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Algebra ● Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13. What is the purpose of this document? To increase student achievement by ensuring educators understand what the standards mean a student must know and be able to do completely and comprehensively. What is in the document? Descriptions of what each standard means a student will know and be able to do. The “unpacking” of the standards done in this document is an effort to answer a simple question “What does this standard mean that a student must know and be able to do?” and to ensure that description is helpful, specific and comprehensive. How do I send feedback? We intend the explanations and examples in this document to be helpful, specific and comprehensive. That said, we believe that as this document is used, teachers and educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us at feedback@dpi.state.nc.us and we will use your input to refine our unpacking of the standards. Thank You! Just want the standards alone? You can find the standards alone at www.corestandards.org. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 1 Seeing Structure in Expressions A-SSE Common Core Cluster Interpret the structure of expressions Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? 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. 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.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 5 expression. For example, the expression $10, 000 (1.055) 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. (Level I) Ex. The expression 150 + 0.10𝑆 models the income earned based on total monthly sales. Interpret the terms and coefficients of the expression in the context of this situation. (Level II) Ex. The expression −4.9𝑡 ! + 17𝑡 + 0.6 describes the height in meters of a basketball t seconds after it has been thrown vertically in the air. Interpret the terms and coefficients of the expression in the context of this situation. (Level III) 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 in context. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 2 A-SSE.1b Students group together parts of an expression to reveal underlying structure. (Level I) Ex. The expression 10 + .50(𝑚 − 5) models the cost of a taxi ride that is m miles long. What does (𝑚 − 5) represent in the context of this situation? (Level II) Ex. What information related to symmetry is revealed by rewriting the quadratic formula as 𝑥 = !! !! ± ! ! !!!" !! ? (Level III) 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. 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. (Level I/II) Ex. The expression 4000𝑝 − 250𝑝 ! represents the income at a concert, where p is the price per ticket. Rewrite this expression in another form to reveal the expression that represents the number of people in attendance based on the price charged. (Level III) 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. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 3 Seeing Structure in Expressions A-SSE Common Core Cluster Write expressions in equivalent forms to solve problems Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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 1.15! can ! !" be rewritten as (1.15 )!"! ≈ (1.012)!"! to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.3a Students factor quadratic expressions and find the zeros of the quadratic function it represents. Zeroes are the x values that yield a y value of 0. They should also explain the meaning of the zeros as they relate to the problem. Ex. If (3𝑚 ! − 15𝑚) is the income gathered at a rock concert, what values of m would produce an income of 0? A-SSE.3b Students rewrite a quadratic expression in the form a- add to limitations in course 3 where limitation a = 2 1 ( x − h ) + k , with a = 1, to identify the vertex of the parabola (h, k), and explain its meaning in context. Ex. The quadratic expression 𝑥 ! − 24𝑥 + 55 models the height of a ball thrown vertically. Find the vertex and interpret its meaning in this context. – context and equation don’t match (thrown ball should have a negative x coefficient) A-SSE.3c Use properties of exponents to write an equivalent form of an exponential function to reveal and explain specific information about the rate of growth or decay. Ex. The equation 𝑦 = 14000(0.8) ! represents the value of an automobile x years after purchase. Find the yearly and the monthly rate of depreciation of the car. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 4 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− r n ) (1− r) Mortgage payments can be found using the formula, P = iA where P represents the payment amount, A 1− (1+ i)−n 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 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? Spreadsheet connection? High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 5 Arithmetic With Polynomials and Rational Expressions A-APR Common Core Cluster Perform arithmetic operations on polynomials Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? 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. (Level III) Ex. If the radius of a circle is ( 5x − 2 ) kilometers, write an expression for the area of the circle. Ex. Explain why (4𝑥 ! + 3)! does not equal (16𝑥 ! + 9). High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 6 Arithmetic With Polynomials and Rational Expressions A-APR Common Core Cluster Understand the relationship between zeros and factors of polynomials Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? 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 factor of p(x). 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(𝑎) = 0, then (𝑥 − 𝑎) is a factor of 𝑓(𝑥), which means that 𝑎 is a root of the function 𝑓(𝑥). This is known as the Factor Theorem. (Level III) Ex. Given 𝑓 𝑥 = 2𝑥 ! + 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, which means that 𝑓 is divisible by 𝑥 − 𝑐 . 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. