Math Curriculum Guide Clovis Municipal Schools Algebra II e ed Spring 201
ALGEBRA II/ Pre-AP Algebra II COMMON CORE STANDARDS 1st Nine Weeks Create equations that describe numbers or relationships A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (piecewise functions this 9 weeks) 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. (linear and quadratic this 9 weeks) Represent and solve equations and inequalities graphically 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. (linear functions this 9 weeks) Create equations that describe numbers or relationships A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Interpret functions that arise in applications in terms of the context F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 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. Analyze functions using different representations F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Perform arithmetic operations on polynomials CMS Factor polynomials of various types (e.g., difference of squares, perfect square trinomials, sum and difference of cubes). Interpret functions that arise in applications in terms of the context F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 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. F-IF.8a 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. Interpret the structure of expressions A-SSE.2 Use the structure of an expression to identify ways to rewrite it. 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. A-SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines. A-SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A-SSE.3c Use the properties of exponents to transform expressions for exponential functions. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).
ALGEBRA II/ Pre-AP Algebra II COMMON CORE STANDARDS 1st Nine Weeks, cont'd Use complex numbers in polynomial identities and equations N-CN.7 Solve quadratic equations with real coefficients that have complex solutions. N-CN.8 (+) Extend polynomial identities to the complex numbers. Perform arithmetic operations with complex numbers N-CN.1 Know there is a complex number "I" such that i2 = 1, and every complex number has the form a + bi with "a" and "b" real. N-CN.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Revised May 2016 Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%). Revised May 2016
ALGEBRA II/ Pre-AP Algebra II COMMON CORE STANDARDS 2nd Nine Weeks Interpret the structure of expressions A-SSE.1 Interpret expressions that represent a quantity in terms of its context. A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. 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. F-IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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. Understand the relationship between zeros and factors of polynomials 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. Represent and solve equations and inequalities graphically 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. (polynomial functions this 9 weeks) Perform arithmetic operations on polynomials CMS Factor polynomials of various types (e.g., difference of squares, perfect square trinomials, sum and difference of cubes). 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. 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 factor of p(x). Use polynomial identities to solve problems A-APR.4 Prove polynomial identities and use them to describe numerical relationships. Use complex numbers in polynomial identities and equations N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Use polynomial identities to solve problems 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) 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. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).
ALGEBRA II/ Pre-AP Algebra II COMMON CORE STANDARDS 2nd Nine Weeks, cont. Understand solving equations as a process of reasoning and explain the reasoning A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (radical equations this 9 weeks) Build a function that models a relationship between two quantities F-BF.1 Write a function that describes a relationship between two quantities. F-BF.1b Combine standard function types using arithmetic operations. Build new functions from existing functions F-BF.4 Find inverse functions. F-BF.4a 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. 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. Revised May 2016 F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (square root, cube root, absolute value this 9 weeks) Write expressions in equivalent forms to solve problems 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. Extend the properties of exponents to rational exponents. N- RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%). Revised May 2016
ALGEBRA II/ Pre-AP Algebra II COMMON CORE STANDARDS 3rd Nine Weeks Interpret the structure of expressions A-SSE.1 Interpret expressions that represent a quantity in terms of its context. A-SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. Rewrite rational expressions 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. 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. (Rational this 9 weeks) Represent and solve equations and inequalities graphically 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. (rational functions this 9 weeks) Understand solving equations as a process of reasoning and explain the reasoning A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (rational equations this 9 weeks) Understand and evaluate random processes underlying statistical experiments S-IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. 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.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.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. S-IC.6 Evaluate reports based on data. 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. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%). Revised May 2016
ALGEBRA II/ Pre-AP Algebra II COMMON CORE STANDARDS 4th Nine Weeks 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. (Exponential this 9 weeks) 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. F-IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent and solve equations and inequalities graphically 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. (logarithmic functions this 9 weeks) 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. F-IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Construct and compare linear, quadratic, and exponential models and solve problems 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. Extend the domain of trigonometric functions using the unit circle 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.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Model periodic phenomena with trigonometric functions F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Prove and apply trigonometric identities F-TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. Major Clusters-areas of intensive focus, where students need fluent understanding and application of the core concepts (approximately 70%). Supporting Clusters-rethinking and linking; areas where some material is being covered, but in a way that applies core understandings (approximately 20%). Additional Clusters-expose students to other subjects, though at a distinct level of depth and intensity (approximately 10%).
