutions to quadraticequations.


“How has your understanding of the quadratic formula changed from the work in this lesson?”

“A classmate who is absent today is not sure where the quadratic formula came from. Whatwould you say to help them understand, in a nutshell, what the formula is all about?”

“If the formula is connected to the steps of completing the square, why not just complete thesquare when we need to solve equations? Why do you think is it helpful to have a formula,even if it involves quite a few operations?”

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19.4 Step by StepCool Down: 5 minutesAddressing

HSA-REI.B.4.a


Here is one solution for solving by completing the square, where each step isshown, but numerical expressions are not evaluated.

1. In Step 2, the equation is multiplied by 4. Why might that be?

2. In Step 5, is added to each side. Why might that be?

3. What happened between Step 5 and Step 6?

Student Response

Sample response:

1. To make the coefficient of a perfect square and the coefficient of an even number.

2. completes the square for the expression on the left side. To keep the two sides equal, itneeds to be added to the other side as well.

3. The perfect square is rewritten in factored form and is rewritten as .

Student Lesson SummaryRecall that any quadratic equation can be solved by completing the square. The quadraticformula is essentially what we get when we put all the steps taken to complete the square for

into a single expression.

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When we expand a squared factor like , the result is . Noticehow the appears in two places. If we replace with another letter like , we have

, which is a recognizable perfect square.

Likewise, if we expand , we have . Replacing with gives, also a recognizable perfect square.

To complete the square is essentially to make one side of the equation have the samestructure as . Substituting a letter for makes it easier to see what isneeded to complete the square. Let’s complete the square for !

Start by subtracting from each side.

Next, let’s multiply both sides by . Onthe left, this gives , a perfect squarefor the coefficient of .

can be written , andcan be written .

Let’s replace with the letter .

is the constant term that completesthe square, so let’s add to each side.

The left side is now a perfect square andcan be written as a squared factor.

The square roots of the expression onthe right are the values of .

Once is isolated, we can write in itsplace and solve for .

The solution is the quadratic formula!


Statementa. The quadratic equation is in the form of . What are

the values of , , and ?

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b. Some steps for solving the equation by completing the square have been started here.In the third line, what might be a good reason for multiplying each side of the equationby 4?

c. Complete the unfinished steps, and explain what happens in each step in the secondhalf of the solution.

d. Substitute the values of , , and into the quadratic formula, , but

do not evaluate any of the expressions. Explain how this expression is related to solvingby completing the square.

Solutiona.

b. Multiplying by 4 makes the coefficient of the squared term a perfect square, which makes iteasier to complete the square.

c.


d. . Sample explanation: The expression that is equal to bundles up all the

steps for solving the equation by completing the square into a single expression. Rather thanevaluating at each step, the calculation is done all at once, at the end.


a. Does the quadratic formula work to solve this equation? Explain or show how you know.

b. Can you solve this equation using square roots? Explain or show how you know.

Solutiona. Yes. Sample response: The quadratic formula works for all quadratic equations. In the

equation , is 1, is 0, and is -39. Substituting those values into the quadraticformula and evaluating the expression gives the solutions.

b. Yes. Sample response: Adding 39 to both sides of the equation gives . The solutions

are .

Problem 3StatementClare is deriving the quadratic formula by solving

by completing the square.

She arrived at this equation.

Briefly describe what she needs to do to finish solving for and then show the steps.


SolutionSample response: She needs to find the square roots of each side, subtract from each side, andthen divide each side by 2.

Problem 4StatementTyler is solving the quadratic equation .

Study his work and explain the mistake he made. Then, solvethe equation correctly.

SolutionTyler needed to add 5 to each side to get or 9 on the right side of the equal sign.


Problem 5StatementSolve the equation by using the quadratic formula. Then, check if your solutions are correctby rewriting the quadratic expression in factored form and using the zero product property.

a.

b.

c.


Solution

a. -1 and 2.5. Using the quadratic formula, which is or

.

To check the solution, the equation can be written as ,which means or .

b. -3 and 7. Using the quadratic formula, which is or

.


c. -1 and . Using the quadratic formula, which is or

.



Problem 6StatementA tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeledby the equation , where is measured in seconds since the ball wasthrown.

a. Find the solutions to without graphing. Show your reasoning.

b. What do the solutions say about the tennis ball?

Solutiona. and . Sample reasoning: can be rewritten as

and then as . This means either or , so or

.

b. Sample response: The ball is 6 feet above the ground after second and second.




a. What are the -intercepts of the graph of this equation? Explain how you know.

b. What is the -coordinate of the vertex of the graph of this equation? Explain how youknow.

c. What is the -coordinate of the vertex? Show your reasoning.

d. Sketch the graph of this equation.

Solutiona. and . Sample explanation: When

is 0, is 0, so is 0. When is 6, is0, so is 0.

b. The -coordinate is 3, because this ishalfway between 0 and 6 which are the

-coordinates of the -intercepts.

c. The -coordinate is 18. When ,.

d. See graph.



Lesson 20: Rational and Irrational Solutions

GoalsExplain (orally and in writing) why the product of a non-zero rational and irrational number isirrational.

Explain (orally and in writing) why the sum of a rational and irrational number is irrational.

Explain (orally and in writing) why the sum or product of two rational numbers is rational.

Learning TargetsI can explain why adding a rational number and an irrational number produces an irrationalnumber.

I can explain why multiplying a rational number (except 0) and an irrational number producesan irrational number.

I can explain why sums or products of two rational numbers are rational.

Lesson NarrativeStudents have seen both rational and irrational solutions when solving quadratic equations in thisunit. Before this lesson, minimal emphasis was placed on reviewing the meaning of rational andirrational numbers (which students first learned in grade 8), or on the fact that certain solutions areirrational. Students may have noticed, however, that it can be tricky to get a sense of the value ofirrational solutions—even if they are exact—because irrational solutions are expressed with thesquare root symbol and their decimal values need to be approximated.

This is the first of two lessons in which students look closely at whether a solution to a quadraticequation is rational or irrational. Distinguishing solutions as rational or irrational does notnecessarily impact students’ ability to solve applied problems about quadratic functions. In thosecases, we deal mostly with finite decimal approximations. The work here extends students’understanding of the real number system. Reasoning about the properties of rational and irrationalnumbers also offers opportunities to construct logical arguments and attend to precision inreasoning (MP3, MP6). Along the way, students also practice solving quadratic equations andfinding zeros of the corresponding functions.

For some solutions to quadratic equations, it is relatively straightforward to classify them as rational

(for example, or -7) or irrational (for example, or ). Other solutions are harder to classify,

however, based on what students have learned up until this point. For instance, is rational

or irrational? What about ? Students begin wondering about these questions and making

conjectures about what category each of these numbers belongs to. In an upcoming lesson, theywill reason about the sums and products of rational and irrational numbers more generally andjustify them as being one type of number or the other.

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Alignments

Building On

8.NS.A.1: Know that numbers that are not rational are called irrational. Understand informallythat every number has a decimal expansion; for rational numbers show that the decimalexpansion repeats eventually, and convert a decimal expansion which repeats eventually intoa rational number.

Addressing



HSN-RN.B.3: Explain why the sum or product of two rational numbers is rational; that the sumof a rational number and an irrational number is irrational; and that the product of a nonzerorational number and an irrational number is irrational.

Building Towards



Graph It


Required Materials






Let’s consider the kinds of numbers we get when solving quadratic equations.

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20.1 Rational or Irrational?Warm Up: 5 minutesThis warm-up refreshes students’ memory about rational and irrational numbers. Students thinkabout the characteristics of each type of number and ways to tell that a number is rational. Thisreview prepares students for the work in this lesson: identifying solutions to quadratic equations asrational or irrational, and thinking about what kinds of numbers are produced when rational andirrational numbers are combined in different ways.

Building On

8.NS.A.1

Building Towards

HSN-RN.B.3

Launch

Remind students that a rational number is “a fraction or its opposite” and that numbers that are notrational are called irrational. (The term rational is derived from the word ratio, as ratios andfractions are closely related ideas.)


Numbers like -1.7, , and are known as rational numbers.

Numbers like are known as irrational numbers.

Here is a list of numbers. Sort them into rational and irrational.

97 -8.2

Student Response

Rational: Irrational:

Activity Synthesis

Ask students to share their sorting results. Then, ask, “What are some ways you can tell a number isirrational?” After each student offers an idea, ask others whether they agree or disagree. Askstudents who disagree for an explanation or a counterexample.

Point out some ways to tell that a number might be irrational:

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It is written with a square root symbol and the number under the symbol is not a perfect

square, or not the square of a recognizable fraction. For example, it is possible to tell that

is rational because . However, is irrational because 18 is not a perfect square.

If we use a calculator to approximate the value of a square root, the digits in the decimalexpansion do not appear to stop or repeat. It is possible, however, that the repetition doesn’t

happen within the number of digits we see. For example, is rational because it equals ,

but the decimal doesn’t start repeating until the seventh digit after the decimal point.

20.2 Suspected Irrational Solutions15 minutesIn this activity, students use algebraic reasoning to solve quadratic equations and to identify thesolutions as rational or irrational. But first, they inspect the zeros of a corresponding function forhints about whether the solutions might be rational. They notice that it is impossible to knowwhether a solution is irrational simply by looking at the decimal approximation of a point shown ona graph.

While solutions obtained by algebraic solving can better show the types of number they are, somesolutions are difficult to classify because they are combinations of rational and irrational numbers,

such as and . Students will investigate these kinds of solutions in the next activity.

Building On

8.NS.A.1

Addressing

HSA-REI.B.4.b

HSF-IF.C.7.a

Building Towards

HSN-RN.B.3


Graph It


Launch

Give students a moment to look at the equations in the activity statement and be prepared to sharewhat they noticed about the two sets of equations. If not mentioned by students, point out that theequations in the two questions describe the same set of functions. Ask students how finding the

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zeros of the equations in the first question relates to solving the equations in the second question.(Both are ways to solve the second set of equations.).

Arrange students in groups of 2. Provide access to devices that can run Desmos or other graphingtechnology. Ask students to take turns graphing and solving algebraically. One partner shouldgraph the first 2 equations while the other solves algebraically, and then switch roles for theremaining questions. Each student should decide whether they think the zeros or the solutions theyfound—by graphing or by algebraic solving—are rational or irrational and then discuss theirthinking.


Conversing: MLR8 Discussion Supports. Use this routine to help students describe their reasoningfor determining whether a solution is rational or irrational. Arrange students in groups of 2.Invite Partner A to begin with this sentence frame: “_____ is rational/irrational, because _____.”Invite the listener, Partner B, to press for additional details referring to specific features of thesolution, such as whether the square root is a rational number. Students should switch rolesfor each equation. This will help students justify how to determine whether solutions arerational or irrational.Design Principle(s): Support sense-making; Cultivate conversation


When solving equations such as algebraically, some students may expandrather than use square roots (perhaps to use the quadratic formula to solve). For students who usethis approach, check to see if they expanded correctly and remind them, if needed, that

. These students may benefit from continued use of arectangle diagram to expand. Emphasize the need to check their answers against the graphicalsolution. When their solutions don’t agree, encourage them to find the error in their work.


1. Graph each quadratic equation using graphing technology. Identify the zeros of thefunction that the graph represents, and say whether you think they might be rational orirrational. Be prepared to explain your reasoning.

equations zeros rational or irrational?


2. Find exact solutions (not approximate solutions) to each equation and show yourreasoning. Then, say whether you think each solution is rational or irrational. Beprepared to explain your reasoning.

a.

b.

c.

d.

Student Response

Answers vary.

1. Sample response:a. The zeros of are -2.828 and 2.828 to 3 decimal places. They could be rational

or irrational. It’s impossible to tell without more information.

b. The zeros of are 4 and 6. Both are rational.

c. The zeros of are 5.586 and 8.414 to 3 decimal places. They could berational or irrational. It’s impossible to tell without more information.

d. The zeros are -8.944 and 8.944 to 3 decimal places. They could be rational orirrational. It’s impossible to tell without more information.

2. Sample response:

a. The solutions are . These are irrational because 8 isn’t a perfect square.

b. The solutions are 4 and 6. Both are rational.

c. The solutions are . Perhaps they are irrational because is irrational, but I’mnot sure.

d. The solutions are . Perhaps they are irrational because is irrational, but I’m notsure.

Activity Synthesis

Invite students to share their responses and explanations. Ask students to compare the zeros theyfound by graphing and the solutions they found algebraically. Discuss with students:

“How do the zeros of the functions and , which you found bygraphing, compare to the solutions to and ?” (The zeros found by graphingare decimal approximations. They show decimal approximations up to 3 decimal places, ormore, depending on the tool or setting used. The solutions found algebraically can be writtenas expressions that use the square root symbol.)

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“Is it easier to tell whether the solutions to an equation are irrational by graphing and findingthe zeros, or by solving algebraically?” (By solving algebraically. Observing the coordinates on agraph doesn’t always help, because it is impossible to tell whether a number is irrational byonly looking at a few digits of its decimal approximation.)

Draw students’ attention to the numerical expressions they encountered while solving theequations:

: This expression is rational because it equals 6.

: This is irrational because 8 is not a perfect square.

: is irrational and 7 is rational. Is the sum rational or irrational?

: 4 is rational and is irrational. Is the product rational or irrational?

Tell students that they will now experiment with the sums and products of rational and irrationalnumbers and investigate what kinds of numbers the sums and products are.


Representation: Develop Language and Symbols. Create a display of important terms andvocabulary. Make a large two-column display, reserving one side for examples of rationalnumbers and another side for examples of irrational numbers. Invite students to suggestlanguage to include that will support their understanding of rational and irrational numbers.Keep the chart visible throughout the lesson, and as examples are discussed, add them to thechart.Supports accessibility for: Conceptual processing; Language

20.3 Experimenting with Rational and IrrationalNumbers15 minutes

Students ended the previous activity wondering whether solutions that look like andare rational or irrational. In this activity, they pursue that question and experiment with adding andmultiplying different types of numbers. The goal here is to make conjectures about the sums andproducts by noticing regularity in repeated reasoning with concrete numbers (MP8). In an upcominglesson, students will reason logically and abstractly about what the sums and products of rationaland irrational numbers must be.