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 7 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. (Level II) Ex. Given the function 𝑦 = 2𝑥 ! + 6𝑥 – 3, list the zeros of the function and sketch the graph. (Level II) Ex. Sketch the graph of the function 𝑓 𝑥 = 𝑥 + 5 ! . What is the multiplicity of the zeros of this function? How does the multiplicity relate to the graph of the function? (Level III) 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. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 8 Arithmetic With Polynomials and Rational Expressions A-APR Common Core Cluster Use polynomial identities to solve problems Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? 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). A-APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1 A-APR.5 The Binomial Theorem describes the algebraic expansion of powers of a binomial. There are patterns that develop with the coefficients and the variables when expanding binomials. Pascal’s triangle is a triangular array that identifies the coefficients of an expanded binomial. The numbers in Pascal’s triangle are also evaluations of combinations, nCr. The values of the combinations correspond with the coefficients of the expanded binomial, which indicates how many times that term will appear in the completely expanded form. This is a connection between Probability and Algebra that should be made explicit. For example, when squaring the binomial (𝑎 + 𝑏), note that the product 𝑎𝑏 occurs twice: 1 The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. (Level III) Ex. Is (2𝑥 − 3)! − 64 equivalent to (2𝑥 − 11)(2𝑥 + 5)? Explain why or why not. Ex. Jessie claims that (𝑥 + 𝑦)! = 𝑥 ! + 2𝑥𝑦 + 𝑦 ! . Is he correct? Prove why or why not. Ex. Prove 𝑥 ! − 𝑦 ! = 𝑥 − 𝑦 (𝑥 ! + 𝑥𝑦 + 𝑦 ! ). Justify each step. ! Ex. Solve the quadratic 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 Justifying each step. What was interesting about the result? (a + b)2 = a 2 + ab + ab + b 2 = a 2 + 2ab + b 2 Using combinatorics, the coefficient of the second term would be 2C1 = 2. (Level III) Ex. Explain how to generate a row of Pascal’s triangle. Ex. What are the coefficients of the expanded terms of (𝑎 + 𝑏)5? Ex. Using the binomial theorem, expand (𝑎 + 𝑏)5. Ex. Why are the coefficients of a binomial expansion equal to values of nCr? High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 9 Arithmetic With Polynomials and Rational Expressions A-APR Common Core Cluster Rewrite rational expressions Common Core Standard 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. Unpacking What does this standard mean that a student will know and be able to do? A-APR.6 Rewrite rational expressions, a( x) r ( x) , in the form q ( x) + using long division, synthetic division or b( x ) b( x ) 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). (Level III) !"# 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 equivalent expression in the form of 𝑞 𝑥 + !(!) !!! !! ! !!!!! !!! can be written with an . Ex. Divide. Write the answer in the form of quotient plus remainder/divisor. 𝑥 ! + 3𝑥 𝑥! − 4 Ex. Use a computer algebra system to rewrite the following rational expression in quotient and remainder form High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 10 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. 9𝑥 ! + 9𝑥 ! − 𝑥 + 2 2 𝑥+ 3 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 rational expression in terms of 𝑥. High School Mathematics Common Core State Standards (! ! !!!!) !! sq. ft. and a height of !! (!!!) ft. Express the width of the rectangle as a Unpacked Content December 10, 2010 11 Creating Equations A-CED Common Core Cluster Create equations that describe numbers or relationships Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? 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 using a table or graph. See A-REI.11. ! ! ! ! ! ! , . Note, the resulting equation can be solved in Level I (Level I) Ex. The Tindell household contains three people of different generations. The total of the ages of the three family members is 85. a. Find reasonable ages for the three Tindells. b. Find another reasonable set of ages for them. c. One student, in solving this problem, wrote C + (C+20)+ (C+56) = 85 1. What does C represent in this equation? 2. What do you think the student had in mind when using the numbers 20 and 56? 3. What set of ages do you think the student came up with? (Level I) Ex. A salesperson earns $700 per month plus 20% of sales. Write an equation to find the minimum amount of sales needed to receive a salary of at least $2500 per month. (Level I) Ex. A scientist has 100 grams of a radioactive substance. Half of it decays every hour. Write an equation to find how long it takes until 25 grams are left. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 12 (Level II) Ex. A pool can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the pool is full, it can be drained by pipe C in 4 hours. If the pool is initially empty and all three pipes are open, write an equation to find how long it takes to fill the pool. (Level III) Ex. A cardboard box company has been contracted to manufacture open-top rectangular storage boxes for small hardware parts. The company has 30 cm x 16 cm cardboard sheets. They plan to cut a square from each corner of the sheet and bend up the sides to form the box. If the company wants to make boxes with the largest possible volume, what should be the dimensions of the square to be cut out? What are the dimensions of the box? What is x cm the maximum volume? x cm 16 cm 30 cm 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. (Level I) Ex. The height of a ball t seconds after it is kicked vertically depends upon the initial height and velocity of the ball and on the downward pull of gravity. Suppose the ball leaves the kicker’s foot at an initial height of 0.7 m with initial upward velocity of 22m/sec. Write an algebraic equation relating flight time t in seconds and height h in meters for this punt. (Level I) Ex. In a woman’s professional tennis tournament, the money a player wins depends on her finishing place in the standings. The first-place finisher wins half of $1,500,000 in total prize money. The second-place finisher wins half of what is left; then the third-place finisher wins half of that, and so on. a. Write a rule to calculate the actual prize money in dollars won by the player finishing in nth place, for any positive integer n. