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North Carolina High School Mathematics Math III Unpacking Document

NC DEPARTMENT OF PUBLIC INSTRUCTION 1

The Real Number System N-‐RN

Common Core Cluster

Use properties of rational and irrational numbers.

Common Core Standard Unpacking What does this standard mean that a student will know and be able to do?

N-‐RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

N.RN.3 Know and justify that when adding or multiplying two rational numbers the result is a rational number Ex: What kind of number (rational or irrational) is the sum of !

! and !

!? What kind of number (rational or

irrational) is the product of !! and !

!? Explain how you know whether the solution would be rational or

irrational without computing. Ex: Prove and illustrate with examples that when adding or multiplying two rational numbers the result is a rational number. N.RN.3 Know and justify that when adding a rational number and an irrational number the result is irrational.

Ex: What kind of number (rational or irrational) is the sum of !! and 2? Explain how you know whether the

solution would be rational or irrational without computing. Ex: Prove and illustrate with examples that when adding a rational number and an irrational number the result is irrational. Ex: Give an example of two different irrational numbers that have a rational number as their sum. Justify your answer.



N.RN.3 Know and justify that when multiplying of a nonzero rational number and an irrational number the result is irrational.

Ex: What kind of number (rational or irrational) is the product of !! and 2? Explain how you know whether

the solution would be rational or irrational without computing. Ex: Prove and illustrate with examples that when multiplying of a nonzero rational number and an irrational number the result is irrational. Ex: Give an example of two different irrational numbers that have a rational number as their product. Justify your answer.

Quantities* N-‐Q

Common Core Cluster

Reason quantitatively and use units to solve problems.


N.Q.1 Use units as a way to understand problems and to guide the solution of multi-‐step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N.Q.1 Interpret units in the context of the problem. For example, students should use the units assigned to quantities in a problem to help identify which variable they correspond to in a formula. Students should also analyze units to determine which operations to use when solving a problem. Given the speed in mph and time traveled in hours, what is the distance traveled? From looking at the units, we can determine that we

must multiply mph times hours to get an answer expressed in miles:

(Note that knowledge of the distance formula is not required to determine the need to multiply in this case.)



N.Q.1 When solving a multi-‐step problem, use units to evaluate the appropriateness of the solution.

Ex. You are mixing concrete for a school project and you have calculated according to the directions that you need 8 gallons of water for your mix. Your bucket is not calibrated, so you do not know how much it holds.

On the other hand, you have just finished a 2 liter bottle of soda. If you use the bottle to measure your water, how many times will you need to fill it? Conversion factor 1 gallon = 3.785 liters.

! !"##$%&!!"#

∙ !.!"# !"#$%&! !"##$%

and ! !"#$%&!!"#

∙ ! !"##$%! !"#$%&

to find the number of bottles needed.

N.Q.1 Choose the appropriate units for a specific formula and interpret the meaning of the unit in that context. Based on the type of quantities represented by variables in a formula, choose the appropriate units to express the variables and interpret the meaning of the units in the context of the relationships that the formula describes. Ex. When finding the area of a circle using the formula 𝐴 = 𝜋𝑟!, which unit of measure would be appropriate for the radius?

a. square feet b. inches c. cubic yards d. pounds

Ex. Based on your answer to the previous question, what units would the area of the circle be measured in?

N.Q.1 Choose and interpret both the scale and the origin in graphs and data displays. When given a graph or data display, read and interpret the scale and origin. When creating a graph or data display, choose a scale that is appropriate for viewing the features of a graph or data display. Understand that using larger values for the tick marks on the scale effectively “zooms out” from the graph and choosing smaller values “zooms in.” Understand that the viewing window does not necessarily show the x-‐ or y-‐axis, but the apparent axes are parallel to the x-‐ and y-‐axes. Hence, the intersection of the apparent axes in the viewing window may not be the origin. Also be aware that apparent intercepts may not correspond to the actual x-‐ or y-‐intercepts of the graph of a function.



Ex. In science class, Parker studied the motion of a simple pendulum. She attached different weights to a string and looked for patterns relating the weight, the length of the string, and the motion of the weight as it moves from side to side. She found that the frequency of a pendulum (the number of swings per unit time) depends only on the length of the string, not the weight attached or the initial starting height of the swing. The function 𝐹 = !"

! models the relationship between the length of the string, 𝐿, measured in meters and the

frequency, 𝐹, measured in swings per second.

a. Write the rule for pendulum frequency as a power function (𝑦 = 𝑎𝑏!). b. Sketch the graph of the frequency function. Choose a scale that is appropriate for viewing the

features of the function , taking into consideration the practical domain. c. Explain what the shape of the function tells about how the frequency changes as the string length

increases. d. Estimate the frequency of a pendulum with string length 1 meter and 0.5 meters. e. Estimate the string length needed to produce a frequency of 1 swing per second.



N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.

N.Q.2 Determine and interpret appropriate quantities when using descriptive modeling. For example, if you want to describe how dangerous the roads are, you may choose to report the number of accidents per year on a particular stretch of interstate. Generally speaking, it would not be appropriate to report the number of exits on that stretch of interstate to describe the level of danger.

Ex. What quantities could you use to describe the best city in North Carolina?

Ex. What quantities could you use to describe the effectiveness of a basketball player?

Ex. If you were opening your very own restaurant, what is everything you must consider for your restaurant to be successful? Now that you have defined the variables, which ones will provide income and which ones will be expenses? Ex. What quantities could you use to describe a safe bungee jump apparatus? If you were to build an in-‐classroom bungee jump apparatus, what units would be best to use for your measurements? Explain. If you were to build a real bungee jump apparatus, what units would it be best to use for your measurements? Explain. How can you relate the model from the classroom to the real life bungee jump? Ex. Elizabeth is working on a crime report for the city in which she lives. She interviewed people to hear their views on crime. Some people thought that building more police stations would result in less crime and that living closer to a police station would result in less crime. Other people Elizabeth interviewed thought that the closeness to the police station was not a playing factor in a neighborhood’s level of crime. Elizabeth wanted to research these claims. She called her local police stations to get data on crimes within the past year, focusing specifically on robberies and how far away each robbery was from the police station.



Number of blocks from police station Number of crimes per block 0-‐5 1.4 6-‐10 1.5

Greater than 10 1.7

1. Analyzing this information, what relationship do you believe there is between closeness to the police station and the amount of crime? Justify your reasoning.

2. Some people believe there are other factors affecting crime rate other than just the distance from the police station. Determine other factors you believe that would affect crime rates.

N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N.Q.3 Determine the accuracy of values based on their limitations in the context of the situation. Understand that the tool used determines the level of accuracy that can be reported for a measurement. For example, when using a ruler, you can only legitimately report accuracy to the nearest division. If I use a ruler that has centimeter divisions to measure the length of my pencil, I can only report its length to the nearest centimeter. Ex. What is the accuracy of a ruler with 16 divisions per inch? Ex. What would an appropriate level of accuracy be when studying the number of Facebook users? Ex. Vivian and John’s mother is a chemist, and she brought home a very delicate and responsive scale. Her children enjoyed learning how to use the device by measuring the weight of pennies one at a time. Here is a list from lightest to heaviest (weights are in milligrams). 2480 2484 2487 2491 2493 2495 2496 2498 2501 2503 2506 2507 2511 2515 2516

1. Given the information above, what do you think is the best estimate of the weight of a penny? Explain your reasoning.

2. Vivian and John’s Aunt Maria claimed she had a penny that was counterfeit. It looked and felt so real that Vivian and John could not believe it was counterfeit. They decided to weigh the penny and discovered the penny weighed 2541 milligrams.

