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Common Core Learning Standards for MathematicsHigh School Algebra 1

Expressions & Equations

Common Core Learning Standards Concepts Embedded Skills VocabularyInterpret the structure of expressions.Focus on quadratic and exponential expressions. For A.SSE.1b, exponents are extended from the integer exponents found in Unit 1 to rational exponents focusing on those that represent square or cube roots.

Polynomial expressions – quadratic and exponential

A.SSE.1a: Define and identify parts of an expression – terms, factors, and coefficients

Classify expressions as monomial, binomial, trinomial.

Term FactorCoefficientExpressionMonomialBinomialTrinomial

A.SSE.1 Interpret expressions that represent a quantity in terms of itscontext.★a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

A.SSE.1b: Describe complex expressions by dissecting the expression into its individual parts (terms, factors, exponents – including fractional exponents).

Identify the factors within a term.

Exponent

A.SSE.2:Identify ways to rewrite an expression.Rewrite an expression in factored or expanded form.

Factored FormExpanded Form

SAMPLE TASKS

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics— Permission to use (not alter) and reproduce for educational purposes only.

1) When you simplify , why do you take the cube root of 27 separately from the cube root of ?

2) a) Write an algebraic expression to model the following phrase: the price of a meal plus a 15% tip for the meal.

b) What could the expression represent, including units?

c) What if the tip is 20% instead of 15% and 3 people are sharing the cost evenly. How can you represent the amount each person pays with a simplified algebraic expression? Identify the units for the expression.

3) Given the expression:

a) What is the degree of this expression?

b) What is the largest coefficient?

c) What is the leading coefficient of the polynomial in standard form?

d) How many terms are in the polynomial?

4) Rewrite the polynomial such that it has more terms than the original expression.


5) Simplify the expression in two different ways. Justify each step.

6) Why is the expression not the correct factored form for ?

7) You purchase n cans of tennis balls at $4.50 per can from an online retailer. There is a tax of 8% on your order. There are also shipping costs of $7.00 per order. Write the expression that can be used to find the total cost. Rewrite the expression in an equivalent form.

8) You burn calories at a rate of 15 calories per minute when running and 6 calories per minute when walking. Suppose you exercise for 60 minutes by running for r minutes and walking for the remaining time. The expression 15r+6(60-r) represents the calories burned. a) What units are associated with r? What units are associated with 15? What does 15r represent? Use unit analysis to explain.

b) What does 60-r represent? What units are associated with 6(60-r)?


c) Simplify 15r+6(60-r) . What units are associated with the expression? Use unit analysis to explain.

9) Can you write the expression using only one exponent? Show how or explain why not?

Common Core Learning Standards Concepts Embedded Skills Vocabulary Write expressions in equivalent forms to solve problems. It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions.For example, development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.

Write expressions in equivalent forms to

solve problems.

A.SSE.3a:Factor a quadratic expression to reveal the zeros (roots).

FactorQuadraticZerosRootsFunction

A.SSE.3bSolve a quadratic equation by completing the square.Identify the max/min of a quadratic equation.

Complete the SquareMaximumMinimumFunction

A.SSE.3cRewrite exponential expressions in equivalent forms using the properties of exponents.

Properties of Exponents


Simplify an exponential expression to determine if the function is growing or decaying.

A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by theexpression.★a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

SAMPLE TASKS1) a) What critical features of the function will be revealed when written in factored form?

b) What critical features of the function will be revealed while completing the square?

2) Given the following expressions, rewrite them in an equivalent form. a) b)

c)


3) A student deposits $500 into an account that earns 3.5% interest compounded quarterly. Describe and correct the student’s error in calculating the account balance after two years.

4) Does the function represent exponential growth or decay? Justify your reasoning.

5) Write the function in an equivalent form with a different base.

Common Core Learning Standards Concepts Embedded Skills VocabularyPerform arithmetic operations on polynomials.Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.

Operations on polynomials

A.APR.1Define a set of numbers as closed under addition, subtraction, and multiplication.Explain why the operation of division is not closed for polynomials.

ClosedPolynomialOperations


Add, subtract, and multiply polynomials.

A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

SAMPLE TASKS1) Given and find:

2) Which of the operations are closed for the set of polynomials? Explain your reasoning.

3) Write the equation in two different equivalent forms. Explain why they are equivalent.

4) To raise funds , a club is publishing and selling a calendar. The club has sold $500 in advertising and will sell copies of the calendar for $20 each. Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics— Permission to use (not alter) and reproduce for educational purposes only.

The cost of printing each calendar is $6. Let c be the number of calendars to be printed and sold. a) Write a rule for the function R(c), which gives the revenue generated by the sale of the calendars.

b) Write a rule for the function E(c), which gives the expense of printing the calendars.

c) Describe how the function P(c), which gives the clubs’ profit from the sale of the calendars, is related to R(c) and E(c). Then write the rule for P(c).