Addressing

HSN-RN.B.3

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•




Launch

Tell students that they will now further investigate what happens when different types of numbersare combined by addition and multiplication. Are the results rational or irrational? Can we come upwith general rules about what types of numbers the sums and products will be?

Consider keeping students in groups of 2. Provide access to calculators for numerical calculations,in case requested. Students who choose to use technology to help them analyze patterns practicechoosing a tool strategically (MP5).


Engagement: Internalize Self Regulation. Chunk this task into more manageable parts todifferentiate the degree of difficulty or complexity. Invite students to begin by sorting andlabeling the initial numbers into rational or irrational. Invite each partner group to choose andrespond to either the sum or the product statements. Provide sentence frames for students touse in their discussion. For example, “I agree/disagree with _____ because . . . .” and “Here is anexample that shows . . . .” Then, have partners switch or select students to share with thewhole class, so that each student has examples and conclusions for all truth statements.Supports accessibility for: Organization; Attention


Here is a list of numbers:

Here are some statements about the sums and products of numbers. For each statement,decide whether it is always true, true for some numbers but not others, or never true.

1. Sums:

a. The sum of two rational numbers is rational.

b. The sum of a rational number and an irrational number is irrational.

c. The sum of two irrational numbers is irrational.

2. Products:

a. The product of two rational numbers is rational.

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b. The product of a rational number and an irrational number is irrational.

c. The product of two irrational numbers is irrational.

Experiment with sums and products of two numbers in the given list to help you decide.

Student Response

1. Sums:a. Always true

b. Always true

c. True for some (such as ) but not for others (such as )

2. Products:a. Always true

b. True for some (many examples) but not for others (such as )

c. True for some (such as ) but not for others (such as )


It can be quite difficult to show that a number is irrational. To do so, we have to explain whythe number is impossible to write as a ratio of two integers. It took mathematiciansthousands of years before they were finally able to show that is irrational, and they stilldon’t know whether or not is irrational.

Here is a way we could show that can’t be rational, and is therefore irrational.

Let's assume that were rational and could be written as a fraction , where and

are non-zero integers.

Let’s also assume that and are integers that no longer have any common factors. Forexample, to express 0.4 as , we write instead of or . That is, we assume that

and are 2 and 5, rather than 4 and 10, or 200 and 500.

1. If , then .

2. Explain why must be an even number.

3. Explain why if is an even number, then itself is also an even number. (If you getstuck, consider squaring a few different integers.)

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4. Because is an even number, then is 2 times another integer, say, . We can write. Substitute for in the equation you wrote in the first question. Then, solve

for .

5. Explain why the resulting equation shows that , and therefore , are also evennumbers.

6. We just arrived at the conclusion that and are even numbers, but given ourassumption about and , it is impossible for this to be true. Explain why this is.

If and cannot both be even, must be equal to some number other than .

Because our original assumption that we could write as a fraction led to a false

conclusion, that assumption must be wrong. In other words, we must not be able to write

as a fraction. This means is irrational!

Student Response

1.

2. Sample response: and be rewritten as . Because is twice another integer, it

is an even number by definition.

3. Sample response: The square of any even number is another even number. If were odd, itssquare would also be odd.

4. , which can be rewritten as $2b^2 = (2k)^2 and then as or .

5. Sample response: is twice an integer, so it is an even number by definition. We cannotsquare an odd number and get an even number, so must be an even number.

6. Sample response: If and were both even, then they still have a common factor, 2. Thismeans that cannot be a fraction where and are integers with no common factors, as

originally assumed.

Activity Synthesis

Display the six statements for all to see. Invite students to share their responses. After each studentshares, ask others whether they agree or disagree. Ask students who disagree for an explanation ora counterexample. Develop a consensus on what the class thinks is true when we combinenumbers by addition and multiplication. Record and display the consensus for all to see, forinstance:

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Explain to students that in an upcoming lesson, they will have a chance to test their conjectures.


Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-classdiscussion. After each student shares, provide the class with the following sentence frames tohelp them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoicestudent ideas to demonstrate mathematical language, and invite students to chorally repeatphrases that include relevant vocabulary in context.Design Principle(s): Support sense-making

Lesson SynthesisTo help students consolidate the insights from this lesson, ask students questions such as:

“How would you explain to a classmate who is absent today how to tell if a number is rationalor irrational?”

“How can we tell from a graph created using graphing technology whether the solutions to aquadratic equation could be rational or irrational?”

“Why is the -intercept information given by a graphing tool not a sure way to tell if thosenumbers are irrational?”

20.4 What Kind of Solutions?Cool Down: 5 minutesBuilding On

8.NS.A.1

Addressing

HSA-REI.B.4.b

Building Towards

HSN-RN.B.3

••••

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Launch

Provide continued access to technology, in case requested.


1. Decide whether the solutions to each equation are rational or irrational. Explain yourreasoning.

a.

b.

2. Someone says that if you add two irrational numbers, you will always get an irrationalsum. How can you convince the person they are wrong?

Student Response

1. Explanations vary. A likely approach is to find the solutions of and and notethat 9 is a perfect square so its square roots are , but 10 is not a perfect square.

a. rational

b. irrational

2. Sample response: We can show a counterexample, for example: .

Student Lesson SummaryThe solutions to quadratic equations can be rational or irrational. Recall that:

Rational numbers are fractions and their opposites. Numbers like 12, -3, , -4.79,

and are rational. ( is a fraction, because it’s equal to . The number -4.79 is

the opposite of 4.79, which is .)

Any number that is not rational is irrational. Some examples are , and .

When an irrational number is written as a decimal, its digits do not stop or repeat, so adecimal can only approximate the value of the number.

How do we know if the solutions to a quadratic equation are rational or irrational?

If we solve a quadratic equation by graphing a corresponding function( ), sometimes we can tell from the -coordinates of the -intercepts. Othertimes, we can't be sure.

Let's solve and by graphing and .

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The graph of crosses the -axis at

-0.7 and 0.7. There are no digits after the 7,suggesting that the -values are exactly

and , which are rational.

To verify that these numbers are exactsolutions to the equation, we can see if theymake the original equation true.

and , so

are exact solutions.

The graph of , created using graphing technology, is shown to cross the -axis at-2.236 and 2.236. It is unclear if the -coordinates stop at three decimal places or if theycontinue. If they stop or eventually make a repeating pattern, the solutions would be rational.If they never stop or make a repeating pattern, the solutions would be irrational.

We can tell, though, that 2.236 is not an exact solution to the equation. Substituting 2.236 forin the original equation gives , which we can tell is close to 0 but is not exactly 0.

This means are not exact solutions, and the solutions may be irrational.

To be certain whether the solutions are rational or irrational, we can solve the equations.

The solutions to are , which are rational.

The solutions to are , which are irrational. (2.236 is an approximation

of , not equal to .)

What about a solution like , which is a sum of a rational number and an irrational

one? Or a solution like , which is a product of a rational number and an irrational

number? Are they rational or irrational?

We will investigate solutions that are sums and products of different types of numbers in anupcoming lesson.


StatementDecide whether each number is rational or irrational.

10 -3

•

•


SolutionOnly and are irrational. The rest are rational numbers.

Problem 2StatementHere are the solutions to some quadratic equations. Select all solutions that are rational.

A.

B.

C.

D.

E.

F.

Solution["A", "B", "C", "E"]

Problem 3StatementSolve each equation. Then, determine if the solutions are rational or irrational.

a.

b.

c.

d.

Solutiona. or , rational

b. or , rational

c. , irrational

d. , irrational


Problem 4StatementHere is a graph of the equation .

a. Based on the graph, what are the solutions to theequation ?

b. Can you tell whether they are rational or irrational? Explain how you know.

c. Solve the equation using a different method and say whether the solutions are rationalor irrational. Explain or show your reasoning.

Solutiona. 2.778 and 3.222

b. It is hard to tell because those numbers do not seem to be the decimal form of a familiarnumber like 2.5 or 3.25.

c. This equation is the same as . The solutions are or . The solutions

are rational because is a perfect square.

Problem 5StatementMatch each equation to an equivalent equation with a perfect square on one side.

A.

B.

C.

D.

E.

F.

1.

2.

3.

4.

5.

6.

Solution


A: 2

B: 5

C: 1

D: 4

E: 3

F: 6


Problem 6StatementTo derive the quadratic formula, we can multiply by an expression so thatthe coefficient of a perfect square and the coefficient of an even number.

a. Which expression, , , or , would you multiply by to get startedderiving the quadratic formula?

b. What does the equation look like when you multiply both sides byyour answer?

Solutiona.

b.


Problem 7StatementHere is a graph the represents .

On the same coordinate plane, sketch andlabel the graph that represents each equation:

a.

b.


Solution


Problem 8StatementWhich quadratic expression is in vertex form?

A.

B.

C.

D.


Problem 9StatementFunction is defined by the expression .

a. Evaluate .

b. Explain why is undefined.

c. Give a possible domain for .

Solutiona. When the input is 12, the output is 0.5.

b. . Dividing a number by 0 is undefined.

c. Sample response: All real numbers except 2.



Lesson 21: Sums and Products of Rational andIrrational Numbers

GoalsExplain (orally and in writing) why the product of a non-zero rational and irrational number isirrational.

Explain (orally and in writing) why the sum of a rational and irrational number is irrational.

Explain (orally and in writing) why the sum or product of two rational numbers is rational.

Lesson NarrativeThis is the second of two lessons that explore the properties of rational and irrational numbers. Inthe first lesson, students classified solutions to quadratic equations as rational or irrational. Alongthe way, they noticed that some solutions are expressions that combine—by addition ormultiplication—two numbers of different types: one rational and the other irrational. They beganexperimenting with concrete examples to find out whether the sums and products are rational orirrational.

In this lesson, students develop logical arguments that can be used to explain why the sums andproducts of rational and irrational numbers are one type or the other (MP6). Constructing logicalarguments encourages students to pay attention to precision (MP6).

Technology isn’t required for this lesson, but there are opportunities for students to choose to useappropriate technology to solve problems. We recommend making technology available.

Alignments

Building On

7.NS.A.1: Apply and extend previous understandings of addition and subtraction to add andsubtract rational numbers; represent addition and subtraction on a horizontal or verticalnumber line diagram.

Addressing


HSN-RN.B: Use properties of rational and irrational numbers.


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Building Towards







Think Pair Share


Let’s make convincing arguments about why the sums and products of rational andirrational numbers always produce certain kinds of numbers.

21.1 Operations on IntegersWarm Up: 5 minutesThis activity prompts students to recognize that the sum of any two integers is always an integerand that the product of any two integers is also always an integer. Students will not be justifying in aformal way as to why these properties are true. (In a future course, when studying polynomials,students will begin considering integers as a closed system under addition, subtraction, andmultiplication.) Some students may simply assert that these are true because they are not able tofind two integers that add up to or multiply to make a non-integer. Others may reason that:

It seems impossible to add two whole amounts and end up with a partial amount in the sum.

It seems impossible to multiply whole-number groups of whole numbers to make a non-wholenumber quantity.

If we use the number line to represent addition of two integers, we will start at an integer andtake whole-number steps, so we cannot end up at a point between integer values.

These are valid conclusions at this stage. Later in the lesson, students will use these conclusions toreason about the sums and products of rational numbers.

Building On

7.NS.A.1

Building Towards

HSN-RN.B.3

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Think Pair Share

Launch

Arrange students in groups of 2. Ask them to think quietly for a couple of minutes before discussingtheir thinking with a partner.


Some students might persist in attempting to find examples of two integers that add up to anon-integer. Rather than giving them more time to find examples, encourage them to think aboutthe placement of integers on a number line and what it might imply about the sum or product ofany two integers.


Here are some examples of integers (positive or negative whole numbers):

-25 -10 -2 -1 0 5 9 40

1. Experiment with adding any two numbers from the list (or other integers of yourchoice). Try to find one or more examples of two integers that:

a. add up to another integer

b. add up to a number that is not an integer

2. Experiment with multiplying any two numbers from the list (or other integers of yourchoice). Try to find one or more examples of two integers that:

a. multiply to make another integer

b. multiply to make a number that is not an integer

Student Response

1. Sample response:a. , , and any sum of two integers.

b. no examples found

2. Sample response:a. , , and any product of two integers.

b. no examples found

Activity Synthesis

Select students or groups to share their examples and their challenges in finding examples for thesecond part of each question. Invite as many possible explanations as time permits as to why they

•


could not find examples of two integers that add or multiply to make a number that is not aninteger. If no students brought up reasons similar to those listed in the Activity Narrative, askstudents to consider them.

Explain that while we have not proven that two integers can never produce a sum or a product thatis not an integer, for now we will accept this to be true.

21.2 Sums and Products of Rational Numbers15 minutesIn this activity, students use logical reasoning to develop a general argument about why the sumand product of two rational numbers are also rational numbers.

Students begin by studying some numerical examples of addition of two rational numbers andarticulating why the sums are rational numbers. Next, they reason more abstractly about rationalnumbers in the form of and (MP2). Students consolidate what they know about the sum and

product of integers (that these are also integers) and about rational numbers (that these arefractions with integers in the numerator and in the denominator) to argue that the result of addingor multiplying any two rational numbers must be another rational number (MP3).

Addressing

HSN-RN.B.3



Think Pair Share

Launch

Remind students that rational numbers are fractions and their opposites. Ask students what kindsof numbers the numerator and denominator in a fraction are. (They are integers and thedenominator is not 0.)

Display the four addition expressions in the first question for all to see. Ask students, “How do weknow that the numbers being added are each a rational number?” (They can be written as a positiveor negative fraction.)

Arrange students in groups of 2. Give students a moment to think quietly about the first questionand then time to discuss their thinking with their partner. Pause for a class discussion beforestudents proceed to the rest of the questions.