3. John said that because the penny weighed more than all of their other measurements, it must be counterfeit. Vivian does not believe it is counterfeit based only on one weight measurement; she believes if they weigh the penny again it might be closer to the weight of the other pennies. Vivian also had a thought that if they measured the weights of more pennies, that Aunt Maria’s penny might not seem so strange.



4. Do you believe this penny is counterfeit and why? If you answered no, then how light or heavy would a penny need to be before you would believe it was counterfeit?

Ex. What would an appropriate level of accuracy be when measuring the length of shore of a beach? Explain your reasoning. Ex. Erica the Engineer is designing a bridge. She correctly computes that the maximum safe load of a bridge being planned will be 1000(99-‐70√2) tons. Fast Frank is the safety supervisor. He is asked to design a sign to tell the drivers how much weight the bridge will hold. Fast Frank uses Erica the Engineer’s expression and uses 1.4 as an approximation for √2, he then creates a sign based on his calculations. The bridge opens to traffic on a bright June morning. Two hours later, it collapses under a load less than a tenth of the weight shown on Fast Frank’s sign. Fast Frank tells the city council that he had simply used Erica the Engineer’s figures. Erica the Engineer reports that she has been over and over his figures and can’t see how they could be wrong. Write a clear explanation for the city council of why the bridge collapsed.

The Complex Number System N.CN

Common Core Cluster

Perform arithmetic operations with complex numbers.


N.CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

N.CN.2 Apply the fact that the complex number i2 = –1. Ex. Ohm’s Law relates the voltage E, current I, and resistance R, in an electrical circuit: E=IR. Respectively, these quantities are measured in volts, amperes, and ohms, respectively.

a. Find the voltage in an electrical circuit with current (2 + 4i) amperes and resistance (5 -‐ 4i) ohms. b. Find the necessary resistance value to produce a voltage that is not complex.



N.CN.2 Use the associative, commutative, and distributive properties, to add, subtract, and multiply complex numbers.

Ex. A. On a single coordinate diagram, locate and label points corresponding to these complex numbers.

4 + 3i -‐1 + 2i -‐3 – 2i 3-‐4i

B. Use the commutative and associative properties of addition and combining like terms to suggest a rule for addition of complex numbers, (a + bi) + (c + di) = (a + c) + (b + d)i. Use this rule to add the complex numbers 2 + 3i to each of the numbers in Part A.

C. Using another color, plot and label the points corresponding to your results in Part B. Then identify the geometric transformation that is accomplished by adding 2 + 3i to every complex number a + bi.

The Complex Number System N.CN

Common Core Cluster

Use complex numbers in polynomial identities and equations.


N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.

N.CN.7 Solve quadratic equations with real coefficients that have solutions of the form a + bi and a – bi. Ex. Use your knowledge of the quadratic formula to write quadratic equations with the following solutions:

a. One real number solution b. Solutions that are complex numbers in the form of a + bi, a ≠ 0, b ≠ 0 c. Solutions that are imaginary numbers bi

Ex. Your class is given the quadratic equation ax2 + bx + c = 0 and we know one solution is 2 + 3i. For the other solution, four of your classmates each came up with a different solution, 3 + 2i, 2 – 3i, -‐2 – 3i, and -‐2 + 3i. Evaluate the solutions each student provided. Which are values for x? Show your process and justify your reasoning.



N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N.CN.9 Understand The Fundamental Theorem of Algebra, which says that the number of complex solutions to a polynomial equation is the same as the degree of the polynomial. Show that this is true for any quadratic polynomial. Ex. Find and describe the solution(s) for the following equation, 4x2 – 12x + 13 = 0. How does The Fundamental Theorem of Algebra apply in this situation?

Ex. Use your knowledge of the quadratic formula to write quadratic equations with the following solutions:

a. One real number solution b. Solutions that are complex numbers in the form of a + bi, a ≠ 0, b ≠ 0 c. Solutions that are imaginary numbers bi

Seeing Structure in Expressions A.SSE

Common Core Cluster

Interpret the structure of expressions


A.SSE.1 Interpret expressions that represent a quantity in terms of its context.« a. Interpret parts of an expression,

such as terms, factors, and coefficients.

A.SSE.1a. Students manipulate the terms, factors, and coefficients in difficult expressions to explain the meaning of the individual parts of the expression. Use them to make sense of the multiple factors and terms of the expression. For example, the expression represents the amount of money I have in an account. My account has a starting value of $10,000 with a 5.5% interest rate every 5 years, where 10,000 and (1+.055) are factors, and the $10,000 does not depend on the amount the account is increased by. More scaffolding needed for quadratic. Ex. A person is walking across a hanging bridge that is suspended over a river. A hanging bridge droops in the middle creating a parabolic shape. The distance in feet from the person crossing the bridge to the river at any point can be described by the expression −0.02𝑥 100 − 𝑥 + 110, where x is the horizontal distance the person has walked from one side of the bridge. Interpret the terms, factors, and coefficients of the expression

$10,000 1.055( )5



in context. (Level III)

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret 𝑃(1 + 𝑟)! as the product of P and a factor not depending on P.

A.SSE.1b Students group together parts of an expression to reveal underlying structure. Ex. A person is walking across a hanging bridge that is suspended over a river. A hanging bridge droops in the middle creating a parabolic shape. The distance in feet from the person crossing the bridge to the river at any point can be described by the expression −0.02𝑥 100 − 𝑥 + 110, where x is the horizontal distance the person has walked from one side of the bridge. Interpret the factors −.02𝑥 and (100 − 𝑥) in the context of this situation. (Level III)

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.2 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.

Ex. The height of a child’s bounce above a trampoline is given by the function 𝑦 = −16𝑡! + 24𝑡 − 3. Rewrite the expression −16𝑡! + 24𝑡 − 3 to reveal the maximum height of the bounce and how long it takes to reach the maximum height. (Level III)



Seeing Structure in Expressions A.SSE

Common Core Cluster

Write expressions in equivalent forms to solve problems.


A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A.SSE.3b Students rewrite a quadratic expression in the form y = a , with a = 1, to identify the vertex of the parabola (h, k), and explain its meaning in context. Ex. If the quadratic expression −𝑥! − 24𝑥 + 55 models the height of a ball thrown vertically, identify the vertex-‐form of the expression and interpret the meaning of the vertex in this context.

A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

A.SSE.4 To derive the formula, expand the finite geometric series to show a few terms, including the last term. Create a new series by multiplying both sides of the original series by the common ratio, r. Subtract the new series from the original series, and solve for Sn. Sn= (a + ar + ar2 + … + arn-‐1) -‐ r Sn= (ar + ar2 + ar3 + … + arn-‐1 + arn) Sn -‐ r Sn = a -‐ arn Sn(1-‐r) = a(1-‐ rn)

Sn = a(1− rn )(1− r)

Mortgage payments can be found using the formula, P = iA1− (1+ i)−n

where P represents the payment

amount, A represents the loan amount, n represents the number of payments, and i is the monthly interest rate. The mortgage payment formula can be derived from the formula for the sum of a finite geometric series

x − h( )2 + k



because the mortgage process can be viewed as a finite series of (Principal + Interest – Payment). Ex. You just bought a $230,000 house, with 10% down on a 30-‐year mortgage with an interest rate of 8.5% per year. What is the monthly payment?