Common Core State s-mathematics Page # : 01 Domain: RN-The Real Number System Cluster: Extend the properties of exponents to rational exponents N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. MP.1 Making sense of problems and persevere in solving them. Algebra 2 Common Core, Pearson 2012 Chapter 6, Section 4 Algebra 2 Common Core, Pearson 2012 Chapter 6, Section 4 TE p. 386 #19-34
Common Core State s-mathematics Page # : 02 Domain: CN- The Complex Number System Cluster: Perform arithmetic operations with complex numbers N-CN.1 Know there is a complex number i such that i2 = 1, and every complex number has the form a + bi with a and b real. MP.6 Attend to precision. - Chapter 4, Section 8 Every number is a complex number or the form a + bi where a and b are elements of the Real Numbers and bi is an element of the Pure Imaginary Numbers. Students should know the sets and subsets of the Complex Number System. The identity, i= -1, is not only used to identify non-real solutions for particular functions, but is also be used to find the identity, i 2 = -1, which is used to simplify expressions. - Chapter 4, Section 8 TE p. 253 #8-12 #49-52 N-CN.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. MP.8 Look for and express regularity in repeated reasoning. - Chapter 4, Section 8 When adding, subtracting, or multiplying Complex Numbers and i 2 remains in the expression, use the identity 1 2 = -1 and the commutative, associative, and distributive properties to simplify the expression further. Example: Simplify the following expression. Justify each step using the commutative, associative and distributive properties. (3-2i)(-7+4i) - Chapter 4, Section 8 TE p. 253 #18-26 #48-55
Common Core State s-mathematics Page # : 03-04 Domain: CN- The Complex Number System Cluster: Use complex numbers in polynomial identities and equations (+) For advanced course: calculus, advanced statistics, etc. N-CN.7 Solve quadratic equations with real coefficients that have complex solutions. MP.1 Make sense of problems and persevere in solving them. - Chapter 4, Section 8 Extend strategies for solving quadratics such as; taking the square root and applying the quadratic formula, to find solutions of the form, a + bi, for quadratic equations. This extension is made when the identity i = -1 is used to simplify radicals having negative numbers under the radical. Examples: Within which number system can x 2 = 2 be solved? Explain how you know. Solve x 2 + 2x + 2 = 0 over the complex numbers. Find all solutions of 2x 2 + 5 = 2x and express them in the form a + bi. - Chapter 4, Section 8 TE p. 253, #39-44 #61-63 N-CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i). - Chapter 4, Section 8 Use polynomial identities to write equivalent expressions in the form of complex numbers. - Chapter 4, Section 8 TE p. 253, #33-38 #64-66
Common Core State s-mathematics Page # : 03-04 Domain: CN- The Complex Number System Cluster: Use complex numbers in polynomial identities and equations (+) For advanced course: calculus, advanced statistics, etc. N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. MP.3 Construct viable arguments and critique the reasoning of others. - Chapter 5, Section 6 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. Examples: How many zeros does -2x 2 + 3x - 8 have? Find all the zeros and explain, orally or in written format, your answer in terms of the Fundamental Theorem of Algebra. How many complex zeros does the following polynomial have? How do you know? p(x)=(x 2-3)(x 2 + 2) (x -3) (2x -1) - Chapter 5, Section 6 TE p. 323, #30-37