Make sure students understand how leads to and can explain why this sum is a

rational number. Remind students of a conclusion from the warm-up: that the sum of two integersis always an integer. Because is an integer, must be an integer, and is therefore a

fraction.

•

••



Some students may not recognize 0.175, 4.175, and -0.75 as rational numbers. Demonstrate thatthese numbers can be written as the fractions , , and (or ).


1. Here are a few examples of adding two rational numbers. Is each sum a rationalnumber? Be prepared to explain how you know.

a.

b.

c.

d. is an integer:

2. Here is a way to explain why the sum of two rational numbers is rational.

Suppose and are fractions. That means that and are integers, and and

are not 0.

a. Find the sum of and . Show your reasoning.

b. In the sum, are the numerator and the denominator integers? How do you know?

c. Use your responses to explain why the sum of is a rational number.

3. Use the same reasoning as in the previous question to explain why the product of tworational numbers, , must be rational.

Student Response

1. All sums are rational. Sample explanations:

a. 4.175 can be written as .

b. is a fraction and 13 and 10 are both integers.

c. 1 can be written as a fraction with the same integer for both numerator anddenominator.

d. is an integer because the sum of two integers is an integer. The denominator isalso an integer, and is a fraction.

2. Sample response:a.


b. The numerator and denominator are both integers. The product of two integers is aninteger, so , and are all integers. The sum of two integers is an integer, so

is an integer.

c. The sum of is a rational number because it can be written as a fraction with an

integer in the numerator and in the denominator, and the denominator is not 0.

3. Sample response: . , and are all integers. Because the products of integers

are also integers, both the numerator and denominator of the product are integers, so theproduct is a fraction.


Consider numbers that are of the form , where and are whole numbers. Let’s callsuch numbers quintegers.

Here are some examples of quintegers:

( , )

( , )

( , )

3 ( , ).

1. When we add two quintegers, will we always get another quinteger? Either prove this, orfind two quintegers whose sum is not a quinteger.

2. When we multiply two quintegers, will we always get another quinteger? Either provethis, or find two quintegers whose product is not a quinteger.

Student Response

1. Yes. Sample response: . Because , , , andare integers, and must also be integers.

2. Yes. Sample response:

Because , , , and are integers, and must also be integers.

Activity Synthesis

The second question guides students through the pieces needed to make an argument that thesum of two rational numbers must be rational. Use the discussion to help students consolidatethese pieces into a logical and coherent argument:

If and are fractions, then its sum is . The expressions , and must be

integers because they are each a product of integers, and consequently must also be

••

••

•


an integer. Because the numerator and the denominator are both integers, the number is afraction and is therefore rational.

Make sure students see how to construct a similar argument for the product of two rationalnumbers, as shown in the student response.


Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. At theappropriate time, give students 2–3 minutes to plan what they will say when they share theirargument for why is a rational number. Encourage students to consider what details are

important to share and to think about how they will explain their reasoning usingmathematical language. Ask students to rehearse, which provides additional opportunities tospeak and clarify their thinking, and will improve the quality of explanations shared during thewhole-class discussion.Design Principle(s): Support sense-making; Maximize meta-awareness

21.3 Sums and Products of Rational and IrrationalNumbers15 minutesIn the previous activity, students justified why the sum and product of two rational numbers arealso rational. In this activity, they develop an argument to explain why the sum and product of arational number and an irrational number are irrational.

Students encounter an argument by contradiction. They learn that we can first assume that thesum is rational. If this is indeed true, when this sum is added to anotherrational number, the result must also be rational. It turns out that this is not the case (adding thissum to a certain rational number produces the original irrational number), which means that thesum cannot be rational.

In the last part of the activity, students make a similar argument for why the product of a rationalnumber and an irrational number is irrational.

Addressing

HSN-RN.B.3



•

•


Launch

Keep students in groups of 2. Ask students to think quietly about the first question beforeconferring with their partner.


Action and Expression: Develop Expression and Communication. To help get students started,display sentence frames that support the process of building an argument. First, invitestudents to respond using frames that explore their initial reaction to the given statements,such as: “We can agree that . . . .” and “If _____ then _____ because . . . .” Then, encouragestudents to use frames that support creating generalizations and push them to explore anyexceptions to their argument, such as: “_____ will always _____ because . . . .” and “Is it alwaystrue that . . . ?”Supports accessibility for: Language; Organization


1. Here is a way to explain why is irrational.

Let be the sum of and , or .

Suppose is rational.

a. Would be rational or irrational? Explain how you know.

b. Evaluate . Is the sum rational or irrational?

c. Use your responses so far to explain why cannot be a rational number, and

therefore cannot be rational.

2. Use the same reasoning as in the earlier question to explain why is irrational.

Student Response

1. Sample response:a. Rational, because is rational and the sum of two rational numbers is rational.

b. . The sum is irrational.

c. The sum of cannot be both rational and irrational, so cannot be rational, which

means cannot be rational.

◦◦


2. Sample response: Let . Suppose is rational. would be rational, because the

product of two rational numbers is rational. Evaluating gives , which is

irrational. ( ). cannot be both rational and irrational, so or

must not be rational. It must be irrational.

Activity Synthesis

As in the previous activity, students are guided through the pieces needed to make a particularargument—that the sum of a rational number and an irrational number must be irrational. Makesure students can consolidate these pieces into a logical and coherent argument:

Let be the sum of (rational) and (irrational).

Suppose is rational. Adding and must produce a rational number because the sum of

two rational values is rational.

Adding and also gives or , which is irrational.

The sum of and cannot be both rational and irrational, so the assumption that is

rational must be false. So (the sum of a rational number and an irrational number) must beirrational.

Make sure students see how to construct a similar argument for the product of a rational numberand an irrational number, as shown in the Student Response.


Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to preparestudents for the whole-class discussion. Give students time to meet with 2–3 partners to shareand get feedback on their responses to the last question. Display feedback prompts that willhelp students strengthen their ideas and clarify their language. For example, "Your explanationtells me . . .", "Can you say more about why . . . ?", and "A detail (or word) you could add is _____,because . . . ." Give students with 3–4 minutes to revise their initial draft based on feedbackfrom their peers. This will help students refine their justifications for why the product of twoirrational numbers cannot be rational.Design Principle(s): Optimize output (for explanation); Cultivate conversation

21.4 Equations with Different Kinds of SolutionsOptional: 30 minutesIn this optional activity, students investigate how the parameters in a quadratic equation affect thenumber of solutions and the kinds of solutions. Students are first given the equation

and are asked to find a value of that would produce a certain number or kind of

••

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solution. To do so, students may take different approaches, each with different degrees ofefficiency.

Monitor for these strategies, from less efficient to more efficient:

Substitute different integers for and calculate the solutions by rewriting in factored form (ifpossible), completing the square, or using the quadratic formula.

Enter the equation in a graphing tool, graph the equation at different integervalues of , and identify the -intercepts (and notice when there is only one or when there arenone).

Use a spreadsheet tool to calculate the solutions at various values of (using the quadraticformula) and choose the values of accordingly.

Students then study their findings and make general observations about which values of produce0, 1, or 2 solutions. Making generalization about the connections between and the number ofsolutions encourages students to look for and make use of structure (MP7).

The last question is an open-ended task and may be challenging. It prompts students to writeoriginal equations that produce specified solutions. To do so effectively and efficiently, studentsneed to make use of structure and any productive strategies seen in the first question and in pastwork (MP7) and to persevere (MP1). If time is limited and if desired, consider returning to thisquestion at another time.

Depending on the strategy used, equations with rational solutions and those with single solutionsmight be harder to write than those with irrational solutions or no solutions. (The former requiresmaking use of structure, while the latter could be achieved by choosing , and somewhatrandomly for and checking by graphing that the zeros appear irrational or arenon-existent.) If desired, differentiate the work by assigning certain questions to certain students orgroups.

Making graphing and spreadsheet technology available gives students an opportunity to chooseappropriate tools strategically (MP5).

Addressing

HSA-REI.B.4.b

HSN-RN.B




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Launch

Tell students that the numbers in a quadratic equation affect the type of solutions and the numberof solutions. Display for all to see. Assign each student an integer between -8 and 8(or another range that includes positive and negative values). Give students a moment to solve theequation with their assigned . Remind students that the solutions can be found by rewriting theequation in factored form, completing the square, using the quadratic formula, or graphing.

Ask students to report how many solutions they found at different values of and what types ofnumbers the solutions are. Record and organize their findings in a table such as this one, displayedfor all to see:

number of solutions rational or irrational?

-8 two irrational

-7 two irrational

...

-2 none

-1 none

0 none

1 none

2 none

3 none

4 one rational

5 two rational

6 two irrational

Tell students that their job in this activity is to write and solve quadratic equations such that eachequation has a particular kind or a particular number of solutions, and to think more generallyabout how the numbers in the equation relate to the solutions.

Arrange students in groups of 2–4. Encourage group members to collaborate and find differentvalues of in the first question so that they have a greater set of data to help them answer thesecond question. Ask them to consider helpful ways to collect and organize their findings and tothink about efficient ways to find the right values beyond trying different values of .



1. Consider the equation . Find a value of so that the equation has:

a. 2 rational solutions

b. 2 irrational solutions

c. 1 solution

d. no solutions

2. Describe all the values of that produce 2, 1, and no solutions.

3. Write a new quadratic equation with each type of solution. Be prepared to explain howyou know that your equation has the specified type and number of solutions.

a. no solutions

b. 2 irrational solutions

c. 2 rational solutions

d. 1 solution

Student Response

1. Sample responses:a. 15, 20

b. 14, -16

c. 12, -12

d. 10, 3

2. For 2 solutions: or . For 1 solution: . For no solutions: .

3. Sample responsesa.

b.

c. or

d. or

Activity Synthesis

Select students or groups to share their strategies for finding the right values for and for knowingwhich values of would produce certain kinds or numbers of solutions.


Ask them to present in the order listed in the Activity Narrative. If only guessing and checking ismentioned by students, bring up at least one other strategy. Connect these strategies to whatstudents learned about the equations and graphs of quadratic functions, for instance:

The solutions to a quadratic equation are represented by the -intercepts ofthe graph of . Graphing can be a handy way to see when an equation has onesolution or no solution. It is less handy for telling if the solutions are rational or irrational.

If a quadratic equation can be written as , we can tell that the solutions willbe rational (provided that the factors do not contain irrational numbers).

A quadratic equation that is of the form has two factors that are the sameexpression, which tells us that there is only one solution.

If time permits, consider demonstrating how graphing or spreadsheet technology could be used tohelp us spot patterns and suggest which values of , and in a quadratic equation lead to certainsolutions.


Representing, Conversing: MLR7 Compare and Connect. Prior to the whole-class discussion, invitestudents to create a visual display that shows their strategies for finding the right values forand for knowing which values of would produce certain kinds or numbers of solutions.Students should consider how to represent their strategy so that other students will be able tounderstand their solution method. Students may wish to add notes or details to their displaysto help communicate their reasoning and thinking. Begin the whole-class discussion byselecting and arranging 2–4 displays for all to see. Give students 1–2 minutes of quiet thinktime to interpret the displays before inviting the authors to share their strategies as describedin the activity narrative.Design Principle(s): Cultivate conversation; Maximize meta-awareness

Lesson SynthesisDisplay a list of sums and products, such as shown:

, where and are integers and is not zero. The sum is rational.

, where , , and are all integers and is not zero. The product is rational.

. The product is irrational.

. The sum is irrational.

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Ask students to choose a sum or a product (or one of each). Then, ask them to practice explainingto 2–3 different partners why the sum or product must be rational or irrational, and to try makingtheir explanation clearer and stronger each time.

21.5 Adding Irrational NumbersCool Down: 5 minutesAddressing

HSN-RN.B.3


Someone says that if you add two rational numbers, and , where all variables are

integers and is not zero, the sum could be rational or irrational.

How could you convince the person that this is not true?

Student Response

Sample response: . We know that and are both integers (so long as

is not 0) and that the sum and product of two integers is also an integer. This makes and

both integers, and it makes a fraction, which is rational.

Student Lesson SummaryWe know that quadratic equations can have rational solutions or irrational solutions. Forexample, the solutions to are -3 and 1, which are rational. The solutions to

are , which are irrational.

Sometimes solutions to equations combine two numbers by addition or multiplication—for

example, and . What kind of number are these expressions?

When we add or multiply two rational numbers, is the result rational or irrational?

The sum of two rational numbers is rational. Here is one way to explain why it is true:

Any two rational numbers can be written and , where are

integers, and and are not zero.

The sum of and is . The denominator is not zero because neither nor

is zero.

Multiplying or adding two integers always gives an integer, so we know thatand are all integers.

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◦

◦


If the numerator and denominator of are integers, then the number is a

fraction, which is rational.

The product of two rational numbers is rational. We can show why in a similar way:

For any two rational numbers and , where are integers, and

and are not zero, the product is .

Multiplying two integers always results in an integer, so both and areintegers, so is a rational number.

What about two irrational numbers?

The sum of two irrational numbers could be either rational or irrational. We can showthis through examples:

and are each irrational, but their sum is 0, which is rational.

and are each irrational, and their sum is irrational.

The product of two irrational numbers could be either rational or irrational. We canshow this through examples:

and are each irrational, but their product is or 4, which is rational.

and are each irrational, and their product is , which is not a perfectsquare and is therefore irrational.

What about a rational number and an irrational number?

The sum of a rational number and an irrational number is irrational. To explain whyrequires a slightly different argument:

Let be a rational number and an irrational number. We want to show thatis irrational.

Suppose represents the sum of and ( ) and suppose is rational.

If is rational, then would also be rational, because the sum of two rationalnumbers is rational.

is not rational, however, because .

cannot be both rational and irrational, which means that our originalassumption that was rational was incorrect. , which is the sum of a rationalnumber and an irrational number, must be irrational.