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 13 b. Graph the relationship that exists between the first 10 finishers and the prize money in dollars. c. What pattern do you notice in the graph? What type of relationship exists between the two variables? (Level II) Ex. The intensity of light radiating from a point source varies inversely as the square of the distance from the source. Write an equation to model the relationship between these quantities given a fixed energy output. need Level III 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. 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. (Level II) 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. c. How many of each type of calculator should the company produce to maximize profit using the stock on hand? (Level II) Ex. If H = kA (T1 − T2 ) , solve for T2 L High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 14 Reasoning with Equations and Inequalities A-REI Common Core Cluster Understand solving equations as a process of reasoning and explain the reasoning Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? A-REI.1 Explain each step in solving a simple equation as following from the 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. 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. Assuming an equation has a solution, construct a convincing argument that justifies each step in the solution process with mathematical properties. 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. (Level I) 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 ± 0i (Level II) Ex. Solve 5 − − ( x + 4) = 2 for x. Ex. Mary solved x = 2 − x for x and got x=-2, and x=1. Are all the values she found solutions to the equation? Why or why not? Ex. Solve 3 x 3 = − for x. x −3 x −3 2 High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 15 Reasoning with Equations and Inequalities A-REI Common Core Cluster Solve equations and inequalities in one variable Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-REI.3 Solve linear equations in one variable, including coefficients represented by letters. (Level I) Ex. Solve, Ax +B =C for x. What are the specific restrictions on A? Ex. What is the difference between solving an equation and simplifying an expression? Ex. Grandma’s house is 20 miles away and Johnny wants to know how long it will take to get there using various modes of transportation. a. Model this situation with an equation where time is a function of rate in miles per hour. b. For each mode of transportation listed below, determine the time it would take to get to Grandma’s. Mode of Transportation Rate of Travel in mph Time of Travel hrs. bike 12mph car 55mph walking 4mph A-REI.3 Solve linear inequalities in one variable, including coefficients represented by letters. (Level I/II) Ex. A parking garage charges $1 for the first half-hour and $0.60 for each additional half-hour or portion thereof. Write an inequality and solve it to find how long you can park if you have only $6.00 in cash. Ex. Compare solving an inequality to solving an equation. Compare solving a linear inequality to solving a linear equation. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 16 A-REI.4a 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. 2 A-REI.4a Transform a quadratic equation written in standard form to an equation in vertex form, (x - p ) = q , by completing the square. (Level II) 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. (Level II/III) Ex. Solve y = ax2 +bx+c for x. What do you notice about your solution? A-REI.4b 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 (Level II) 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 a ± bi , where a and b are real numbers. Students should understand that the solutions are always complex numbers of the form a ± bi . Real solutions are produced when b = 0, and pure imaginary solutions are found when a = 0. The value of the discriminant, b 2 − 4ac , determines how many and what type of solutions the quadratic equation has. (Level II) Ex. Ryan used the quadratic formula to solve an equation and x = 8 + (−8)2 − 4(1)(−2) was his result. 2(1) 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? High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 17 Reasoning with Equations and Inequalities A-REI Common Core Cluster Solve systems of equations Common Core Standard A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Common Unpacking What does this standard mean that a student will know and be able to do? A-REI.5 Solve systems of equations using the elimination method (sometimes called linear combinations). Since equations that are multiples of each other have the same solution, one equation in a system can be multiplied by a constant to produce another equation with the same solution. For example, in the system. Core plus: Jenna Conrad and Andrea 3x + 2y = 6 x - 4y = 2 the first equation can be multiplied by 2 to generate 6x + 4y = 12. Then the two equations are added to eliminate y. 6x + 4y = 12 x - 4y = 2 7x = 14 7 7 x = 2 Then the x-coordinate of 2 can be substituted in either equation to retrieve the y-coordinate, resulting in (2, 0) as the solution for this system. (Level II) Ex. Solve the system: -3x + 5y = 6 2x + y = 6 High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 18 A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.6 Solve systems of equations using graphs. (Level I/II) Ex. The equations y = 18 + .4m and y = 11.2 + .54m Give the lengths or two different springs in centimeters, as mass is added in grams, m, to each separately. a. Graph each equation. b. When are the springs the same length? c. When is one spring at least 10cm longer than the other? d. Write a sentence comparing the two springs. 