Arithmetic With Polynomials and Rational Expressions A.APR

Common Core Cluster

Perform arithmetic operations on 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.

A.APR.1 The Closure Property means that when adding, subtracting or multiplying polynomials, the sum, difference, or product is also a polynomial. Polynomials are not closed under division because in some cases the result is a rational expression. Ex. Needed


Common Core Cluster

Understand the relationship between zeros and factors of polynomials


A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a

A.APR.2 The Remainder Theorem states that if a polynomial, p(x) is divided by a monomial, (x – c), the remainder is the same as if you evaluate the polynomial for c , i.e. calculate p(c). If the remainder when dividing by (x -‐ c) is 0, or p(c) = 0, then (x – c) is a factor of the polynomial. If f (u) = 0, then (x– u) is a factor of f(x), which means that is a root of the function f(x). This is known as



factor of p(x). the Factor Theorem.



Ex. Given f(x) = 2x2 + 6x -‐ 20, determine whether -‐ 5 is a root of the function, then write the function in factored form. Ex. Compare the process of synthetic division to the process of long division for dividing polynomials. Ex. Assume that (x-‐c) is a factor of f. Explain why it must be true that f c = 0. Ex. Assume we know that f c = 0. Explain why it must be true that (x-‐c) is a factor of f.

A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A.APR.3 Find the zeros of a polynomial when the polynomial is factored. Then use the zeros to sketch the graph. Ex. For a certain polynomial function, x = 3 is a zero with multiplicity two, x = 1 is a zero with multiplicity three, and x = -‐3 is a zero with multiplicity one. Write a possible equation for this function and sketch its graph.


Common Core Cluster

Use polynomial identities to solve problems


A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

A.APR.4 Prove polynomial identities algebraically by showing steps and providing reasons or explanations. The following examples are meant to be investigated by students considering analogous problems, and trying special cases and simpler forms of the original problem in order to gain insight into its solution(s). Ex. Is (2𝑥 − 3)! − 64 equivalent to (2𝑥 − 11)(2𝑥 + 5)? Explain your reasoning. Ex. Jessie thinks that (𝑥 + 𝑦)! = 𝑥! + 2𝑥𝑦 + 𝑦!. Is he correct? Explain how you know. Ex. Prove 𝑥! − 𝑦! = 𝑥 − 𝑦 (𝑥! + 𝑥𝑦 + 𝑦!). Justify each step. Ex. Solve the quadratic ax2 + bx + c = 0 for x, justifying each step. What was interesting about the result?




Common Core Cluster

Rewrite rational expressions


A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

A.APR.6 Rewrite rational expressions, , in the form using long division, synthetic division

or with expressions that pose difficulty by hand, use a computer algebra system such as the TI Inspire CAS or Ipad applications. When dividing a polynomial by a polynomial, the new form is the quotient plus the remainder divided by the divisor. This process should be connected to dividing with numbers. The quotient represents the number of times something will divide, plus the parts or pieces remaining. Know that the degree of the quotient is less than the degree of the dividend. Connect division of polynomials to the remainder theorem when 𝑏(𝑥) is in the form (𝑥 − 𝑐). Ex. We know from arithmetic, that a fraction like !"#

!" indicates the division of 327 by 10. The result can be

expressed 32 R 7 or as 32 + !!". Use division of polynomials to show that !!

!!!!!!!!!

can be written with an

equivalent expression in the form of 𝑞 𝑥 + !(!)!!!

. Ex. Divide. Write the answer in the form of quotient plus remainder/divisor.

𝑥! + 3𝑥𝑥! − 4

Ex. Use a computer algebra system to rewrite the following rational expression in quotient and remainder form

9𝑥! + 9𝑥! − 𝑥 + 2

𝑥 + 23

)()(xbxa

)()(

+)(xbxr

xq



A.APR.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

A.APR.7 When performing any operation on a rational expression, the result is always another rational expression, which is the Closure Property for rational expressions. Compare this to the Closure Property for polynomials. Perform operations with rational expressions, division by nonzero rational expressions only. Ex. A rectangle has an area of (!

!!!!!)!!

sq. ft. and a height of !!

(!!!)ft. Express the width of the rectangle as a

rational expression in terms of 𝑥.

Creating Equations A.CED

Common Core Cluster

Create equations that describe numbers or relationships


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.1 From contextual situations, write equations and inequalities in one variable and use them to solve problems. Include one-‐variable equations that arise from functions by the selection of a particular target y-‐value. For example, in the radioactive decay problem below, 25 would be substituted for y in the equation 𝑦 = 100 !

!

!, which results in the one-‐variable equation 25 = 100 !

!

!. Note, the resulting

equation can be solved in Level I using a table or graph. See A-‐REI.11. Ex. Needed



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.2 Given a contextual situation, write equations in two variables that represent the relationship that exists between the quantities. Also graph the equation with appropriate labels and scales. Make sure students are exposed to a variety of equations arising from the functions they have studied. Ex. Needed



Reasoning with Equations and Inequalities A.REI

Common Core Cluster

Understand solving equations as a process of reasoning and explain the reasoning


A.REI.1 Explain each step in solving a simple equation as following from the

A.REI.1 Relate the concept of equality to the concrete representation of the balance of two equal quantities. Properties of equality are ways of transforming equations while still maintaining equality/balance.

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-‐ viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A.CED.3 When given a problem situation involving limits or restrictions, represent the situation symbolically using an equation or inequality. Interpret the solution(s) in the context of the problem. When given a real world situation involving multiple restrictions, develop a system of equations and/or inequalities that models the situation. In the case of linear programming, use the Objective Equation and the Corner Principle to determine the solution to the problem. Ex. Imagine that you are a production manager at a calculator company. Your company makes two types of calculators, a scientific calculator and a graphing calculator.

a. Each model uses the same plastic case and the same circuits. However, the graphing calculator requires 20 circuits and the scientific calculator requires only 10. The company has 240 plastic cases and 3200 circuits in stock. Graph the system of inequalities that represents these constraints.

b. The profit on a scientific calculator is $8.00, while the profit on a graphing calculator is $16.00. Write an equation that describes the company’s profit from calculator sales.

How many of each type of calculator should the company produce to maximize profit using the stock on hand?

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.4 Solve multi-‐variable formulas or literal equations, for a specific variable. Explicitly connect this to the process of solving equations using inverse operations.

Ex. If H =kA T1 −T2( )

L , solve for T2



equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process with mathematical properties. Ex. Solve the following equations for x. Use mathematical properties to justify each step in the process.

a. 5(x+3)-‐3x=55 b.

A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A.REI.2 Solve simple rational and radical equations in one variable and provide examples of how extraneous solutions arise. Add context.

Ex. Solve for x.

Ex. Mary solved for x and got x=-‐2, and x=1. Show how she might have solved the equation and whether you agree with her solutions. Is there any reason whey you might believe one of your solutions is impossible?

Ex. Solve for x. Could x have a value of 3? Explain your reasoning.


Common Core Cluster

Solve equations and inequalities in one variable


A.REI.4 Solve equations and inequalities 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.

A.REI.4a Transform a quadratic equation written in standard form to an equation in vertex form, -‐ , by completing the square.

Ex. Write the quadratic equation, y= -‐2x2 – 16x -‐20 in vertex form. What is the vertex of the graph of the equation?