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•

◦

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The product of a non-zero rational number and an irrational number is irrational. Wecan show why this is true in a similar way:

Let be rational and irrational. We want to show that is irrational.

Suppose is the product of and ( ) and suppose is rational.

If is rational, then would also be rational because the product of two

rational numbers is rational.

is not rational, however, because .

cannot be both rational and irrational, which means our original assumption

that was rational was false. , which is the product of a rational number and anirrational number, must be irrational.


StatementMatch each expression to an equivalent expression.

A.

B.

C.

D.

E.

1. 3 and 7

2. and

3. -6 and 0

4. and

5. and

Solution

A: 2

B: 5

C: 4

D: 1

E: 3


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Problem 2StatementConsider the statement: "An irrational number multiplied by an irrational number alwaysmakes an irrational product."

Select all the examples that show that this statement is false.

A.

B.

C.

D.

E.

F.

G.

Solution["C", "D", "F"]

Problem 3Statement

a. Where is the vertex of the graph that represents ?

b. Does the graph open up or down? Explain how you know.

Solutiona.

b. The graph opens up, because the squared term has a positive coefficient of 1.


Problem 4StatementHere are the solutions to some quadratic equations. Decide if the solutions are rational orirrational.


Solution

and are irrational. The others are rational.

Problem 5StatementFind an example that shows that the statement is false.

a. An irrational number multiplied by an irrational number always makes an irrationalproduct.

b. A rational number multiplied by an irrational number never gives a rational product.

c. Adding an irrational number to an irrational number always gives an irrational sum.

SolutionSample responses:

a.

b.

c.

Problem 6StatementWhich equation is equivalent to but has a perfect square on one side?

A.

B.

C.

D.

SolutionC



Problem 7StatementA student who used the quadratic formula to solve said that the solutions are

and .

a. What equations can we graph to check those solutions? What features of the graphdo we analyze?

b. How do we look for and on a graph?

Solutiona. Sample responses:

We can graph and find the -intercepts because their -coordinates aresolutions to the equation .

We can graph and and find the -coordinates of the intersectionpoints because they are solutions to the equation .

b. Sample response: We find the decimal approximations of and , whichare (to 3 decimal places) 4.236 and -0.236. Then, we see if the points where the graph crossesthe -axis (or crosses , if graphing and ) have roughly thosedecimal values for the -coordinates.


Problem 8StatementHere are 4 graphs. Match each graph with a quadratic equation that it represents.

▪

▪


Graph A Graph B

Graph C Graph D

A. Graph A

B. Graph B

C. Graph C

D. Graph D

1.

2.

3.

4.

Solution

A: 4

B: 2

C: 3

D: 1



Lesson 22: Rewriting Quadratic Expressions inVertex Form

GoalsAnalyze and explain (orally and in writing) the steps for completing the square and understandhow they transform a quadratic expression from standard to vertex form.

Identify the vertex of a graph of a quadratic function when the expression that defines it is invertex form.

Write equivalent quadratic expressions in vertex form by completing the square.

Learning TargetsI can identify the vertex of the graph of a quadratic function when the expression that definesit is written in vertex form.

I know the meaning of the term “vertex form” and can recognize examples of quadraticexpressions written in this form.

When given a quadratic expression in standard form, I can rewrite it in vertex form.

Lesson NarrativePreviously, students used completing the square to rewrite quadratic expressions as perfectsquares so they could solve equations. They also completed the square to derive the quadraticformula, which makes it possible to solve any quadratic equation. In this lesson, students encounteranother use for completing the square—it can be used to rewrite a quadratic expression fromstandard form to vertex form.

Students explored the vertex form in a previous unit on quadratic functions. The lesson begins witha review of the form, its advantage, and its connections to the graph. Then, students recall how totransform expressions in vertex form into standard form, and then experiment with transformingthe same expressions back to vertex form. Students notice that to take an expression fromstandard form to vertex form is essentially to complete the square, while being careful not tochange the value of the expression.

To transform expressions into vertex form, students need to look for and make use of structure(MP7).

Alignments

Building On


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Addressing


.

HSA-SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explainproperties of the quantity represented by the expression.

HSA-SSE.B.3.b: Complete the square in a quadratic expression to reveal the maximum orminimum value of the function it defines.

HSF-IF.C: Analyze functions using different representations.

Building Towards



Graph It

MLR4: Information Gap Cards

Notice and Wonder

Think Pair Share

Required Materials

Graphing technologyExamples of graphing technology are:a handheld graphing calculator, a computer witha graphing calculator application installed, andan internet-enabled device with access to a sitelike desmos.com/calculator or geogebra.org/graphing. For students using the digitalmaterials, a separate graphing calculator tool

isn't necessary; interactive applets areembedded throughout, and a graphingcalculator tool is accessible on the studentdigital toolkit page.

Pre-printed slips, cut from copies of theblackline master




Let’s see what else completing the square can help us do.

22.1 Three Expressions, One FunctionWarm Up: 5 minutes

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•


This warm-up reminds students about features of the graph that are visible in the different forms ofexpressions defining a quadratic function.

Building On

HSF-IF.C.7.a


These expressions each define the same function.

Without graphing or doing any calculations, determine where the following features would beon a graph that represents the function.

1. the vertex

2. the -intercepts

3. the -intercept

Student Response

1. vertex:

2. -intercepts: and

3. -intercept:

Activity Synthesis

Invite students to share how they would locate the specified features on a graph. Make surestudents are reminded that:

The constant term in the standard form tells us the -intercept.

The factored form shows us the -intercepts.

The vertex form reveals the vertex.

Consider using graphing technology to demonstrate that the three expressions appear to producethe same graph. (We can verify algebraically that the three expressions define the same function,but we can’t be sure that the three expressions define the same function just by looking at thegraph.) Label the vertex, -intercepts, and -intercept.

22.2 Back and Forth15 minutesThe goal of this activity is for students to recognize rewriting a quadratic expression from standardform to vertex form essentially entails completing the square.

•

•••


Students had quite a bit of experience completing the square, mostly in the context of solvingequations, in which they know to add or subtract the same number from both sides of an equationto keep the equation true. Here students are dealing with expressions and need to be careful tokeep each expression equivalent to the original. If they add a number to the expression, they needto remember to subtract the same number to keep the value of the expression unchanged.

Addressing

HSA-SSE.A.2

Building Towards

HSA-SSE.B.3.b


Graph It

Think Pair Share

Launch

Arrange students in groups of 2. Give students quiet time to think about the first two questions andthen time to share their thinking with a partner.

Pause for a class discussion after the second question and invite one or more students todemonstrate their strategy. Highlight approaches that involve completing the square. If no studentstook that approach, display and discuss the following:

Then, ask students to proceed with the rest of the activity. Provide access to devices that can runDesmos or other graphing technology.


Representation: Internalize Comprehension. Use color and annotations to illustrate studentthinking. As students share their reasoning about how to convert standard form back intovertex form, scribe their thinking on a visible display. Use annotations and labels to show thestructure of the steps using the form included in the launch. For example, labeling the first stepas “standard form,” labeling +9 as “add nine to complete the square,” and drawing an arrow tothe -9 and labeling it as “keeping it equivalent.” Continue to add in any key phrases studentscontributed that illustrate the steps.Supports accessibility for: Visual-spatial processing; Conceptual processing

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••



Some students may wonder why they need to both add and subtract a number from the standardform expression in order to complete the square. In previous lessons, students completed thesquare while solving an equation, often adding the same number to each side of the equal sign tomaintain equality. Here, there is no equal sign. Emphasize that each move must generate anequivalent expression. Show a few expressions such as 5, , , , , and

. Ask them which ones are equal to 5. Next, ask them to write 2 more expressions thatinclude the number 5 and also equal 5. Encourage students to notice that the sum of the numbersexcluding 5 must be 0.


1. Here are two expressions in vertex form. Rewrite each expression in standard form.Show your reasoning.

a.

b.

2. Think about the steps you took, and about reversing them. Try converting one or bothof the expressions in standard form back into vertex form. Explain how you go aboutconverting the expressions.

3. Test your strategy by rewriting in vertex form.

4. Let’s check the expression you rewrote in vertex form.

a. Use graphing technology to graph both and your new expression.Does it appear that they define the same function?

b. If you convert your expression in vertex form back into standard form, do you get?

Student Response

1. Each expression in standard form:a.

b.

2. Sample response: To convert back to vertex form, I can complete the square byadding 9 and then subtracting 9 so that the expression remains unchanged. The part that is aperfect square, , can be written as , and then there is the , whichequals -7.

3.

4. Sample response:a. Yes, both expressions seem to produce the same graph.


b. , which is .

Activity Synthesis

Invite students to share their graph and response to the last question. Emphasize that whilegraphing is a quick way to check whether two expressions define the same function, it is not alwaysreliable. If we make an algebraic error, the graph would almost certainly show it. But having twographs that appear to be identical does not prove that two expressions are indeed equivalent. We canonly be sure that two expressions define the same function by showing equivalence algebraically.For example, when students convert the expression in vertex form back to standard form, does itproduce the original expression?

Make sure students understand that whatever operation is performed on an expression tocomplete the square, it should not change the value of the expression. Adding opposites (forexample, 9 and -9, or -25 and 25), or adding and subtracting the same number, has the effect ofadding 0, which keeps the original and the transformed expressions equivalent.

22.3 Inconvenient Coefficients15 minutesIn the first activity, students rewrote quadratic expressions whose squared term has a coefficient of1 into vertex form. They did so by completing the square. In this activity, they transformexpressions whose squared term has a coefficient other than 1 into vertex form.

Students learn that one way to deal with an inconvenient in is to rewrite theexpression as a product of and an expression (which now has a leading coefficient of 1), completethe square for the latter, and redistribute afterward to obtain an equivalent expression in vertexform.

Addressing

HSA-SSE.A.2

HSA-SSE.B.3.b


Notice and Wonder

Launch

Keep students in groups of 2. Display these expressions for all to see and explain that these aresome other expressions to rewrite in vertex form so that we could identify the vertex of their graph.

••

•


Give students 1 minute of quiet think time and ask them to be prepared to share at least one thingthey notice and one thing they wonder.

Students may notice:

In all of the expressions, the coefficient of the squared term is not 1.

Each of the expressions contains terms that have a common factor.

The coefficients of some squared terms are negative.

The last expression has no constant term.


How to write these expressions in vertex form.

Whether it is still possible to write the last expression in vertex form given that it has noconstant term.

Invite a few students to share what they noticed and wondered, and then to begin the activity.

Pause for a class discussion after the first question. Display the worked example. Ask students toshare their explanation for each step and record their explanation for all to see. Make sure studentsunderstand the rationale for each step and how to check that their expression is equivalent to theoriginal (by converting it back into standard form).

Students have learned in an earlier unit that a positive in means an upward-openinggraph and may offer this explanation for the last part of the question. It is not necessary to focus onthe direction of the opening of the parabola here, as students will explore it further in an upcominglesson.


Engagement: Internalize Self Regulation. Chunk this task into more manageable parts todifferentiate the degree of difficulty or complexity. Allow students to discuss the steps andthen provide students with printed slips of explanations from the student responses. Dividethe slips between partners and encourage them to work together to identify which slipmatches with each step. Allow them to keep the steps nearby and put them next to their stepsas they check their work and use them as a guide.Supports accessibility for: Organization; Attention


1. a. Here is one way to rewrite in vertex form. Study the steps and writea brief explanation of what is happening at each step.

••••

••


b. What is the vertex of the graph that represents this expression?

c. Does the graph open upward or downward? Explain how you know.

2. Rewrite each expression in vertex form. Show your reasoning.

a.

b.

c.

Student Response

1. a. Sample response:

b.

c. Opens upward. Sample reasoning: In an earlier unit, we saw that when the coefficient ofthe squared term is positive, the graph opens upward.

2. a.


b.

c.


1. Write in vertex form without completing the square. (Hint: Thinkabout finding the zeros of the function.) Explain your reasoning.

2. Write in vertex form without completing the square. Explainyour reasoning.

Student Response

1. . Sample explanation: The zeros of are 3 and 9. Because the graph of aquadratic function has a vertical line of symmetry across the vertex, the vertex of the graphmust have -coordinate 6. The -coordinate of the vertex is -18 because . Thismeans can be expressed as . If we expand , we can see thatthe coefficient of is 2. This means that in the vertex form has to be 2 as well.

2. . Sample reasoning: Although 3 and 9 are not zeros of , when is 3 andwhen is 9, is 21, or . Because the graphs of quadratic functions aresymmetric, the -coordinate of the vertex is still 6. The -coordinate of the vertex is 3 because

. The value of is 2 for the same reason as in function .

Activity Synthesis

Select students to display their responses for all to see. Discuss questions such as:

“How did you know what factor to use to rewrite the original expression?” (Find a factor that allterms in an expression have in common. Use the coefficient of the squared term so that, afterthe expression is rewritten, the squared term has a coefficient of 1.)

“In , both 2 and 4 are common factors. Is one number better than the other forour purposes here?” (4 is a better factor to use because it allows the squared term to have acoefficient of 1, which makes it easier to complete the square. If we use 2, the squared termstill has an inconvenient coefficient that is neither 1 nor a perfect square.)

“How can we check if the expression in vertex form is equivalent to the original expression?”(One way is to convert it back into standard form and see if it is the same expression as theoriginal.)

22.4 Info Gap: Features of FunctionsOptional: 20 minutesThis optional Info Gap activity gives students an opportunity to determine and request theinformation needed to write expressions that define quadratic functions with certain graphicalfeatures. To do so, students need to consider what they learned about the structure of quadraticexpressions in various forms (MP7).

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•


The Info Gap structure requires students to make sense of problems by determining whatinformation is necessary, and then to ask for information they need to solve it. This may takeseveral rounds of discussion if their first requests do not yield the information they need (MP1). Italso allows them to refine the language they use and ask increasingly more precise questions untilthey get the information they need (MP6).

Because students are expected to make use of structure and construct logical arguments abouthow the structure helps them write expressions, technology is not an appropriate tool.