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. A-REI.7 Solve a system containing a linear equation and a quadratic equation in two variables (conic sections possible) graphically and symbolically. Add context or analysis A-REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.8 Write a system of linear equations as a single matrix equation. Add context or analysis (Level I) Ex. Solve the following system graphically and symbolically: x2 + y2 = 1 y=x Ex. Write the system of equations as a matrix equation. x+2y-z=1 2x-y+3z=2 2x+y+z=-1 A-REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). A-REI.9 Find the inverse of the coefficient matrix in the equation, if it exits. Use the inverse of the coefficient matrix to solve the system. Use technology for matrices with dimensions 3 by 3 or greater. The inverse of a matrix exists if and only if the matrix is square and the determinate is not 0. The process of finding the inverse of a 2x2 matrix is easily done by hand, however technology should be used to find the inverse of 3x3s or greater. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 19 Solving a matrix equation is similar to solving linear equations in that we multiply both sides by the inverse matrix to find the solution. However the inverse matrix must be multiplied on the left on each side, because matrix multiplication is not commutative. After multiplying, the left side results in the identity matrix multiplied by the variable matrix and the right side yields the solutions to the variables in the system of equations. The product of the identity matrix and any other matrix with the same dimensions is that matrix. Ex. AX = B A-1A Reasoning with Equations and Inequalities A-REI Common Core Cluster Represent and solve equations and inequalities graphically Unpacking Common Core Standard What does this standard mean that a student will know and be able to do? A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 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 true. Add context or analysis Ex. Which of the following points lie on the graph of the equation −5𝑥 + 2𝑦 = 20? a. (4, 0) b. (0, 10) c. (-1, 7.5) d. (2.3, 5) How many solutions does this equation have? Justify your conclusion. High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 20 A-REI.11 Explain why the xcoordinates 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.12 Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. 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. (Level 1/II) Ex. John and Jerry both have jobs working at the town carnival. They have different employers, so their daily wages are calculated differently. John’s earnings are represented by the equation, p(x) = 2x and Jerry’s by g(x) = 10 + 0.25x. a. What does the variable x represent? b. If they begin work next Monday, Michelle told them that Friday would be the only day they made the same amount of money. Is she correct in her assumption? Explain your reasoning. c. When will Jerry earn more money than John? When will John earn more money than Jerry? During what day will their earnings be the same? Justify your conclusions. A-REI.12 By graphing a two variable inequality, students understand that the solutions to this inequality are all the ordered pairs located on a portion or side of the coordinate plane that, when substituted into the inequality, make the equation true. Students should be able to graph the inequality, specifying whether the points on the boundary line are also solutions by using a dotted or solid line. Using a variety of methods, which include selecting and substituting test points into the inequality, students should be able to determine which portion or side of the graph contains the ordered pairs that are the solutions to the original inequality. (Level III) Ex. Michelle has $80 total for shopping. In her favorite store, each shirt costs $12 and each pair of earrings cost $4. a. Write an inequality representing the total number of shirts and earrings she can purchase without spending more than $80. b. Graph this inequality on a coordinate plane. c. Are the points on the boundary line included in the solution set? Explain why or why not. d. Which portion of the plane contains the solutions to the inequality? Demonstrate this appropriately on your graph. e. William claims that all of the solutions to this inequality are reasonable solutions. Do you agree with him? Development an argument to support your position? High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 21 A-REI.12 When given a system of linear inequalities, students will understand the need to graph the linear inequalities separately, finding the solutions for each. When shading each inequality’s half-plane or solution set, students understand that the over-lapping region of the half-planes represents all the ordered pairs shared by both linear inequalities. Therefore, when these ordered pairs are substituted into each inequality, their values will be congruent, and therefore, these ordered pairs are solutions to both linear inequalities. (Level III) Ex. The Flatbread Pizza Palace makes gourmet flatbread pizzas for sale to hotel chains. They only sell vegetarian and pepperoni pizzas to the hotels. Their business planning has the following constraints and objective: • Each vegetarian pizza takes 15 minutes of labor and each pepperoni pizza takes 8 minutes of labor. At most, the plant has 4,800 minutes of labor available each day. • The restaurant freezer can hold a total of at most 580 pizzas per day. • The pepperoni flatbread pizza is more popular than the vegetarian pizza, so the plant makes at most 290 vegetarian pizzas each day. • Each pepperoni pizza sold earns Flatbread Pizza Palace $4 profit and each vegetarian pizza sold earns them $3.25 profit. a. What are the variables in this situation? b. Write a system of linear equations representing each constraint. c. Write the objective function that will maximize the profit for Flatbread Pizza Palace. d. Graph the constraints High School Mathematics Common Core State Standards Unpacked Content December 10, 2010 22