Modeling Common Core State s-mathematics Page # : 05-06 Domain: SSE-Seeing Structure in Expressions Cluster: Interpret the structure of expressions A-SSE.1 Interpret expressions that represent a quantity in terms of its context. MP.1 Make sense of problems and persevere in solving them. - Chapter 5, Section 1 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 $10,000(1.055) 5 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. Students should understand the vocabulary for the parts that make up the whole expression and be able to identify those parts and interpret their meaning in terms of a context. - Chapter 5, Section 1 TE p. 285, #8-19 #41-46 A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. MP.1 Make sense of problems and persevere in solving them. - Chapter 5, Section 1 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 $10,000 (1.055) 5 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. - Chapter 5, Section 1 TE p. 285, #8-19 #41-46
Modeling Common Core State s-mathematics Page # : 05-06 Domain: SSE-Seeing Structure in Expressions Cluster: Interpret the structure of expressions A-SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 +r)n as the product of P and a factor not depending on P. MP.1 Make sense of problems and persevere in solving them. - Chapter 8, Section 4 Students group together parts of an expression to reveal underlying structure. - Chapter 8, Section 4 TE p. 531, #14-19 #27-35 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). - Chapter 4, Section 4 Students rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression. Students should extract the greatest common factor (whether a constant, a variable, or a combination of each). If the remaining expression is quadratic, students should factor the expression further. Example: Factor x 3-2x 2-35x - Chapter 4, Section 4 TE p. 222 #73-78 p. 223, #91-92
Common Core State s-mathematics Grade: PreAPAlgebraII Page # : 07 Domain: SSE-Seeing Structure in Expressions Cluster: Interpret the structure of expressions A.SSE.3 Choose and produce an equivalent form of the 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 t can be rewritten as (1.15 1/12) ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. MP.1 Making sense of problems and persevere in solving them. Algebra 2 Common Core, Pearson 2012 - Chapter 4, Sections 4 & 6 Algebra 2 Common Core, Pearson 2012 - Chapter 4, Section 4 TE p. 221 #58-69, #73-78, #83-88 Chapter 4, Section 6 p.238 #46-51
Modeling Common Core State s-mathematics Page # : 08 Domain: SSE- Seeing Structure in Expressions Cluster: Write expressions in equivalent forms to solve problems 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. MP.3 Construct viable arguments and critique the reasoning of others. MP.8 Look for and express regularity in repeated reasoning. - Chapter 9, Section 5 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 S n. Example: In February, the Bezanson family starts saving for a trip to Australia in September. The Bezanson s expect their vacation to cost $5375. They start with $525. Each month they plan to deposit 20% more than the previous month. Will they have enough money for their trip? - Chapter 9, Section 5 TE p. 599, #8-16 #32-37 #41-45
Common Core State s-mathematics Page # : 09 Domain: APR- Arithmetic with Polynomials and Rational Expressions 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. www.kutasoftware.com/free.html (Algebra 1 worksheets - polynomials) 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. Example: If the radius of a circle is (5x-2) kilometers, write an expression for the area of the circle. Example: Explain why (4x 2 +3) 2 does not equal (16x 4 +9) CMS Factor polynomials of various types (e.g., difference of squares, perfect square trinomials, sum and difference of cubes). - Chapter 4, Section 4 - Chapter 5, Section 3 (sum/difference of cubes) - Chapter 4, Section 4 page 221 #14-46 p. 222 #58-69 - Chapter 5 Section 3 page 301 #10-12 #39-50 MP.8 Look for and express regularity in repeated reasoning.