Here is the text of the cards for reference and planning. Note that the questions on card 1 havemany possible correct answers, but no possible expressions for and can define the samefunction.

Addressing

HSA-SSE.A.2

HSA-SSE.B.3

HSF-IF.C


MLR4: Information Gap Cards

Launch

Tell students they will continue to write expressions in different forms that define quadraticfunctions. Explain the info gap structure, and consider demonstrating the protocol if students areunfamiliar with it.

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•


Arrange students in groups of 2. In each group, distribute a problem card to one student and a datacard to the other student. After you review their work on the first problem, give them the cards fora second problem and instruct them to switch roles.

Since this activity was designed to be completed without technology, ask students to put away anydevices.


Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussinginformation necessary to solve problems involving features of quadratic functions. Displayquestions or question starters for students who need a starting point such as: “Can you tell me. . . (specific piece of information)”, and “Why do you need to know . . . (that piece ofinformation)?"Design Principle(s): Cultivate Conversation


Engagement: Develop Effort and Persistence. Display or provide students with a physical copy ofthe written directions. Check for understanding by inviting students to rephrase directions intheir own words. Keep the display of directions visible throughout the activity.Supports accessibility for: Memory; Organization


Your teacher will give you either a problem card or a data card. Do not show or read yourcard to your partner.


If your teacher gives you the data card:

1. Silently read the information on yourcard.

2. Ask your partner “What specificinformation do you need?” and wait foryour partner to ask for information. Onlygive information that is on your card. (Donot figure out anything for your partner!)

3. Before telling your partner theinformation, ask “Why do you need toknow (that piece of information)?”

4. Read the problem card, and solve theproblem independently.

5. Share the data card, and discuss yourreasoning.

If your teacher gives you the problem card:

1. Silently read your card and think aboutwhat information you need to answerthe question.

2. Ask your partner for the specificinformation that you need.

3. Explain to your partner how you areusing the information to solve theproblem.

4. When you have enough information,share the problem card with yourpartner, and solve the problemindependently.

5. Read the data card, and discuss yourreasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cardsand repeat the activity, trading roles with your partner.

Student Response

Card 1:

1. Possible response:

2. Possible response:

3. Rewriting in standard form gives , but is .

Card 2:

1. The zeros of are 3 and 7.

2. The vertex of the graph of is .

3. Rewriting in standard form gives , and in standard form is .

Activity Synthesis

After students have completed their work, discuss the correct answers to the questions and anydifficulties that came up.

Highlight for students that different forms of quadratic expressions are useful in different ways, soit helps to be able to move flexibly across forms. For example, a quadratic expression in factoredform makes it straightforward to tell the zeros of the function that the expression defines and the

-intercepts of its graph. The vertex form makes it easy to identify the coordinates of the vertex of agraph of function.


To know whether two expressions define the same function, we can rewrite the expression in anequivalent form. There are many tools at our disposal. For instance, we can rewrite an expressioninto factored form, apply the distributive property to expand a factored expression, rearrange partsof an expression, combine like terms, or complete the square.

Lesson SynthesisDisplay the equation for all to see. Ask students to think about features ofa graph of function , as many as they can, without creating a graph. To scaffold this work, considerdisplaying the list of features of a graph.

-intercept

-intercepts

coordinates of vertex

Solicit responses from students on how they found each feature of a graph of .

Highlight that the -intercepts and the coordinates of the vertex aren’t easy to spot in the standardform of the expression as given, but we have techniques to rewrite the expression in other forms.

To rewrite in factored form, we can use the distributive property twice:

Once the expression is in factored form, we can tell that the -intercepts of a graph of are atand .

To rewrite in vertex from, we can complete the square:

Once the expression is in vertex form, we can tell that the vertex of a graph of is at

Time permitting, demonstrate that with all of this information, we can now sketch a reasonablegraph of by hand.

22.5 Rewrite This ExpressionCool Down: 5 minutes

•••

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Addressing

HSA-SSE.B.3.b

Launch

Ask students to put away their devices for the cool-down.


A quadratic function is defined by the expression .

Rewrite the expression in vertex form and give the coordinates of the vertex. Show yourreasoning.

Student Response

. The vertex is at .

Student Lesson SummaryRemember that a quadratic function can be defined by equivalent expressions in differentforms, which enable us to see different features of its graph. For example, these expressionsdefine the same function:

From factored form, we can tell that the-intercepts are and .

From standard form, we can tell that the-intercept is .

From vertex form, we can tell that thevertex is .

Recall that a function expressed in vertex form is written as: . The values ofand reveal the vertex of the graph: are the coordinates of the vertex. In this example,

is 1, is 5, and is -4.

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If we have an expression in vertex form, we can rewrite it in standard form by using thedistributive property and combining like terms.

Let’s say we want to rewrite in standard form.

If we have an expression in standard form, we can rewrite it in vertex form bycompleting the square.

Let’s rewrite in vertex form.

A perfect square would be , so we need to add 1. Adding 1, however,would change the expression. To keep the new expression equivalent to the originalone, we will need to both add 1 and subtract 1.

Let’s rewrite another expression in vertex form: .

To make it easier to complete the square, we can use the distributive property torewrite the expression with -2 as a factor, which gives .

For the expression in the parentheses to be a perfect square, we need . Wehave 15 in the expression, so we can subtract 6 from it to get 9, and then add 6 again tokeep the value of the expression unchanged. Then, we can rewrite infactored form.

This expression is not yet in vertex form, however. To finish up, we need to apply thedistributive property again so that the expression is of the form :

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When written in this form, we can see that the vertex of the graph representingis .

Glossaryvertex form (of a quadratic expression)


StatementThe following quadratic expressions all define the same function.

Select all of the statements that are true about the graph of this function.

A. The -intercept is .

B. The vertex is .

C. The -intercepts are and .

D. The -intercepts are and .

E. The -intercept is .

F. The -intercept is .

G. The vertex is .

Solution["B", "C", "F"]

Problem 2StatementThe following expressions all define the same quadratic function.

•


a. What is the -intercept of the graph ofthe function?

b. What are the -intercepts of the graph?

c. What is the vertex of the graph?

d. Sketch a graph of the function withoutgraphing technology. Make surethe -intercepts, -intercept, and vertexare plotted accurately.

Solutiona.

b. and

c.

d. See graph.

Problem 3StatementHere is one way an expression in standard form is rewritten into vertex form.


a. In step 1, where did the number come from?

b. In step 1, why was added and then subtracted?

c. What happened in step 2?

d. What happened in step 3?

e. What does the last expression tell us about the graph of a function defined by thisexpression?

SolutionSample responses:

a. is half of -7 being squared.

b. is added to to complete the square. It is subtracted to keep the value of the

expression unchanged.

c. The expression , which is a perfect square, is written as a squared expression.

d. The 6 is written as so that it has a common denominator as , which makes it easier to

subtract.

e. The vertex of the graph is at .

Problem 4StatementRewrite each quadratic expression in vertex form.

a.

b.

c.

Solutiona.

b.

c.


Problem 5Statement

a. Give an example that shows that the sum of two irrational numbers can be rational.

b. Give an example that shows that the sum of two irrational numbers can be irrational.

Solutiona. Sample response: is equal to 0, which is a rational number.

b. Sample response:


Problem 6Statement

a. Give an example that shows that the product of two irrational numbers can be rational.

b. Give an example that shows that the product of two irrational numbers can beirrational.

Solutiona. Sample response: is equal to 1, which is rational.

b. Sample response: which is the same as the approximate value of .


Problem 7StatementSelect all the equations with irrational solutions.


A.

B.

C.

D.

E.

F.

G.

Solution["C", "F", "G"](From Unit 7, Lesson 15.)

Problem 8Statement

a. What are the coordinates of the vertex of the graph of the function defined by?

b. Find the coordinates of two other points on the graph.

c. Sketch the graph of .


Solutiona.

b. Sample response: and

c. See graph.


Problem 9StatementHow is the graph of the equation related to the graph of the equation

?

A. The graph of is the same as the graph of but is shifted 1 unit tothe right and 4 units up.

B. The graph of is the same as the graph of but is shifted 1 unit tothe left and 4 units up.

C. The graph of is the same as the graph of but is shifted 1 unit tothe right and 4 units down.

D. The graph of is the same as the graph of but is shifted 1 unit tothe left and 4 units down.



Lesson 23: Using Quadratic Expressions in VertexForm to Solve Problems

GoalsExplain (orally and in writing) how an expression in vertex form can show whether the vertexof a graph represents the maximum or minimum of a quadratic function.

Rewrite a quadratic expression in vertex form to identify the maximum or minimum value ofthe function the expression defines.

Use the structure of a quadratic expression in vertex form to determine whether the vertex ofits graph represents the maximum or minimum of the quadratic function.

Learning TargetsI can find the maximum or minimum of a function by writing the quadratic expression thatdefines it in vertex form.

When given a quadratic function in vertex form, I can explain why the vertex is a maximum orminimum.

Lesson NarrativeIn a previous lesson, students recalled that a quadratic expression in vertex form can help usidentify the vertex of a graph of a quadratic function. They then used completing the square torewrite expressions from both standard and factored forms into vertex form. In this lesson, theyuse the vertex form to determine the maximum or minimum value of a function and to solveproblems.

This is not the first time that students find a maximum or minimum value of a quadratic function. Inan earlier unit on quadratic functions, students had a brief encounter with this idea, including withthe use of the vertex form to determine a maximum or minimum.

At that time, however, students did not yet know how to rewrite expressions in vertex form, so theycould only use an expression to determine maximum or minimum if the given expression is alreadyin vertex form. (Otherwise, students would have had to graph the expression or analyze a table ofvalues.) Now that they can rewrite a given expression into vertex form, students can find amaximum or minimum of a function regardless of form and solve new kinds of problems.

An increased emphasis on using the structure of the vertex form to explain maximums andminimums also distinguishes the work in this lesson from earlier work. Previously, students mayhave relied on their observation of graphs, or recalling that the graph of an equation

opens upward when is positive and downward when is negative. Here, theyreason that:

is always zero or positive.

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When , the expression is 0 because . When it is any other value,the expression has a value greater than 0.

When is positive, is positive except when (at which point it is 0). This means 0is the lowest possible value.

When is negative, is negative except when (at which point it is 0). Thismeans 0 is the highest possible value.

As students reason about and explain why a vertex is a maximum or a minimum, they practiceconstructing logical arguments (MP3) and being precise in their communication (MP6).

Alignments

Building On

HSF-IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpretstatements that use function notation in terms of a context.

Addressing


HSF-IF.C: Analyze functions using different representations.

HSF-IF.C.9: Compare properties of two functions each represented in a different way(algebraically, graphically, numerically in tables, or by verbal descriptions). For example, givena graph of one quadratic function and an algebraic expression for another, say which has thelarger maximum.

Building Towards



MLR6: Three Reads


Think Pair Share


Let’s find the maximum or minimum value of a quadratic function.

23.1 Values of a FunctionWarm Up: 10 minutes

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In this activity, students recall the meaning of maximum or minimum value of a function, which theylearned in a previous unit. They also practice interpreting the language related to maximum andminimum values of functions.

Building On

HSF-IF.A.2

Addressing

HSF-IF.C

Building Towards

HSA-SSE.B.3.b

Launch

Ask students to describe some situations in which people use the words minimum and maximum. Forexample, we might say there is a minimum age for voting or for getting a driver’s license, or thatroads and highways have maximum speed limits.

Then, ask students what the words “minimum” and “maximum” mean more generally. We mightthink of a minimum as the least, the least possible, or the least allowable value, and a maximum asthe greatest, the greatest possible, or the greatest allowable value.


Some students may struggle to relate the -coordinates of the points on a graph with the outputs ofa function. Earlier in the course, students learned that the graph of a function is the graph of theequation . Consider having students label the coordinates of points on each graph and thencomplete the statements such as “The point on the graph means .” Another approachwould be to have students organize the points on the graphs into tables with headers andand and .


Here are graphs that representtwo functions, and , definedby:

1. can be expressed in words as “the value of when is 1.” Find or compute:

a. the value of when is 1

b.

•

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•


c.

2. Can you find an value that would make :

a. Less than 1?

b. Greater than 10,000?

3. can be expressed in words as “the value of when is 9.” Find or compute:

a. the value of when is 9

b.

c.

4. Can you find an value that would make :

a. Greater than 7?

b. Less than -10,000?

Student Response

1. Function :a. 10

b. 2

c. 37

2. Sample response:a. No. From the graph, it seems that 1 is the least value of and that occurs when . All

other values produce outputs that are greater than 1.

b. Yes, there are lots of values that produce an output greater than 10,000. For example,both -100 or 110 would make greater than 10,000.

3. Function :a. -2

b. 6

c. -93

4. Sample response:a. No. From the graph, it seems that 7 is the greatest value of and that occurs when

. All other values produce outputs that are less than 7.

b. Yes, there are lots of values that produce an output less than -10,000. For example,both -200 or 200 would make less than -10,000.


Activity Synthesis


“Why does not have a maximum value?” (We can always use larger values of in both thepositive and negative directions to get greater and greater values of .)

“Why does not have a minimum value?” (We can always find an input that makes the value ofless and less.)

Emphasize that we can find an input that makes the value of as great as we want and that makesas small as we want.

Remind students that:

A maximum value of a function is a value of a function that is greater than or equal to all theother values. It corresponds to the highest value on the graph of the function.

A minimum value of a function is a value of a function that is less than or equal to all the othervalues. It corresponds to the lowest point on the graph of the function.

For quadratic functions, there is only one maximum or minimum value.

23.2 Maximums and Minimums15 minutesThe goal of this activity is to use the vertex form to find out if a vertex represents the minimum orthe maximum value of the function. To do this, students rely on the behavior of a quadraticfunction, the structure of the expression, and some properties of operations (MP7).

Because using structure is central to the work here, graphing technology is not an appropriate tool.