5) Given the following binomials find the product using two methods:

Common Core Learning Standards Concepts Embedded Skills VocabularyCreate equations that describe numbers or relationships. Extend work on linear and exponential equations in Unit 1 to quadratic equations. Extend A.CED.4 to formulas involving squared variables.

Represent a problem situation by creating an equation, inequality or

graph.

A.CED.1 Create a linear, exponential, quadratic or rational equation or inequality in one variable from a word problem.Solve a linear, exponential, quadratic or rational equation or inequality from a word problem.

EquationInequalityLinearExponentialQuadraticRational


A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

A.CED.2Write linear, quadratic and exponential equations using two or more variables.Rewrite linear, quadratic and exponential equations in terms of “y”.Graph linear, quadratic and exponential relationships from the equation or a table of values using an appropriate scale.Identify variables as independent or dependent.Identify parts of the coordinate plane (axes and quadrants).

Coordinate planeAxesQuadrantsScale

A.CED.4Manipulate equations using inverse operations to solve for a given variable.

Inverse operationsVariable

SAMPLE TASKS1) Marvin is building a deck. He wants the deck to be 30 square feet in total. If Marvin wants the length to be 7 feet longer than the width, write an equation to model the situation. Use your equation to find the length and width of the deck.


2) The sum of the length and width of a rectangle is 6. Write an equation to represent the area of the rectangle and graph the equation on the coordinate axes. For what dimensions will the rectangle have the maximum area.

3) Write an inequality that reflects the given number line. Describe a real world situation that can be represented with the graph above.

4) The hard drive on your computer has a capacity of 120 GB. You have used 85 GB . You want to save some home videos to your hard drive. Write and solve an inequality to find the possible sizes of the home video collection you can save?

5) A sales associate in a shoe store earns $325 each week, plus a commission equal to 4% of her sales. This week her goal is to earn at least $475. At least how many dollars’ worth of shoes must the associate sell in order to reach her goal?


6) In a math class, two students were asked to go to the board to solve the Area formula for r, radius, in terms of A and . Which student arrived at the incorrect answer? Explain why the student you chose is incorrect.

Student A Student B

7) Given the formula for degrees Celsius solve for F in terms of C.


Common Core Learning Standards Concepts Embedded Skills VocabularySolve equations and inequalities in one variable.Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.

Solve quadratic equations in one

variable.

A.REI.4 aSolve quadratic equations by completing the square.Derive the quadratic formula by completing the square.

Complete the squareDeriveRootsSolutionsZeros

A.REI.4 Solve quadratic equations in one variable.a. Use the method of completing the square to transform any

quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

A.REI.4 bSolve a quadratic equation using the most efficient method (square roots, factor, complete the square, quadratic formula) based on the original form of the equation.Classify the roots of a quadratic equation as real or complex.Write the solution to a quadratic equation as .

Real solutionComplex solution

SAMPLE TASKS1) Joe wants to solve the equation by completing the square. Solve the equation using Joe’s suggested method. Explain to Joe another way to solve the equation by showing him an alternative method.


2) a) Can you solve by factoring? Explain.

b) Use completing the square to explain why has no solution.

c) What method would you use to solve the equation ? Explain why you would chose this method.

3) A person standing on a second floor balcony drops keys to a friend standing below the balcony. The keys are dropped at a height of 10 ft. The

height (in feet) of the keys as they fall is given by the function where t is time (in seconds) since the keys were dropped. The friend catches the keys at a height of 4 feet. Write an equation to find the elapsed time before the keys are caught. Use an appropriate method to solve your equation.


4) Two equations are solved using the quadratic formula. The solutions are as follows.

Which of the given solutions would be considered “non-real”? Explain your reasoning.

Common Core Learning Standards Concepts Embedded Skills VocabularySolve systems of equations.Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding theintersections between x2+y2=1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32+42=52.

Solve systems of equations.

A.REI.7 Solve a linear-quadratic system of equations algebraically.Solve a linear-quadratic system of equations graphically.Identify the points of intersection algebraically and graphically as the solution of the system.Solve a system of equations involving a line and a circle graphically and algebraically by identifying the points of intersection.

IntersectionSystem of equationsLinear equationQuadratic equationAlgebraicallyGraphicallyPoint of intersectionSolution

A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.


SAMPLE TASKS1) Solve the system of equations graphically and algebraically.

y=−x2+4 x−3x+ y=1


2) A rocket is launched and travels a path represented byh=−16 t2+128 t . The ascent of a plane is represented by h=32t. After how many seconds do the rocket and plane crash? Create a graphical representation and explain.

3) Anicia looks at the following system of equations and determines that there is one real solution. x2+ y2=25 4 y=3 x

Explain if Anicia is correct or incorrect. Find the solution(s) to the system to support your conclusion.

4) a) Solve the system of equations algebraically.


b) If the linear function was how many solutions would there be? Justify your answer.

c) When solving algebraically why do you substitute the x values into the equation of the line instead of the equation of the parabola?

d) Where do you find the solution to a system of equations on a graph.

e) Describe how to check the solutions are correct.


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