A.REI.4a Derive the quadratic formula by completing the square on the standard form of a quadratic equation. Add context or analysis. Ex. Solve y = ax2 +bx+ c for x. What is surprising about the solution?

a± 0i

5− − x + 4( ) = 2

x = 2− x

3x −3

=x

x −3−32

(x qp =)2



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.4b Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. Add context or analysis Ex. Find the solution to the following quadratic equations:

a. x2 – 7x -‐18 = 0 b. x2 = 81 c. x2-‐ 10x + 5 = 0

A.REI.4b Use the quadratic formula to solve any quadratic equation, recognizing the formula always produces solutions. Write the solutions in the form , where a and b are real numbers. Students should understand that the solutions are always complex numbers of the form . Real solutions are produced when b = 0, and pure imaginary solutions are found when a = 0. The value of the

discriminant, , determines how many and what type of solutions the quadratic equation has.

Ex. Ryan used the quadratic formula to solve an equation and was his result.

a. Write the quadratic equation Ryan started with. b. Simplify the expression to find the solutions. c. What are the x-‐intercepts of the graph of this quadratic function?


Common Core Cluster

Represent and solve equations and inequalities graphically


A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the

A.REI.10 The solutions to equations in two variables can be shown in a coordinate plane where every ordered pair that appears on the graph of the equation is a solution. Understand that all points on the graph of a two-‐variable equation are solutions because when substituted into the equation, they make the equation

bia ±

bia ±

b2 − 4ac

x = 8+ (−8)2 − 4(1)(−2)2(1)



coordinate plane, often forming a curve (which could be a line).

true. Add context or analysis

Ex. Needed

A.REI.11 Explain why the x-‐coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

A.REI.11 Construct an argument to demonstrate understanding that the solution to every equation can be found by treating each side of the equation as separate functions that are set equal to each other, f(x) = g(x). Allow y1=f (x) and y2= g(x) and find their intersection(s). The x-‐coordinate of the point of intersection is the value at which these two functions are equivalent, therefore the solution(s) to the original equation. Students should understand that this can be treated as a system of equations and should also include the use of technology to justify their argument using graphs, tables of values, or successive approximations. Ex. Needed

Interpreting Functions F.IF

Common Core Cluster

Understand the concept of a function and use function notation.


F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.2 Students should continue to use function notation throughout high school mathematics. F.IF.2 Students should be comfortable finding output given input (i.e. f(3) = ?) and finding inputs given outputs (f(x) = 10). Ex, Needed




Common Core Cluster

Interpret functions that arise in applications in terms of the context.


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

F.IF.4 This standard should be taught alongside the specific function your class is studying. Students should be able to move fluidly between graphs, tables, and words and understand the interplay between the different representations. When given a table or graph of a function that models a real-‐life situation, explain the meaning of the characteristics of the graph in the context of the problem. At the course one level, the focus is on linear, exponentials, and quadratics

o Linear – x/y-‐intercepts and slope as increasing/decreasing at a constant rate. o Exponential-‐ y-‐intercept and increasing at an increasing rate or decreasing at a decreasing rate. o Quadratics – x-‐intercepts/zeroes, y-‐intercepts, vertex, the effects of the coefficient of x2 on the

concavity of the graph, symmetry of a parabola. At the course two level, the focus is on power functions, and inverse functions.

o Power functions – the effects of a positive/negative coefficient, the effects of the exponent on end behavior.

o Inverse functions – understanding the effects on the graph of having a variable in the denominator (asymptotes).

At the course three level, in addition to the previous course work, students should focus on polynomial and trigonometric functions.

o Polynomials – emphasis should be on the commonalities of quadratics and power functions. o Trigonometric functions – intercepts; intervals where the function is increasing, decreasing, positive,

or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Include amplitude, frequency, and midline (F-‐TF 5).

Note – This standard should be seen as related to F.IF.7 with the key difference being students can interpret from a graph or sketch graphs from a verbal description of key features.



Ex. Insert picture of a polynomial function graph from a real-‐life situation. a. What are the x-‐intercepts and y-‐intercepts and explain them in the context of the problem. b. Identify any maximums or minimums and explain their meaning in the context of the problem. c. Describe the intervals of increase and decrease and explain them in the context of the problem.

F-‐IF.4 When given a verbal description of the relationship between two quantities, sketch a graph of the relationship, showing key features. Ex. Jackie found that for a period of 7 days, the daily high was 85° and the low was 65°. He also noticed that the sunrise and sunset temperature was 75°. Sketch a graph showing the fluctuation temperatures during those 7 days.

F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

F.IF.5 Given a function and context, determine the practical domain of the function as input values that make sense to the constraints of the problem context. Ex. Needed




Common Core Cluster

Analyze functions using different representations.


F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and graph trigonometric functions, showing period, midline, and amplitude.

F.IF.7 Part a., b., and c. are learned by students sequentially in Courses I – III. Part e. is carried through Courses I – III with a focus on exponential in Course I and moving towards logarithms in Courses II and III. This standard should be seen as related to F.IF.4 with the key difference being students can create graphs from the symbolic function in this standard. Ex. A roller coaster’s track design can be modeled by the polynomial f(x)=x4-‐8x3+16x2. Analyze the graph of this function and describe the ride of the roller coaster. Is there a possible error to using this function to model the roller coaster? Why or why not?




Common Core Cluster

Analyze functions using different representations.


F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

F.IF.8 a is carried throughout Courses I -‐ III . In Course I and II, students will be limited to factoring quadratic functions. In Course III, students will be able to complete the square. Students should take a quadratic function and manipulate it in a different form (standard, factored, and vertex) so that they can show and explain special properties of the function such as; zeros, extreme values, and symmetry. Students should be able to distinguish when a particular form is more revealing of special properties given the context of the situation. An exemplar lesson plan of F.IF.8 and F.IF.9 is available from the Shell Center for Mathematics Education titled L20: Forming Quadratics at http://map.mathshell.org/materials/lessons.php?taskid=224 Ex. Coyote was chasing road runner, seeing no easy escape, Road Runner jumped off a cliff towering above the roaring river below. Molly mathematician was observing the chase and obtained a digital picture of this fall. Using her mathematical knowledge, Molly modeled the Road Runner’s fall with the following quadratic functions:

h(t) = -‐16t2 + 32t + 48 h(t) = -‐16(t+ 1)(t – 3) h(t) = -‐16(t – 1)2 + 64

a. Why does Molly have three equations? b. Could you convince others that all three of these rules are mathematically equivalent? c. Which of the rules would be most helpful in answering each of these questions? Explain.

i. What is the maximum height the Road Runner reaches and when will it occur? ii. When would the Road Runner splash into the river? iii. At what height was the Road Runner when he jumped off the cliff?



Ex. Suppose a single bacterium lands on one of your teeth and starts increasing by 5% each hour. Three students wrote equations to represent the growth of the bacteria. Their equations to model the situation are shown below.

Carly wrote the equation B = (1.05)h to represent the number of bacteria growth each hour, h.

Anna wrote the equation B = (1.05)24d to represent the number of bacteria at the end of the day, d.

Felix wrote the equation B = (3.23)d to represent the number of bacteria at the end of the day, d. All 3 students provide a correct model. Use properties of exponents to provide a convincing argument that the three equations are equivalent expressions.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F.IF.9 This standard includes comparing two different functions in two different forms and comparing one function in two different forms. An exemplar lesson plan of F.IF.8 and F.IF.9 is available from the Shell Center for Mathematics Education titled L20: Forming Quadratics at http://map.mathshell.org/materials/lessons.php?taskid=224 Ex: A herd of horses at Corolla Beach was first counted at 100 heads. Repopulation efforts have yielded a net growth of 16% yearly of the existing horse population. Simultaneously, biologists have recorded the sea turtle population growth in the following table: Year 0 1 2 5 10 Number of Sea Turtles

75 90 108 186 464

Which population is growing at a faster rate? Explain your reasoning.