Building On

HSF-IF.A.2

Addressing

HSA-SSE.B.3.b



Think Pair Share

Launch

Arrange students in groups of 2 and ask students to keep their materials closed.

Display the equation for all to see. Ask students how they would determine,without graphing, if the vertex of the graph, , is a maximum or a minimum. Give partners a

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moment to discuss their thinking. Solicit a few ideas from the class before asking students toproceed to the activity.

Ask students to take a couple of minutes of quiet think time to make sense of the line of reasoningin the first question and then discuss their understanding with their partner.


Conversing: MLR8 Discussion Supports. Use this routine to help partners explain their reasoningfor the first question. Invite Partner A to begin with this sentence frame: “The point is amaximum value because . . .”, “First, I _____ because. . .”, or “I noticed _____ so I . . . .” Invite thelistener, Partner B, to press for additional details by asking “What happens when ?”, “Whathappens when you square ?”, or “Can you say that a different way?” This will helpstudents justify if a vertex represents the minimum or maximum value of a function.Design Principle(s): Support sense-making; Cultivate conversation


Representation: Internalize Comprehension. To support working memory, provide students withsticky notes or mini whiteboards. After students decide whether the vertex is a maximum orminimum, encourage students to prepare for the discussion by drawing a rough sketch of thegraphs, or by making tables to reinforce understanding.Supports accessibility for: Memory; Organization


1. The graph that represents has its vertex at . Here is one way toshow, without graphing, that corresponds to the minimum value of .

When , the value of is 0, because .

Squaring any number always results in a positive number, so when is any valueother than 8, will be a number other than 0, and when squared,will be positive.

Any positive number is greater than 0, so when , the value of will begreater than when . In other words, has the least value when .

Use similar reasoning to explain why the point corresponds to the maximum valueof , defined by .

2. Here are some quadratic functions, and the coordinates of the vertex of the graph ofeach. Determine if the vertex corresponds to the maximum or the minimum value ofthe function. Be prepared to explain how you know.

◦◦

◦


equationcoordinates of

the vertexmaximum or minimum?

Student Response

1. Sample response:When , the value of is 0, because .

Squaring any number always results in a positive number, so when is any value otherthan 4 (either greater or less), will be a number other than 0. When squared,

will be positive but then it gets multiplied to a negative number, so the productwill be negative.

Any negative number is less than 0, so when , the value of will be lessthan when . In other words, has the greatest value when .

2.equation coordinates of the vertex maximum or minimum?

maximum

minimum

minimum

minimum

maximum


Here is a portion of the graph of function , defined by .

◦◦

◦


is a rectangle. Points and coincide with the-intercepts of the graph, and segment just touches the

vertex of the graph.

Find the area of . Show your reasoning.

Student Response

54 square units. Sample reasoning:

The expression can be rewritten in factored form as , so the -intercepts are 4and 10, which means the length of is 6 units. The -coordinate of the vertex is halfwaybetween 4 and 10, which is 7. Substituting 7 for in gives or 9, so theheight of the rectangle is 9. The area is or 54 square units.

The expression can be written in vertex form as , so its vertex isat and the height of the rectangle is 9 units. Completing the square for

gives the solutions and , which correspond to the -interceptsat points and . This means the width of the rectangle is or 6 units.

Activity Synthesis

Focus the discussion on students’ responses to the first question. Invite students to share theirexplanations and highlight reasoning that makes use of structure (as shown in the sampleresponse).

Squaring an expression results in a positive value or a zero value.

Zero is the least possible value of a squared expression.

Multiplying a squared expression (which is either positive or zero) by a negative number giveseither a negative or zero value.

The graph of a quadratic function is symmetrical across the vertex. If an input value on oneside of the vertex produces an output that is less than , then represents themaximum value of the function. Similarly, if an input value on one side of the vertexproduces an output that is more than , then represents the minimum value of thefunction.

23.3 All the World’s a Stage10 minutesStudents may intuitively think of graphing the function for performance A, comparing the twographs, and seeing which one has a greater value at its vertex. While this strategy is both effectiveand efficient, the task asks students to decide which function has a greater maximum withoutgraphing, in order to encourage them to reason algebraically.

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Monitor for students taking different approaches. Some possible strategies:

Use the factored form of the expression for performance A to find the -intercepts. Then, findthe midpoint of the two intercepts, identify its value, evaluate the expression at that value,and compare the output with the value of the vertex of the other graph.

Convert the expression representing performance A to vertex form, identify the vertex, andcompare the value to that of the vertex on the graph for performance B.

Addressing

HSF-IF.C.9


MLR6: Three Reads

Think Pair Share

Launch

Arrange students in groups of 2. Give students time to read the task statement and think quietlyabout how they would go about comparing the two functions. Then, ask them to share theirthoughts with their partner before beginning to work on the problem.


Reading: MLR6 Three Reads. Use this routine to support reading comprehension, withoutsolving, for students. Use the first read to orient students to the situation. Ask students todescribe what the situation is about without using numbers (two performances occurred andgenerated revenue). Use the second read to identify quantities and relationships. Ask studentswhat can be counted or measured without focusing on the values. Listen for, and amplify, theimportant quantities that vary in relation to each other in this situation: ticket price andrevenue. After the third read, ask students to brainstorm possible strategies to answer thequestion, “Without creating a graph to represent the revenue from Performance A, determinewhich performance has the greater maximum revenue. Explain or show your reasoning.” Thishelps students connect the language in the word problem and the reasoning needed to solvethe problem while keeping the cognitive demand of the task.Design Principle: Support sense-making

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Action and Expression: Internalize Executive Functions. To support development of organizationalskills, check in with students within the first 2–3 minutes of work time. Look for students whoare working to identify the zeros or convert the equation for Performance A into vertex form. Ifstudents are stuck, remind them of previous experiences converting expressions into vertexform, and encourage them to write the vertex of Performance B next to the graph to compare.If a display was created in the previous lesson, use the display as a reference.Supports accessibility for: Memory; Organization


A function , defined by , describes the revenue collected from the sales oftickets for Performance A, a musical.

The graph represents a function that models the revenue collected from the sales of ticketsfor Performance B, a Shakespearean comedy.

In both functions, represents the price ofone ticket, and both revenues and prices aremeasured in dollars.

Without creating a graph of , determinewhich performance gives the greatermaximum revenue when tickets are dollarseach. Explain or show your reasoning.

Student Response

Performance A. Sample reasoning:

The maximum revenue for Performance B is $900 based on the vertex of its graph. ForPerformance A, the expression is in factored form, which tells us that thehorizontal intercepts are and . The horizontal coordinate of the vertex is halfwaybetween 0 and 8, which is 4. Substituting 4 for in gives 1,200, so the vertex is at

. 1,200 is greater than 900.

The maximum revenue for Performance B is $900 based on the vertex of its graph. Theexpression can be rewritten in vertex form: , which tells usthat the vertex of a graph of the function is at and that the maximum revenue is$1,200.

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Activity Synthesis

Invite previously identified students to share their solutions. Ask students to explain why theydecided to take the steps that they did. Highlight any connections made between the structure ofan expression defining a function, points on its graph, and the meaning of any values in thissituation.

Lesson SynthesisThe purpose of this lesson is for students to understand what is meant by a minimum or maximumvalue of a function and ways to approach finding a maximum or minimum value given a functionexpressed in any form. Display this expression for all to see:

Ask students to consider how they would go about deciding whether the function had a maximumor minimum value, and how they would determine what the maximum or minimum value is. It isnot necessaty to actually determine this value for the example. After a few minutes of quiet thinktime, invite them to share their approach with a partner. Time permitting, select a few students toshare with the class. Some possible approaches are:

Since the coefficient of is negative, I know the graph of the function opens downward, sothe function has a maximum value.

I could use technology to graph the function and see if the graph has a largest -coordinate, sothat the function has a maximum value, or a smallest -coordinate, so the function has aminimum value.

I could use the quadratic equation to find the zeros, and since I know the vertex is exactly inbetween the zeros, I could detemine the value of the midpoint of the two zeros and thensubstitute that value into the function to determine its output. Then I could compare thisoutput to any other output value and see if it was larger or smaller.

I could rewrite the expression in vertex form to see the coordinates of the vertex. Then, I couldthink about values on either side, and whether their corresponding values were greaterthan or less than the -coordinate of the vertex.

23.4 Looking for The Greatest or the LeastCool Down: 5 minutesAddressing

HSA-SSE.B.3.b

Launch

Ask students to put away their devices for the cool-down.


1. Without graphing, find the vertex of the graph of a quadratic function defined by. Show your reasoning.

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2. Does the -coordinate of the vertex correspond to a maximum or a minimum value ofthe function? Explain how you know.

Student Response

1. The vertex is at . Sample reasoning:

2. A maximum value. The squared term has a coefficient of -1. The squared term will be 0 whenis -7. For all other values of , the squared term will be subtracted from -11, resulting inoutputs that are less than -11.

Student Lesson SummaryAny quadratic function has either a maximum or a minimum value. We can tell whether aquadratic function has a maximum or a minimum by observing the vertex of its graph.

Here are graphs representing functions and , defined by and.

The vertex of the graph of is andthe graph is a U shape that opensdownward.

No other points on the graph of (nomatter how much we zoom out) arehigher than , so we can say thathas a maximum of 4, and that this occurswhen .

The vertex of the graph of is atand the graph is a U shape that opensupward.

No other points on the graph (no matterhow much we zoom out) are lower than

, so we can say that has aminimum of -10, and that this occurswhen .

We know that a quadratic expression in vertex form can reveal the vertex of the graph, so wedon’t actually have to graph the expression. But how do we know, without graphing, if thevertex corresponds to a maximum or a minimum value of a function?

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The vertex form can give us that information as well!

To see if is a minimum or maximum of , we can rewrite in vertex form,which is . Let’s look at the squared term in .

When , is 0, so is also 0.

When is not -3, the expression will be a non-zero number, and will bepositive (squaring any number gives a positive result).

Because a squared number cannot have a value less than 0, has the least valuewhen .

To see if is a minimum or maximum of , let’s look at the squared term in .

When , is 0, so is also 0.

When is not -5, the expression will be non-zero, so will be positive. Theexpression has a negative coefficient of -1, however. Multiplying(which is positive when ) by a negative number results in a negative number.

Because a negative number is always less than 0, the value of will alwaysbe less when than when . This means gives the greatest value of .

Glossarymaximum

minimum


StatementHere is a graph of a quadratic function .What is the minimum value of ?

Solution0

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Problem 2StatementThe graph that represents has its vertex at .

Explain how we can tell from the expression that -4 is the minimum value ofrather than the maximum value.

SolutionSample response: The expression is a squared expression, so its value will always bepositive or zero. The least value it can have is zero, and this happens when . . Allother values of make greater than zero.

Problem 3StatementEach expression here defines a quadratic function. Find the vertex of the graph of thefunction. Then, state whether the vertex corresponds to the maximum or the minimum valueof the function.

a.

b.

c.

d.

e.

f.

Solutiona. , minimum

b. , minimum

c. , maximum

d. , minimum

e. , maximum

f. , minimum



a. Can we use the quadratic formula to solve this equation? Explain or show how youknow.

b. Is it easier to solve this equation by completing the square or by rewriting it in factoredform and using the zero product property? Explain or show your reasoning.

Solutiona. Yes. Sample response: Yes, but the equation must be in rewritten in standard form,

. Subtracting gives the equation , is 1, is -12, and is 0.Substituting those values into the quadratic formula and evaluating the expression gives thesolutions.

b. Sample response: It would be quicker to solve this equation by rewriting in factored form andusing the zero product property. Subtracting gives the equation . Thisequation in factored form is . The solutions are and .


Problem 5StatementMatch each equation to the number of solutions it has.

A.

B.

C.

1. no solutions

2. 1 solution

3. 2 solutions

Solution

A: 3

B: 2

C: 1



Problem 6StatementWhich equation has irrational solutions?

A.

B.

C.

D.


Problem 7StatementLet represent an irrational number and let represent a rational number. Decide if eachstatement is true or false. Explain your thinking.

a. can be rational.

b. can be rational.

c. can be rational.

Solutiona. True. Sample response: Zero times any irrational number will be zero, which is rational.

b. True. Sample response:

c. True. Sample response: The product of any two fractions is another fraction.


Problem 8StatementHere are graphs of the equations , , and .


a. How do the 3 graphs compare?

b. How does the -3 in affect the graph?

c. How does the +7 in affect the graph?

Solutiona. Sample response: The shape of the graphs are the same but they are in different locations.

b. Sample response: Subtracting 3 from the before squaring shifts the graph of to the rightby 3 units.

c. Sample response: Adding 7 to the squared term shifts the graph of up by 7 units.


Problem 9StatementThree $5,000 loans have different annual interest rates. Loan A charges 10.5% annualinterest, Loan B charges 15.75%, and Loan C charges 18.25%.

a. If we graph the amount owed as a function of years without payment, what would thethree graphs look like? Describe or sketch your prediction.

b. Use technology to graph each function. Based on your graphs, if no payments are made,about how many years will it take for the unpaid balance of each loan to triple?

Solutiona. Sample response: All three loans will begin at $5,000, but the

balance for Loan C will grow faster than the balance for LoanB, and both will grow faster than the balance for Loan A.

b. Loan A: about 11 years, Loan B: about 8 years (less than 8 onthe graph, but the interest is annual), Loan C: about 7 years(less than 7 on the graph, but the interest is annual)



Lesson 24: Using Quadratic Equations to ModelSituations and Solve Problems

GoalsChoose and write the appropriate form for expressing a quadratic function to solve a problem.

Interpret features of graphs and expressions that represent quadratic functions to gaininformation about the situations being modeled.

Learning TargetsI can interpret information about a quadratic function given its equation or a graph.

I can rewrite quadratic functions in different but equivalent forms of my choosing and use thatform to solve problems.

In situations modeled by quadratic functions, I can decide which form to use depending on thequestions being asked.

Lesson NarrativeIn this culminating lesson, students synthesize methods of solving quadratic equations andgraphing quadratic functions to answer questions about quadratic functions within a context. Theyuse tools learned throughout this unit to grapple with solving problems, without scaffolding, abouta quadratic function that represents a context, and finding the points of intersection of a parabolaand a line. Since this work requires using what students know to tackle an unfamiliar problem,students need to make sense of problems and demonstrate perseverance (MP1).

The lesson consists of two substantial problems. You might decide to have all students attemptboth problems and select students with different approaches to share their solution with the class.Alternatively, you might allow students to choose one of the two problems and prepare a visualdisplay of their solution, conducting a gallery walk or a group presentation at the end of the lesson.

Technology isn’t required for this lesson, but there are opportunities for students to choose to useappropriate technology to solve problems. We recommend making technology available.

Alignments

Building On

HSF-LE.A.2: Construct linear and exponential functions, including arithmetic and geometricsequences, given a graph, a description of a relationship, or two input-output pairs (includereading these from a table).

Addressing

HSA-REI.B.4.b: Solve quadratic equations by inspection (e.g., for ), taking square roots,completing the square, the quadratic formula and factoring, as appropriate to the initial form

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of the equation. Recognize when the quadratic formula gives complex solutions and writethem as for real numbers and .

HSA-REI.C.7: Solve a simple system consisting of a linear equation and a quadratic equation intwo variables algebraically and graphically. For example, find the points of intersectionbetween the line and the circle .

HSF-IF.C.8.a: Use the process of factoring and completing the square in a quadratic function toshow zeros, extreme values, and symmetry of the graph, and interpret these in terms of acontext.


MLR5: Co-Craft Questions


Required Materials

Sticky notesTools for creating a visual displayAny way for students to create work that can beeasily displayed to the class. Examples: chart

paper and markers, whiteboard space andmarkers, shared online drawing tool, access to adocument camera.


The tools for creating a visual display and sticky notes are only required if you are doing thesuggested gallery walk in the lesson synthesis.


Let’s analyze a situation modeled by a quadratic equation.

24.1 Equations of Two Lines and A CurveWarm Up: 10 minutesThis warm-up activates some familiar skills for writing and solving equations, which will be usefulfor specific tasks throughout the lesson.

It may have been a while since students thought about writing an equation for a line passingthrough two points. The two questions here are intentionally quite straightforward. Monitor forstudents taking different approaches, such as:

plotting the points and determining the slope and -intercept of a line passing through thepoints

computing the slope by finding the quotient of the difference between the -coordinates anddifference between the -coordinates

considering what operation on each -coordinate would produce its corresponding-coordinate

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Students have not yet solved a quadratic equation like the one in the second question, but theyhave learned and extensively practiced the skills needed to solve it. The two main anticipatedapproaches are:

reasoning algebraically, by performing the same operation to each side of the equation,applying the distributive property to expand factored expressions, combining like terms,rewriting an expression in factored form, and applying the zero product property

graphing and and observing the -coordinate of each point ofintersection

Building On

HSF-LE.A.2

Addressing

HSA-REI.B.4.b


1. Write an equation representing the line that passes through each pair of points.

a. and

b. and

2. Solve this equation: . Show your reasoning.

Student Response

1. Equations (or their equivalents):a.

b.

2. 0 and 5. Sample reasoning:

Activity Synthesis

Invite students taking different approaches to share their work. Ensure that students see more thanone way to think about the equation representing a line for the first question, and recognize thatthe second equation can be solved algebraically.

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24.2 The Dive15 minutesMonitor for students taking different approaches, such as:

Graphing the function and finding and interpreting points on the graph.

Evaluating the function at relevant values.

Writing and solving equations whose solutions answer the questions. Some equations can besolved by rewriting it as a factored expression equal to 0, while some must be solved bycompleting the square or using the quadratic formula.

Rewriting the given expression in a different form—for example, rewriting it in vertex form tofind the maximum value of the function.

If any students take a graphing approach and finish quickly, challenge them to verify their solutionto each problem by using a second method that does not rely on graphing.

Addressing

HSA-REI.B.4.b

HSF-IF.C.8.a


MLR5: Co-Craft Questions

Launch

Ask students to read the task quietly and sketch a rough graph showing the diver’s height as afunction of time. When a diver jumps off a board, her height increases for a short time, and thendecreases until the time at which she enters the water, or the time at which her height above thewater is 0 meters.

Display one of these sketches for all to see, and ask what information might be added to the graphthat we know so far. Generally, we want to make sure students understand that the values alongthe horizontal axis correspond to the number of seconds after the jump, and values along thevertical axis correspond to the height in meters of the diver above the water.

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Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret contextsinvolving quadratic functions, and to increase awareness of language used to talk aboutquadratic functions. Display only the first sentence of this problem (“The function , defined by

, models the height of a diver above the water (in meters), secondsafter the diver leaves the board.”), and ask students to write down possible mathematicalquestions that could be asked about the situation. Invite students to compare their questionsbefore revealing the remainder of the question. Listen for and amplify any questions involvingthe vertex, zeros, and intercepts and their meaning in the situation. This will build studentunderstanding of the language of quadratic functions and help ensure students interpret thetask context correctly.Design Principle(s): Maximize meta-awareness; Support sense-making


Representation: Develop Language and Symbols. Create a display of important terms andvocabulary. During the launch, take time to review terms students will need to access for thisactivity. Invite students to suggest language or diagrams to include that will support theirdecision making in problem solving approaches for quadratic equations. Invite students tobrainstorm the situations from previous units and approaches they used to model and solve.As students contribute methods to the display, elicit student responses that reflect keyunderstandings about how and when to use various approaches and methods. For example, ifstudents name using the zero product property, elicit responses that clarify situations in whichit may be helpful, and under what conditions it is easier to use.Supports accessibility for: Conceptual processing; Language


If students have trouble getting started, tell them that sometimes it is helpful to restate a questionto make it about the function or its graph. For example, “When does the diver hit the water?” can berestated as “At what time is the diver 0 meters above the water?” or “What is the positive horizontalintercept of a graph representing the function?” Ask students to think about how they might restatesome questions about the situation as questions about the function or its graph and write thesedown or share them with a partner before they get to work.


The function , defined by , models the height of a diver above thewater (in meters), seconds after the diver leaves the board. For each question, explain howyou know.


1. How high above the water is the diving board?

2. When does the diver hit the water?

3. At what point during her descent toward the water is the diver at the same height as thediving board?

4. When does the diver reach the maximum height of the dive?

5. What is the maximum height the diver reaches during the dive?

Student Response

1. 7.5 meters

2. About 2.58 seconds after leaving the board. Sample reasoning: the height of the diver whenhitting the water is 0 meters, so we can solve . Using the quadraticformula, we get:

is about 15.8, so the solutions are approximately which are about -0.58 and

2.58.

3. 2 seconds after leaving the board. Sample reasoning: The diving board is 7.5 meters above theground, so we can solve , which is equivalent to or

. The solutions are or . When , the diver is still at the board. When, the diver has gone up and is on the way down.

4. 1 second after leaving the board. Sample reasoning: The maximum height corresponds to thevertex of the graph representing the function, so the maximum height occurs halfwaybetween -0.58 and 2.58, or halfway between 0 and 2.

5. 12.5 meters. Sample explanations:Substituting 1 into the expression gives or

, which is 12.5.

Rewriting the expression in vertex form gives , whichtells us that the vertex of the graph is at .

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Another diver jumps off a platform, rather than a springboard. The platform is also 7.5meters above the water, but this diver hits the water after about 1 second.

Write an equation that would approximately model her height over the water, , in meters,seconds after she has left the platform. Include the term , which accounts for the effect ofgravity.

Student Response

Sample response: . The constant term for will be 7.5 because the diver starts7.5 meters above the water. We want to select a number so that takes the value 0when . The equation represents this constraint. Solving it gives .(The 2.5 represents the initial velocity of the jumper.)

Activity Synthesis

Display the graph for all to see and ask students to interpret the graph: which numbers answerwhich questions? Once this is settled, invite students to share other approaches to solving theproblems besides interpreting coordinates on the graph.

24.3 A Linear Function and A Quadratic Function10 minutesThe purpose of this activity is for students to use what they know to solve an unfamiliar problem.Since it is unfamiliar, students need to make sense of the problem and demonstrate perseverence(MP1). This is a preview of solving a system consisting of a linear equation and a quadratic equationalgebraically and graphically, which students will study more in depth in a future course.

Building On

HSF-LE.A.2•


Addressing

HSA-REI.B.4.b

HSA-REI.C.7




Here are graphs of a linear function and a quadratic function.The quadratic function is defined by the expression

.

Find the coordinates of , and without using graphingtechnology. Show your reasoning.

Student Response

. Sample reasoning: The quadratic expression can be rewritten instandard form as . The constant term 11 means the -intercept is .

and . Sample reasoning: The line has a -intercept of and a slopeof 1, so its equation is . and are the points where the two functions have thesame values, so at those points. Solving the equation gives and

.

Substituting 3 and 6 for in either expression gives -4 and -1, respectively.

Activity Synthesis

Invite one or more students to demonstrate their solution, or if they got stuck, demonstrate theprogress that they made.

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Speaking: MLR8 Discussion Supports. Use this to routine amplify students’ use of mathematicallanguage to communicate reasoning about finding the coordinates of , , and . Whenstudents share their progress, solutions, and reasoning, remind them to use words such asvertex form, standard form, -intercept, and slope. Invite students to restate what they heardto a partner before selecting one or two students to share with the class. Ask the originalspeaker if their peer was accurately able to restate their thinking. This provides more studentswith an opportunity to produce language as they interpret the reasoning of others.Design Principle(s): Optimize output (for explanation)

Lesson SynthesisInvite students to create a visual display of one of their solutions to one of the tasks. Then, displaythese around the room and provide each student with a few sticky notes. Invite them to observeone or more of their classmates’ solutions, and leave a sticky note if they have a question orobservation to share. After this gallery walk, allow students time to review the feedback theyrecieved on their display and invite students to share anything new they learned or questions theyhave after seeing some of their classmates’ work.

24.4 Profit from A River CruiseCool Down: 5 minutesAddressing

HSF-IF.C.8.a


A travel company uses a quadratic function to model the profit, in dollars, that it expects toearn from selling tickets to a river cruise at dollars per person. The expression

defines this function.

Without graphing, find the price that would generate the maximum profit. Then, determinethat maximum profit.

Student Response

The price of $50 per ticket would generate a maximum profit of $1,600. Sample reasoning:can be rewritten in vertex form as , so the vertex of the graph

is at . This means the maximum profit, $1,600, can be expected when tickets are pricedat $50 each.

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Student Lesson SummaryCertain real-world situations can be modeled by quadratic functions, and these functions canbe represented by equations. Sometimes, all the skills we have developed are needed tomake sense of these situations. When we have a mathematical model and the skills to usethe model to answer questions, we are able to gain useful or interesting insights about thesituation.

Suppose we have a model for the height of a launched object, , as a function of time ,defined by . We can answer questions such as these about theobject’s flight:

From what height was the objectlaunched?

(An expression in standard form can help uswith this question. Or, we can evaluate tofind the answer.)

At what time did it hit the ground? (When an object hits the ground, its height is0, so we can find the zeros using one of themethods we learned: graphing, rewriting infactored form, completing the square, or usingthe quadratic formula.)

What was its maximum height, and atwhat time did it reach the maximumheight?

(We can rewrite the expression in vertex form,but we can also use the zeros of the functionor a graph to do so.)

Sometimes, relationships between quantities can beeffectively communicated with graphs and expressions ratherthan with words. For example, these graphs represent a linearfunction, , and a quadratic function, , with the samevariables for their input and output.

If we know the expressions that define these functions, we can use our knowledgeof quadratic equations to answer questions such as:

Will the two functions ever have thesame value?

(Yes. We can see that their graphs intersect ina couple of places.)

If so, at what input values does thathappen? What are the output values theyhave in common?

(To find out, we can write and solve thisequation: .)

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StatementThe function represents the height of an object seconds after it is launched into the air.The function is defined by . Height is measured in meters.

Answer each question without graphing. Explain or show your reasoning.

a. After how many seconds does the object reach a height of 33 meters?

b. When does the object reach its maximum height?

c. What is the maximum height the object reaches?

Solutiona. After 1 second and 3 seconds. Sample reasoning: Solving gives two

solutions: 1 and 3.

b. 2 seconds after launch. Sample reasoning: The horizontal coordinate of the vertex of thegraph would be halfway between the time the object is at 33 meters high, or halfway between1 and 3, which is 2.

c. 38 meters. Sample reasoning. Substituting 2 for in the original expression gives 38.

Problem 2StatementThe graphs that represent a linear function and a quadratic function are shown here.

The quadratic function is defined by .

Find the coordinates of without usinggraphing technology. Show your reasoning.

Solution. Sample reasoning: The linear graph has a slope of -2 and a -intercept of 5. It can be defined

by . We can find the intersections of the two graphs by writing and solving:.


Problem 3StatementDiego finds his neighbor's baseball in his yard, about 10 feet away from a five-foot fence. Hewants to return the ball to his neighbors, so he tosses the baseball in the direction of thefence.

Function , defined by , gives the height of the ball as a functionof the horizontal distance away from Diego.

Does the ball clear the fence? Explain or show your reasoning.

SolutionNo. Sample reasoning:

. 4.7 feet is a little lower than the fence.

The graph of (graphed using technology) shows a vertical value of 4.7 when the horizontalvalue is 10.

Solving or gives us the horizontaldistance at which the baseball will be 5 feet above the ground. Using the quadratic formula(with , , and ) gives or . Because the ball is 5 feethigh and on its way down when it is about 9.6 feet from Diego, it will not clear the fence.

Problem 4StatementClare says, “I know that is an irrational number, so its decimal never repeats orterminates. I also know that is a rational number, so its decimal repeats or terminates. But

I don’t know how to add or multiply these decimals, so I am not sure if and

are rational or irrational."

a. Here is an argument that explains why is irrational. Complete the missing parts

of the argument.

i. Let . If were rational, then would also be rational because . . . .

ii. But is not rational because . . . .

iii. Since is not rational, it must be . . . .

b. Use the same type of argument to explain why is irrational.