Building Functions F.BF

Common Core Cluster

Build a function that models a relationship between two quantities.


F.BF.1 Write a function that describes a relationship between two quantities.* a. Determine an explicit expression,

a recursive process, or steps for calculation from a context.

b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Ex: Ten bacteria are placed in a test tube and each one splits in two after one minute. After 1 minute, the resulting 10 bacteria each split in two, creating 20 bacteria. This process continues for one hour until test tube is filled up.

a. How many bacteria are in the test tube after 5 minutes? 15 minutes? b. Describe how you can take any current number of bacteria to find the number of bacteria at the next

minute (this is writing a NOW -‐ NEXT rule). c. Write an “N = …” rule that gives the number of bacteria after each minute. d. How many bacteria are in the test tube after one hour? e. For further research, Dr. Bland removes 5 bacteria after each minute from the original test tube to

start a new cell culture. How does this affect your rule in parts b. and c.?



F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

Ex. The diagram below represents a famous mathematical fractal called Koch’s Curve. Each new stage is formed by replacing the middle third of each line segment with a “tent”. The process continues indefinitely.

(a) Create a sketch of stage 3. (b) Record the number resulting line segments in the table below.

Stage 0 1 2 3 4 5 6 7 # of Line Segments

1 4

(c) Write a rule that shows how the number of line segments for the next step depends on the prior

number of line segments (Hint: think Now Next) (d) Write a rule to determine the number of line segments at any stage that does not require knowing the

previous step. Start your rule with y=. (e) Compare the effectiveness of the two rules you developed in parts (c) and (d). Make sure to include

conjectures about the pros and cons of each rule.





Building Functions F.BF

Common Core Cluster

Build new functions from existing functions.


F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

F.BF.3 Identify the effect of transformations to functions in multiple modalities. Student should be fluent in representations of functions as equations, tables, graphs, and descriptions and understand the interplay among these representations. Ex. Fill in all missing components to the below table.

Description of Change

Original Function Output

Multiply the original function

by 5 x f(x) f(5x) -‐3 9 -‐2 4 -‐1 1 5 0 0 1 1 2 4 9 3 9

(a) Graph and label each of the function outputs with the corresponding x-‐values on the same set of axis

in three different colors. (b) Carefully explain the relationship between the original function and both transformed functions.

Ex. Given the function f(x)=-‐.5x2+x2+9, g(x)=5f(x), h(x)=f(3x)

(a) Using technology plot the above functions in different colors. Analyze g(x) and h(x) in comparison to f(x). (b) For each of the new functions (g(x) and h(x)) compare the local minimum, local maximum, and zeros

to the original function f(x). (c) Complete a table of values for f(x), h(x), and g(x) for x=-‐5,-‐4,-‐3,-‐2,-‐1,0,1,2,3,4,5



(d) Explain correspondences between the function representations, graphs created in part (a), verbal descriptions.



F.BF.3 Given the graphs of the original function and a transformation, determine the value of (k). Ex. Below are graphs representing the height of a two Farris Wheels at the county fair.

(a) Explain how the speed is different on the blue Ferris Wheel from that of the red Ferris Wheel. How

does this impact the positions of the local minimums and maximums. (b) Write the rule for the red graph assuming that the blue graph is represented f(x) Note: The intent of this question is NOT to develop a sophisticated trigonometric equation. It is acceptable to answer part (b) in terms of f(x).



F.BF.4 Find inverse functions. a. Solve an equation of the form f(x) =

c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 for x > 0 or f(x) = (x+1)/(x–1) for x � ̸ 1.

b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-‐invertible function by restricting the domain.

Ex. At Cosmo Creamery the sundae is $3.50 plus an additional $.30 per topping. (a) Write a function f(x), to model the cost of a Sundae with x toppings. (b) Find f(5) and explain its meeting in this context. (c) If deluxe Sundae costs $5.60, how many toppings does the deluxe Sundae include? (d) Write a rule that explains how to determine the number of toppings given the cost. (e) Re-‐write f(2)=4.10 with the rule developed in part(d)

Ex. Needed

F.BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

F.BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Ex. Needed



Linear, Quadratic, and Exponential Models* F.LE

Common Core Cluster

Construct and compare linear, quadratic, and exponential models and solve problems


F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Ex. Freddy and Fannie Frugal has $10 to invest and is considering three different investment plans. Plan 1 Guarantees that the Frugals will earn $10.00 in interest every year. Plan 2 Guarantees that the Frugals will earn 10% interest compounded annually on the account. Plan 3 has a different approach. Rather the earning interest yearly, this plan has a lump sum payout at

the day of withdraw that depends on the total time the investment is in the bank. The Frugals will receive the original $10 investment plus an interest payment equal to the number of years invested squared in.

Table

Year Plan 1 Plan 2 Plan 3 0 10 10 10 1 11 2 30 3 13.31 4 5 10 15 20 50 60 70

Utilizing technology create sketch of all three plans on the same axis. Indicate key intersection point(s).



Ex. Mathematical Models: Create a mathematical model for each of the following situations. Questions: Making sense of the quantities and relationships in the other three quadrants, answer the following problem situations.

(a) If the Frugals know that their anticipated twins (Mae and Mac) are going to arrive in the next 10 years, which plan would make the most money for the Frugals to renovate the nursery?

(b) Rather than using the investment for nursery renovations, they elect to save for the twin’s college education. Which would be the best plan?

(c) The Frugals then consider providing a retirement account for the twins. Which plan would be the most beneficial if the retirement age is 65?

F.LE.4 For exponential models, express as a logarithm the solution to abct =d where a,c, and d are numbers and the base b Is 2,10,or e; evaluate the logarithm using technology.

F.LE.4 In developing the idea of a logarithm, students should experiment (guess and check) first with rewriting numbers in the form of 10x as demonstrated below. Students should discern the structure of a logarithm and create their own definition. Ex. Using technology, rewrite each of the following numbers in the form of 10x. 10 100 1000000 562 1981 0.19 -‐250 Encourage students to be as exact as possible utilizing multiple decimal place values. Students should also realize that rewriting negative numbers is impossible without changing the form of 10x . Ex. On your calculator evaluate the following expressions Log(10) Log(100) Log(1000000) Log(562) Log(1981) Log(0.19) Log(-‐250)

(a) What are similarities between value of the exponent in part 1 and the value of the logarithm in part 2. (b) Create a definition that generalizes the structure that you see.



F.LE.4 A similar approach should be taken in rewriting values in terms of alternative bases. A biologist has 200 particles of bacteria. One hour later the amount of bacterial is 800 particles. The function b(t)=200(10.6t).

(a) From the observations of bacteria populations, how does the population appear to be growing? (b) Evaluate 106. How is this related to your answer in part (a)? (c) What are the benefits of writing a growth model with a base of 10. How does this relate to previous

work that has been done? (d) Sometime during the next day the biologist observes that there are 12,000,000 bacteria cells, how many

hours had the culture grown since the original observation. Your answer should be as exact as possible.