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◦


Solutiona. i. The sum of rational numbers is rational.

ii. It equals .

iii. It is irrational.

b. Let . If were rational, then would also be rational because products of rational

numbers are rational. But is not rational because it equals . Since is not rational, it

must be irrational.


Problem 5StatementThe following expressions all define the same quadratic function.

a. What is the -intercept of the graphof the function?

b. What are the -intercepts of the graph?

c. What is the vertex of the graph?

d. Sketch a graph of the quadratic functionwithout using technology. Make sure the

-intercepts, -intercept, and vertex areplotted accurately.

Solutiona.

b. and

c.

d. See graph.



Problem 6StatementHere are two quadratic functions: and .

Andre says that both and have a minimum value, and that the minimum value of is lessthan that of . Do you agree? Explain your reasoning.

SolutionAgree. Sample explanations:

The squared term in each expression has a positive coefficient, so the graph would be anupward-opening graph with the vertex being a minimum point. The constant term in the

expression for is less than the constant term 1 in that of .

The value of is 0 at and greater at all other values, so the functions are at theirminimum when . The vertex of the graph of is . The vertex of the graph of is

. The -coordinate of the vertex for the graph of is less than that of .


Problem 7StatementFunction is defined by the equation

.

Function is represented by this graph.

Which function has the smaller minimum?Explain your reasoning.

SolutionFunction . Sample response: The vertex of the graph of function is . The -coordinate ofthe vertex of the graph of is about 7, which is greater than -3, so -3 is the smaller minimum value.


Problem 8StatementWithout using graphing technology, sketch a graph that represents each quadratic function.Make sure the -intercepts, -intercept, and vertex are plotted accurately.

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Solution




Glossaryabsolute valueThe absolute value of a number is its distance from 0 on the number line.

associationIn statistics we say that there is an association between two variables if the two variables arestatistically related to each other; if the value of one of the variables can be used to estimate thevalue of the other.

average rate of changeThe average rate of change of a function between inputs and is the change in the outputs

divided by the change in the inputs: . It is the slope of the line joining and

on the graph.

bell-shaped distributionA distribution whose dot plot or histogram takes the form of a bell with most of the data clusterednear the center and fewer points farther from the center.

bimodal distributionA distribution with two very common data values seen in a dot plot or histogram as distinct peaks.In the dot plot shown, the two common data values are 2 and 7,


categorical dataCategorical data are data where the values are categories. For example, the breeds of 10 differentdogs are categorical data. Another example is the colors of 100 different flowers.

categorical variableA variable that takes on values which can be divided into groups or categories. For example, color isa categorical variable which can take on the values, red, blue, green, etc.

causal relationshipA relationship is one in which a change in one of the variables causes a change in the other variable.

coefficientIn an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by.If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.

The coefficient of in the expression is . The coefficient of in the expression is 1.

completing the squareCompleting the square in a quadratic expression means transforming it into the form ,where , , and are constants.

Completing the square in a quadratic equation means transforming into the form .

constant termIn an expression like the number 2 is called the constant term because it doesn't changewhen changes.

In the expression the constant term is -8, because we think of the expression as . Inthe expression the constant term is 12.

constraintA limitation on the possible values of variables in a model, often expressed by an equation orinequality or by specifying that the value must be an integer. For example, distance above theground , in meters, might be constrained to be non-negative, expressed by .

correlation coefficientA number between -1 and 1 that describes the strength and direction of a linear associationbetween two numerical variables. The sign of the correlation coefficient is the same as the sign ofthe slope of the best fit line. The closer the correlation coefficient is to 0, the weaker the linearrelationship. When the correlation coefficient is closer to 1 or -1, the linear model fits the databetter.

The first figure shows a correlation coefficient which is close to 1, the second a correlationcoefficient which is positive but closer to 0, and the third a correlation coefficient which is close to-1.

Unit 7: Quadratic Equations Glossary 425

decreasing (function)A function is decreasing if its outputs get smaller as the inputs getlarger, resulting in a downward sloping graph as you move fromleft to right.

A function can also be decreasing just for a restricted range ofinputs. For example the function given by , whosegraph is shown, is decreasing for because the graph slopesdownward to the right of the vertical axis.

dependent variableA variable representing the output of a function.

The equation defines as a function of . The variable is the independent variable,because you can choose any value for it. The variable is called the dependent variable, because itdepends on . Once you have chosen a value for , the value of is determined.

The equation defines as a function of . The variable is the independent variable,because you can choose any value for it. The variable is called the dependent variable, because itdepends on x. Once you have chosen a value for x, the value of is determined.

distributionFor a numerical or categorical data set, the distribution tells you how many of each value or eachcategory there are in the data set.

domainThe domain of a function is the set of all of its possible input values.

eliminationA method of solving a system of two equations in two variables where you add or subtract amultiple of one equation to another in order to get an equation with only one of the variables (thuseliminating the other variable).

equivalent equationsEquations that have the exact same solutions are equivalent equations.


equivalent systemsTwo systems are equivalent if they share the exact same solution set.

exponential functionAn exponential function is a function that has a constant growth factor. Another way to say this isthat it grows by equal factors over equal intervals. For example, defines anexponential function. Any time increases by 1, increases by a factor of 3.

factored form (of a quadratic expression)A quadratic expression that is written as the product of a constant times two linear factors is said tobe in factored form. For example, and are both in factored form.

five-number summaryThe five-number summary of a data set consists of the minimum, the three quartiles, and themaximum. It is often indicated by a box plot like the one shown, where the minimum is 2, the threequartiles are 4, 4.5, and 6.5, and the maximum is 9.

functionA function takes inputs from one set and assigns them to outputs from another set, assigningexactly one output to each input.

function notationFunction notation is a way of writing the outputs of a function that you have given a name to. Ifthe function is named and is an input, then denotes the corresponding output.

growth factorIn an exponential function, the output is multiplied by the same factor every time the inputincreases by one. The multiplier is called the growth factor.

growth rateIn an exponential function, the growth rate is the fraction or percentage of the output that getsadded every time the input is increased by one. If the growth rate is 20% or 0.2, then the growthfactor is 1.2.

horizontal interceptThe horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If theaxis is labeled with the variable , the horizontal intercept is also called the -intercept. Thehorizontal intercept of the graph of is .

The term is sometimes used to refer only to the -coordinate of the point where the graph crossesthe horizontal axis.


increasing (function)A function is increasing if its outputs get larger as the inputs getlarger, resulting in an upward sloping graph as you move from leftto right.

A function can also be increasing just for a restricted range ofinputs. For example the function given by , whosegraph is shown, is increasing for because the graph slopesupward to the left of the vertical axis.

independent variableA variable representing the input of a function.

The equation defines as a function of . The variable is the independent variable,because you can choose any value for it. The variable is called the dependent variable, because itdepends on . Once you have chosen a value for , the value of is determined.

inverse (function)Two functions are inverses to each other if their input-output pairs are reversed, so that if onefunctions takes as input and gives as an output, then the other function takes as an input andgives as an output.

You can sometimes find an inverse function by reversing the processes that define the first functionin order to define the second function.

irrational numberAn irrational number is a number that is not rational. That is, it cannot be expressed as a positive ornegative fraction, or zero.

linear functionA linear function is a function that has a constant rate of change. Another way to say this is that itgrows by equal differences over equal intervals. For example, defines a linearfunction. Any time increases by 1, increases by 4.

linear termThe linear term in a quadratic expression (In standard form) , where , , and areconstants, is the term . (If the expression is not in standard form, it may need to be rewritten instandard form first.)

maximumA maximum of a function is a value of the function that is greater than or equal to all the othervalues. The maximum of the graph of the function is the corresponding highest point on the graph.


minimumA minimum of a function is a value of the function that is less than or equal to all the other values.The minimum of the graph of the function is the corresponding lowest point on the graph.

modelA mathematical or statistical representation of a problem from science, technology, engineering,work, or everyday life, used to solve problems and make decisions.

negative relationshipA relationship between two numerical variables is negative if anincrease in the data for one variable tends to be paired with adecrease in the data for the other variable.

non-statistical questionA non-statistical question is a question which can be answered by a specific measurement orprocedure where no variability is anticipated, for example:

How high is that building?

If I run at 2 meters per second, how long will it take me to run 100 meters?

numerical dataNumerical data, also called measurement or quantitative data, are data where the values arenumbers, measurements, or quantities. For example, the weights of 10 different dogs arenumerical data.

outlierA data value that is unusual in that it differs quite a bit from the other values in the data set. In thebox plot shown, the minimum, 0, and the maximum, 44, are both outliers.

perfect squareA perfect square is an expression that is something times itself. Usually we are interested insituations where the something is a rational number or an expression with rational coefficients.

••


piecewise functionA piecewise function is a function defined using different expressions for different intervals in itsdomain.

positive relationshipA relationship between two numerical variables is positive if anincrease in the data for one variable tends to be paired with anincrease in the data for the other variable.

quadratic equationAn equation that is equivalent to one of the form , where , , and are constantsand .

quadratic expressionA quadratic expression in is one that is equivalent to an expression of the form ,where , , and are constants and .

quadratic formula

The formula that gives the solutions of the quadratic equation ,

where is not 0.

quadratic functionA function where the output is given by a quadratic expression in the input.

rangeThe range of a function is the set of all of its possible output values.

rational numberA rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point onthe number line that you get by dividing the unit interval into equal parts and finding the pointthat is of them from 0. We can always write a fraction in the form where and are whole

numbers, with not equal to 0, but there are other ways to write them. For example, 0.7 is afraction because it is the point on the number line you get by dividing the unit interval into 10 equalparts and finding the point that is 7 of those parts away from 0. We can also write this number as

.

The numbers , , and are all rational numbers. The numbers and are not rational

numbers, because they cannot be written as fractions or their opposites.


relative frequency tableA version of a two-way table in which the value in each cell is divided by the total number ofresponses in the entire table or by the total number of responses in a row or a column.

The table illustrates the first type for the relationship between the condition of a textbook and itsprice for 120 of the books at a college bookstore.

$10 or less more than $10 but less than $30 $30 or more

new 0.025 0.075 0.225

used 0.275 0.300 0.100

residualThe difference between the -value for a point in a scatter plotand the value predicted by a linear model. The lengths of thedashed lines in the figure are the residuals for each data point.

skewed distributionA distribution where one side of the distribution has more values farther from the bulk of the datathan the other side, so that the mean is not equal to the median. In the dot plot shown, the datavalues on the left, such as 1, 2, and 3, are further from the bulk of the data than the data values onthe right.


solution to a system of equationsA coordinate pair that makes both equations in the system true.

On the graph shown of the equations in a system, the solution isthe point where the graphs intersect.

solutions to a system of inequalitiesAll pairs of values that make the inequalities in a system true are solutions to the system. Thesolutions to a system of inequalities can be represented by the points in the region where thegraphs of the two inequalities overlap.

standard deviationA measure of the variability, or spread, of a distribution, calculated by a method similar to themethod for calculating the MAD (mean absolute deviation). The exact method is studied in moreadvanced courses.

standard form (of a quadratic expression)The standard form of a quadratic expression in is , where , , and are constants,and is not 0.

statisticA quantity that is calculated from sample data, such as mean, median, or MAD (mean absolutedeviation).

statistical questionA statistical question is a question that can only be answered by using data and where we expectthe data to have variability, for example:

Who is the most popular musical artist at your school?

When do students in your class typically eat dinner?

Which classroom in your school has the most books?

•••


strong relationshipA relationship between two numerical variables is strong if thedata is tightly clustered around the best fit line.

substitutionSubstitution is replacing a variable with an expression it is equal to.

symmetric distributionA distribution with a vertical line of symmetry in the center of the graphical representation, sothat the mean is equal to the median. In the dot plot shown, the distribution is symmetric about thedata value 5.

system of equationsTwo or more equations that represent the constraints in the same situation form a system ofequations.

system of inequalitiesTwo or more inequalities that represent the constraints in the same situation form a system ofinequalities.

two-way tableA way of organizing data from two categorical variables in order to investigate the associationbetween them.

has a cell phone does not have a cell phone

10–12 years old 25 35

13–15 years old 38 12

16–18 years old 52 8

uniform distributionA distribution which has the data values evenly distributed throughout the range of the data.


variable (statistics)A characteristic of individuals in a population that can take on different values

vertex (of a graph)The vertex of the graph of a quadratic function or of an absolute value function is the point wherethe graph changes from increasing to decreasing or vice versa. It is the highest or lowest point onthe graph.

vertex form (of a quadratic expression)The vertex form of a quadratic expression in is , where , , and are constants, and

is not 0.

vertical interceptThe vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis islabeled with the variable , the vertical intercept is also called the -intercept.

Also, the term is sometimes used to mean just the -coordinate of the point where the graphcrosses the vertical axis. The vertical intercept of the graph of is , or just -5.

weak relationshipA relationship between two numerical variables is weak if the datais loosely spread around the best fit line.


zero (of a function)A zero of a function is an input that yields an output of zero. If other words, if then is azero of .

zero product propertyThe zero product property says that if the product of two numbers is 0, then one of the numbersmust be 0.


Attributions“Notice and Wonder” and “I Notice/I Wonder” are trademarks of the National Council of Teachers ofMathematics, reflecting approaches developed by the Math Forum (http://www.nctm.org/mathforum/), and used here with permission.

Images that are not the original work of Illustrative Mathematics are in the public domain orreleased under a Creative Commons Attribution (CC-BY) license, and include an appropriatecitation. Images that are the original work of Illustrative Mathematics do not include such a citation.

Image AttributionsFlowers Painting Road Street Art London, by Max Pixel. Public Domain. https://www.maxpixel.net/Flowers-Painting-Road-Street-Art-Art-London-1011410.


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