F.LE.4 It is acceptable for students to use decimal approximations in evaluating logarithms initially, however students should be encourages to attend to precision, and begin leaving answer in terms of logarithms. Carbon-‐14 is a common form of carbon, which decays over time. Carbon dating is commonly used in determining the age of fossilize matter. The amount carbon 14 measured in grams contained by a fossilized flower is modeled by the equation f(t)=5e-‐ct. This equation models the amount of carbon-‐14 from the time the flower dies (t=0).

(a) At the time of death how many grams of carbon-‐14 did the flower contain? (b) The half -‐life of carbon-‐14 is approximately 5,730 years. This is the time that it takes for the initial amount

to become half of its original value. Use this information to find the value of constant c in the function. (c) If there are currently .5g of carbon-‐14, approximately when did the plant die?



Trigonometric Functions F.TF

Common Core Cluster

Extend the domain of trigonometric functions using the unit circle.


F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Ex. A central angle of a circle intercepts an arc on the same circle. The length of the arc is some fraction of the circumference of the circle and is related to the radian measure of the central angle. To make sense of how a radian is defined, conduct the following investigation.

1. Construct three concentric circles with radii of 10, 20, and 30 cm with center O. 2. Extend a ray from the center of the circle through all three circles. Label the intersections of the

ray and the circles as B1,B2.,B3 respectively. 3. Using a piece of string capture the length of the radius OB1. Use this length to create arc𝐴!𝐵! so

that the length of𝐴!𝐵! is equal to the radius OB1. 4. Draw an angle A1OB1 and record its measure in degrees. 5. Repeat steps 3 and 4 for the remaining two circles. 6. Describe the relationship that exists if the intercepted arc and the radius are the same length.

Generalizing your answer for a circle with center O and radius r. What would be the approximate degree measure of angle AOB if arc AB is r units long?

F.TF.1 These steps lead the students to the definition of a radian. A radian is the measure of the central angle of a circle that intercepts an arc of the same measure. It is acceptable to let students define 1 radian as about 57o. Through classroom discussion (not direct instruction) the instruction should guide students to the more precise conversion factor of 360° = 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠.

Ex. In a circle with a radius of 5cm and is centered at point O angle AOB intercepts arc AB. Arc AB has a length of 10cm. What are the radian and degree measures of angle AOB. Ex. In a circle with a radius of 8cm and is centered at point O angle AOB intercepts arc AB. Arc AB has a length of 16cm. What are the radian and degree measures of angle AOB.



Ex. Two students are discussing moving between degrees and radians. Critique and use their arguments in the questions that follow their discussion. James I remember that 360o = 2Pi radians In finding the radian measure of 120o I divided 120 by 360 to get 1/3. Since my angle was 1/3 of the whole circle and that there are 2Pi radians in a circle I think that the angle will be 1/3 of 2Pi which is 2Pi/3

Jim If 360 degrees equals 2Pi radians then if I divide both sides 360, then 1 degree is equal to Pi/180. There 120 degrees times Pi/180 would be radian measure, which reduces to 2Pi/3.

(a) Using James’ method find the radian measure of 30 degrees. (b) Using Jim’s method find the radian measure of 30 degrees. (c) Which method is your preferred method? Why?

F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

F.TF.2 This standard should be approached by students by rotating a circular object (ie Ferris Wheel) around and recording the angle formed with the origin, the height off the ground, and the horizontal height from the center of the circle. Below are brief instructions of how to set this up. Construct a coordinate axes on a large sheet of paper. From the origin mark off “famous” unit circle angles. (This will create a large asterisk like drawing.) On a separate piece of a paper, construct a large “unit” circle. Attach the center of the circle to the origin of the axes with a brad for rotational purpose. Students should draw or place a sticker of a “rider” at (1,0). While not imperative, this would work better if the grid was placed on a 10mm graph paper and the circle constructed with a transparency. Ex. With your rotational circle, record the following data points: Angle Formed by extending ray from center of circle to “rider.”

Height of “rider” from x-‐axis (ground)

Horizontal distance “rider” is from origin.

Pi/6 Pi/4

a) On a coordinate axes, graph the following points (angle formed, height) b) On a separate axes, graph the following points (angle formed, horizontal distance) c) For each angle, find the ratio of the height to the horizontal distance. d) On yet another axes, graph the following points (angle formed, ratio in part (c))




Common Core Cluster

Model periodic phenomena with trigonometric functions


F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*

F.TF.5 Use sine and cosine to model periodic phenomena such as the ocean’s tide or the rotation of a Ferris wheel. F.TF.5 Given the amplitude; frequency; and midline in situations or graphs, determine a trigonometric function used to model the situation. Ex. At midnight the water at a particular beach is at high tide. At the same time a gauge at the end of a pier reads 10 feet. Low tide is reached at 6 AM when the gauge reads 4ft.

(a) Choose which trig function would be the best fit for this model (assuming midnight is t=0). Justify your choice using specific characteristics of trigonometric function graphs.

(b) Determine the midline, amplitude and frequency using the above tidal information. You must show all computations and explain why you performed each computation.

(c) Write a function based on parts one and two to represent the above tidal information. (d) If the times for high and low tides are reversed what (if anything) would change in the equation

from part (c). Justify your conclusion. (e) If you were instructed to let t=0 represent 9pm, would your function in part (a) still be the most

convenient choice? Why or why not? If not, convince you teacher what a better choice would be.


Common Core Cluster

Prove and apply trigonometric identities.


F.TF.8 Prove the Pythagorean F.TF.8. Use the following questions to follow up with the table developed in T.TF.2.



identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios.

Ex.

(a) Verify that at every height and horizontal value on the circle the radius is constant at 1. (Students should either construct the triangle within the circle and apply the Pythagorean theorem)

(b) What trigonometric function represents the height of the rider at any angle? How do you know? (c) What trigonometric function represents the horizontal distance of the rider at any angle?

How do you know? (d) Rather than writing x and y to determine the radius at any angle, use the results from (b) and (c) to

rewrite the Pythagorean Theorem relationship in terms of trigonometric functions.



Congruence G.CO

Common Core Cluster

Experiment with transformations in the plane.


G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Definitions used to begin building blocks for proof –Infuse these definitions into proofs and other problems. Pay attention to Mathematical practice 3 “Construct viable arguments and critique the reasoning of others.” Understand and use stated assumptions, definitions and previously established results in constructing arguments. Also, mathematical practice #6 Attend to precision: Communicate precisely to others and use clear definitions in discussion with others and in their own reasoning. Ex. Needed

Congruence G.CO

Common Core Cluster

Prove geometric theorems. Using logic and deductive reasoning, algebraic and geometric properties, definitions, and proven theorems to draw conclusions. (Encourage multiple ways of writing proofs, to include paragraph, flow charts, and two-‐column format). These proof standards should be woven throughout the course. Students should be making arguments about content throughout their geometry experience. The focus in the G.CO.9-‐11 is not the particular content items that they are proving. However, the focus is on the idea that students are proving geometric properties. Pay close attention to the mathematical practices especially number three, “Construct viable argument and critique the reasoning of others.”


G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles

G.CO.9 In 8th grade, students have already experimented with these angle/line properties (8.G.5). The focus at this level is to prove these properties, not just to use and know them. Ex. Prove that any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the line.



are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Ex. A carpenter is framing a wall and wants to make sure his the edges of his wall are parallel. He is using a cross-‐brace as show in the diagram below. What are several different ways he could verify that the edges are parallel? Can you write a formal argument to show that these sides are parallel? Pair up with another student who created a different argument than yours, and critique their reasoning. Did you need to modify the diagram in anyway to help your argument?

G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G.CO.10 Using any method you choose, construct the medians of a triangle. Each median is divided up by the centroid. Investigate the relationships of the distances of these segments. Can you create a deductive argument to justify why these relationships are true? Can you prove why the medians all meet at one point for all triangles? Extension: using coordinate geometry, how can you calculate the coordinate of the centroid? Can you provide an algebraic argument for why this works for any triangle? Using Interactive Geometry Software or tracing paper, investigate the relationships of sides and angles when you connect the midpoints of the sides of a triangle. Using coordinates can you justify why the segment that connects the midpoints of two of the sides is parallel to the opposite side. If you have not done so already, can you generalize your argument and show that it works for all cases? Using coordinates justify that the segment that connects the midpoints of two of the sides is half the length of the opposite side. If you have not done so already, can you generalize your argument and show that it works for all cases? Ex. Needed

G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

G.CO.11 Ex. Jerry’s laying out the foundation for a rectangular outdoor tool shed. By regulation, he must ensure that it indeed fulfills the definition of a rectangle. The only tools he has with him are pegs (for nailing in the ground to mark the corners), string and a tape measure. Create a plan for Jerry to follow so that he can be sure his foundation is rectangular. Justify why your plan works. Discuss your method with another student to make sure your plan is error proof. Connect this standard with G.CO.8 and use triangle congruency criteria to determine all of the properties of parallelograms and special parallelograms.



Congruence G.CO

Common Core Cluster

Make geometric constructions. Create formal geometric constructions using a compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software.


G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.CO.12 Copy a congruent segment G.CO.12 Copy a congruent angle. G.CO.12 Bisect a segment G.CO.12 Bisect an angle Ex. Use various geometric tools to preform the above geometric constructions. Explain special characteristics that arise as a result of the construction. G.CO.12 Construct perpendicular lines, including the perpendicular bisector of a line segment. Ex. Using a compass and straightedge, construct a perpendicular bisector of a segment. Prove/justify why this process provides the perpendicular bisector. Given a triangle, construct the circumcenter and justify/prove why the process/construction gives the point which is equidistant from the vertices. Given a triangle, construct the incenter and justify/prove why the process/construction gives the point which is equidistant from the sides of the triangle. ***It makes sense to combine this standard with G-‐CO.9 and 10 and have students make arguments about why these constructions work. G.CO.12 Construct a line parallel to a given line through a point not on the line. Ex. Needed



Similarity, Right Triangles, and Trigonometry G.SRT

Common Core Cluster

Understand similarity in terms of similarity transformations.


G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G.SRT.2 Use the idea of geometric transformations to develop the definition of similarity. G.SRT.2 Given two figures determine whether they are similar and explain their similarity based on the congruency of corresponding angles and the proportionality of corresponding sides. Instructional note: The ideas of congruency and similarity are related. It is important for students to connect that congruency is a special case of similarity with a scale factor of one. Therefore these similarity rules can be expanded to work for congruency in triangles. AA similarity is the foundation for ASA and AAS congruency theorems. Knowing from the definition of a dilation, angle measures are preserved and sides change by a multiplication of scale factor k. Ex. Needed

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G.SRT.3 Use the properties of similarity transformations to develop the criteria for proving similar triangles by AA, SSS, and SAS. Connect this standard with standard G.SRT.4

Ex. Given that MNP is a dilation of ABC with scale factor k, use properties of dilations to show that the AA criterion is sufficient to prove similarity.



Similarity, Right Triangles, and Trigonometry G.SRT

Common Core Cluster

Prove theorems involving similarity.


G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G.SRT.4 Connect this standard with G.SRT.3

Ex. Line segment

€

AE is the distance across The French Broad River. John wants to find this distance so he paces of

€

ED and it is 60 paces,

€

DC is 120 paces and

€

EB is 80 paces. How could John use similar triangles to find the width of the river? Can you calculate the distance of

€

AB ? Create a formal argument to show that

€

AEED

=ABBC

G.SRT.4 Start with a right triangle and construct the altitude to the hypotenuse. This produces three similar right triangles whose proportional relationships can lead to a proof of the Pythagorean theorem. Reference students’ experience with the Pythagorean Theorem in 8th grade in 8.G.6-‐8.

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G.SRT.5 Connect with G.CO.8 and G.CO.11. and use triangle congruency criteria to determine all of the properties of parallelograms and special parallelograms.



Ex. Sam claims that there is a SSSS congruency criteria for quadrilaterals? Do you agree or disagree? Justify your answer. If you disagree, can you provide a counterexample? If you agree, can you prove it?



Circles G.C

Common Core Cluster

Understand and apply theorems about circles.


G.C.1 Prove that all circles are similar.

G.C.1 Show that the ratio of the radiuscircumference

will be the scale factor for any circle. Or, show that the ratio

of the circumference

radius is the same for any circle, namely

€

π .

Ex. Needed

G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Ex. An archeologist dug up an edge piece of a circular plate. He wants to know what the original diameter of the plate was before it broke. However, the piece of pottery does not display the center of the plate. How could he find the original dimensions? Ex. Apply what you know about chords in circles. You are given two circles of diameters 24cm and 10cm. Locate two chords of equal length, 8 cm for each circle. Compare their distances from the centers of the circles. Ex. Using Interactive Geometry Software investigate measurements of central angles and inscribed angles that intercept the same arc. What is the relationship of these angle measures? Then, explain why inscribed angles on a diameter are always 90°. Also, make an argument for why the opposite angles in a quadrilateral inscribed in a circle are supplementary. Ex. A discus athlete is performing in a local competition. Sketch an overhead view that shows the ball’s circular path, the target, and the position at the moment that she releases the discus.



G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Given a triangle, construct the circumcenter and justify/prove why the process/construction gives the point which is equidistant from the vertices. Given a triangle, construct the incenter and justify/prove why the process/construction gives the point which is equidistant from the sides of the triangle. ***It makes sense to combine this standard with G-‐CO.9 and 10 and have students make arguments about why these constructions work. Resource from Shell Centre (http://map.mathshell.org.uk/materials/download.php?fileid=764) Listed under ‘Circles in Triangles.’ Construct an argument to show that opposite angles in an inscribed quadrilateral are supplementary.

Circles G.C

Common Core Cluster

Find arc lengths and areas of sectors of circles.


G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G.C.5 Use similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the radius, identifying the constant of proportionality as the radian measure of the angle. G.C.5 Find the arc length of a circle. G.C.5 Using similarity, derive the formula for the area of a sector. G.C.5 Find the area of a sector in a circle. Ex. Needed



Expressing Geometric Properties with Equations G.GPE

Common Core Cluster

Translate between the geometric description and the equation for a conic section.


G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

G.GPE.1 Use the Pythagorean Theorem to derive the equation of a circle, given the center and the radius. G.GPE.1 Given an equation of a circle, complete the square to find the center and radius of a circle. Ex. Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is that given distance from the given point (based on the Pythagorean theorem).

G.GPE.2 Derive the equation of a parabola given a focus and a directrix.

G.GPE.2 Given a focus and directrix, derive the equation of a parabola Parabola is defined as “the set of all points P in a plane equidistant from a fixed line and a fixed point in the plane.” The fixed line is called the directrix, and the fixed point is called the focus. Ex. Derive the equation of the parabola that has the focus (1, 4) and the directrix x=-‐5. Ex. Derive the equation of the parabola that has the focus (2, 1) and the directrix y=-‐4. Ex. Derive the equation of the parabola that has the focus (-‐3, -‐2) and the vertex (1, -‐2).



Modeling with Geometry G.MG

Common Core Cluster

Apply geometric concepts in modeling situations.


G.MG.3 Apply geometric methods to solve design problems (e.g. designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

G.MG.3 Solve design problems by designing an object or structure that satisfies certain constraints, such as minimizing cost or working with a grid system based on ratios (i.e., The enlargement of a picture using a grid and ratios and proportions) Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-‐deep end. (a) Sketch the pool and indicate all measures on the sketch. (b) How much water is needed to fill the pool to the top? To a level 6 inches below the top? (c) One gallon of pool paint covers approximately 75 sq feet of surface. How many gallons of paint are

needed to paint the inside walls of the pool? If the pool paint comes in 5-‐gallon cans, how many cans are needed?

(d) How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all sides?

(e) How many 6-‐inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool? If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed?



Interpreting Categorical and Quantitative Data S.ID

Common Core Cluster

Summarize, represent, and interpret data on a single count or measurement variable.


S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

S.ID.4 Understand the characteristics of the Standard Normal Distribution (µ = 0, σ = 1), including symmetry, the Empirical Rule (68-‐95-‐99.7 rule) and the fact that the mean = median = mode. Ex. Describe the characteristics of the Standard Normal Distribution. S.ID.4 Find the number of standard deviations a value is from the mean by calculating its z-‐score. Ex. IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15. How many standard deviations below the mean is an IQ score of 75? S.ID.4 Use the Empirical Rule to estimate population percentages for a set of data that is approximately normally distributed. Ex. IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15. What percent of the population has IQ scores above 115? What percent have IQ scores between 85 and 130? S.ID.4 Understand that population percentages correspond to areas under the normal distribution curve between given values. S.ID.4 Use a table of z-‐scores, spreadsheets and calculators to find areas under the curve to estimate population percentages. Interpret these percentages in context. Ex. The heights of adult males are approximately normally distributed with µ = 70 in. and σ = 3 in. What percentage of adult males are between 5 ½ and 6 ½ feet tall? What percentage of adult males are 7 ft tall or taller?



S.ID.4 Understand that the use of the normal distribution to estimate population percentages is only appropriate for mound-‐shaped, symmetrical distributions (not skewed distributions). Ex. Which of the following sets of data are approximately normally distributed?

weights of adult females incomes of NFL football players diameters of trees in Umstead Park shoe sizes of sophomore males at your school female ages at marriage



Making Inferences and Justifying Conclusions S.IC

Common Core Cluster

Understand and evaluate random processes underlying statistical experiments.


S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S.IC.1 Explain the difference between a population parameter and a sample statistic. Ex. What is the difference between a population parameter and a sample statistic? S.IC.1 Understand that random samples tend to be representative of the population they are drawn from and therefore we can draw conclusions about the population based on the sample. If conclusions cannot be drawn from the random sample, discuss why and propose a better way to select a random sample. Ex. Describe the process of statistical inference. Ex. For statistical inference, why is it important that a sample be representative of the population it is drawn from? S.IC.1 Demonstrate understanding of the different kinds of sampling methods (simple random sample, systematic random sample, stratified random sample, cluster or multistage sample, convenience sample). Ex. What kind of sampling method would you use for each of the following situations? Explain why in each case.

To determine which gubernatorial candidate voters are most likely to choose in the next election. To determine the quality of potato chips being produced at a factory. To determine the average size of bass fish in a lake. To determine the average number of televisions per household in the US.



Making Inferences and Justifying Conclusions S.IC

Common Core Cluster

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.


S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S.IC.3 Students should understand that sample surveys are used to collect data from human subjects to describe the population of interest. Experiments and observational studies are used to establish a cause and effect relationship. In an experiment, a treatment is imposed on the experimental units. In an observational study, the treatment is not imposed but the relationship between the variables of interest is observed (e.g. smoking and birth defects). S.IC.3 Students should understand that in sample surveys randomization occurs when the sample is selected. For surveys, randomization ensures that the sample is representative of the population it is drawn from. With experiments, randomization occurs when experimental units are assigned to treatments (randomized comparative experiment) in order to ensure that the treatment groups are equivalent. In an observational study, there is no random assignment of treatments. (For example, when looking at the relationship between smoking and cancer, we do not “assign” people to be smokers, they choose to be one or the other. We then observe the rate of cancer for smokers and for non –smokers and compare.) S.IC.3 Given a situation, decide whether an experiment or observational study is more appropriate to establish a cause and effect relationship. Ex. For each of the following situations, decide whether an experiment or an observational study is more appropriate to determine if there is a cause and effect relationship:

a. Cell phone use and brain tumors b. Use of a fertilizer and growth of plants



S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

S.IC.4 Given data from a sample survey, calculate the sample mean or sample proportion and use it to estimate the population value. Ex. You conduct a survey at your school to determine who is most likely to be voted as Prom Queen. You randomly survey 20 students and receive the responses below. Gabby Sarah Gabby Lola Anna Anna Gabby Lola Anna Sarah Gabby Lola Anna Gabby Gabby Sarah Anna Gabby Lola Sarah How likely is it that Gabby will be Prom Queen? Create an argument justifying your conclusion. S.IC.4 Use simulation to collect multiple samples. Calculate the sample mean or proportion for each and use this information to determine a reasonable margin of error for the population estimate. Ex. For the situation above, use technology to generate random numbers to simulate multiple samples of 20 votes. Calculate the proportion of votes for Gabby in each sample and use this information to determine a reasonable margin of error for the proportion of students who are going to vote for Gabby for Prom Queen.

S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

S.IC.5 Determine whether one treatment is more effective than another treatment in a randomized experiment. Ex. Sal purchased two types of plant fertilizer and conducted an experiment to see which fertilizer would be best to use in his greenhouse. He planted 20 seedlings and used Fertilizer A on ten of them and Fertilizer B on the other ten. He measured the height of each plant after two weeks. Use the data below to determine which fertilizer Sal should use. Write a letter to Sal describing your recommendation. Be sure to explain fully how you arrived at your conclusion.

Plant Height (cm) Fertilizer A 23.4 30.1 28.5 26.3 32.0 29.6 26.8 25.2 27.5 30.8 Fertilizer B 19.8 25.7 29.0 23.2 27.8 31.1 26.5 24.7 21.3 25.6

S.IC.5 Use simulations to generate data simulating application of two treatments. Use results to evaluate significance of differences. Ex. Use the list above to generate simulated treatment results by randomly selecting ten plant heights from



the twenty plant heights listed. (Mix them all together.) Calculate the average plant height for each “treatment” of ten plants. Find the difference between consecutive pairs of “treatment” averages and compare. Does your simulated data provide evidence that the difference in average plant heights using Fertilizer A and Fertilizer B is significant? Explain.

S.IC.6 Evaluate reports based on data.

S.IC.6 Evaluate reports based on data on multiple aspects (e.g. experimental design, controlling for lurking variables, representativeness of samples, choice of summary statistics, etc.) Ex. Find a statistical report based on an experiment and evaluate the experimental design.



Using Probability to Make Decisions S.MD

Common Core Cluster

Use probability to evaluate outcomes of decisions.


S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S.MD.6 Make decisions based on expected values. Use expected values to compare long term benefits of several situations. Ex. Needed

S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

S.MD.7 Explain in context decisions made based on expected values. Ex. Needed

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