Common Core State s-mathematics Page # : 10 Domain: APR- Arithmetic with Polynomials and Rational Expressions 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 factor of p (x). MP.3 Construct viable arguments and critique the reasoning of others. MP.8 Look for and express regularity in repeated reasoning. - Chapter 5, Section 4 The Remainder theorem says that if a polynomial p(x) is divided by x a, then the remainder is the constant p(a). That is, p(x)=q(x)(x-a)+p(a). So if p(a) = 0 then p(x) = q(x)(x-a). Let p(x)=x3-3x4 +8x2-9x +30> Evaluate p(-2). What does your answer tell you about the factors of p(x)? [Answer: p(-2) = 0 so x+2 is a factor.] - Chapter 5, Section 4 TE p. 309, #9-16 #32-39 #44-48 *EMPHASIZE LONG DIVISION 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. MP.1 Make sense of problems and persevere in solving them. MP.8 Look for and express regularity in repeated reasoning. - Chapter 5, Section 2 Graphing calculators or programs can be used to generate graphs of polynomial functions. Example: Factor the expression x 3 + 4x 2-59x -126 and explain how your answer can be used to solve the equation. Explain why the solutions to this equation are the same as the x-intercepts of the graph of the function f(x) = x 3 + rx 2-59x -126. - Chapter 5, Section 2 TE p. 293, #13-34
Common Core State s-mathematics Page # : 11 Domain: APR- Arithmetic with Polynomials and Rational Expressions Cluster: Use polynomial identities to solve problems (+) For advanced course: calculus, advanced statistics, etc. 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. MP.8 Look for and express regularity in repeated reasoning. - Chapter 5, Concept Byte with Lesson 5-5 Examples: Use the distributive law to explain why x 2 y 2 = (x y)(x + y) for any two numbers x and y. Derive the identity (x y) 2 = x 2 2xy + y 2 from (x + y) 2 = x 2 + 2xy + y 2 by replacing y by y. Use an identity to explain the pattern 22 12 = 3 32 22 = 5 42 32 = 7 52 42 = 9 [Answer: (n + 1) 2 - n 2 = 2n + 1 for any whole number n.] - Chapter 5, Concept Byte with Lesson 5-5 TE p. 318, #1-5 *EMPHASIZE PATTERNS FOR FACTORING SUM/DIFFERENCE OF CUBES 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. - Chapter 5, Section 7 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, n C r. 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: (a+b) 2 =a 2 +ab+ab+b 2 =a 2 +2ab+b 2 Using combinatorics, the coefficient of the second term would be 2C 1 =2. Examples: Use Pascal s Triangle to expand the expression (2x - 1) 4. Find the middle term in the expansion of (x 2 + 2) 18 - Chapter 5, Section 7 TE p. 329, #8-23 #32-53
Common Core State s-mathematics Page # : 12 Domain: APR- Arithmetic with Polynomials and Rational Expressions Cluster: Rewrite rational expressions (+) For advanced course: calculus, advanced statistics, etc. 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. MP.8 Look for and express regularity in repeated reasoning. - Chapter 5, Section 4 The polynomial q(x) is called the quotient and the polynomial r(x) is called the remainder. Expressing a rational expression in this form allows one to see different properties of the graph, such as horizontal asymptotes. - Chapter 5, Section 4 TE p. 308, #9-16 #40-56 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. MP.8 Look for and express regularity in repeated reasoning. - Chapter 8, Sections 4 and 5 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. - Chapter 8, TE p. 531, #14-25 #27-49; TE p. 539, #11-21 #22-47
Modeling Common Core State s-mathematics Page # : 13-14 Domain: CED- Creating Equations 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. MP.1 Make sense of problems and persevere in solving them. - Chapter 2, Section 5 (Linear) - Chapter 4, Section 2 (Quadratic) - Chapter 7, Section 1 (Exponential) - Chapter 8, Section 5 (Rational) Equations can represent real world and mathematical problems. Include equations and inequalities that arise when comparing the values of two different functions, such as one describing linear growth and one describing exponential growth. Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet of a piece of lava t seconds after it is ejected from the volcano is given by h(t) = -t 2 +16t + 936. After how many seconds does the lava reach its maximum height of 1000 feet? - Chapter 2, Section 5 TE p. 97 #14 #18,19 - Chapter 4, Section 2 TE p. 207, #38, #43 - Chapter 7, Section 1 TE p. 439, 27-45 - Chapter 8, Section 5 TE p. 539, #30,37 A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MP.1 Make sense of problems and persevere in solving them. - Chapter 2, Section 4 (Linear) - Chapter 3, Section 1 (Linear System) - Chapter 4, Section 3 (Quadratic) 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. - Chapter 2, Section 4 (Linear) TE p. 86, #30-31 - Chapter 3, Section 1 (Linear System) TE p. 139, #40,52 - Chapter 4 Section 3 (Quadratic) p. 213, #25-26 - Chapter 4, Section 3 (Quadratic) TE p. 269, #26
Modeling Common Core State s-mathematics Page # : 13-14 Domain: CED- Creating Equations Cluster: Create equations that describe numbers or relationships A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. MP.1 Make sense of problems and persevere in solving them. - Chapter 3, Section 4 (Linear) - Chapter 4, Section 9 (quadratic/linear) 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. Example: A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8. Write a system of inequalities to represent the situation. Graph the inequalities. If the club buys 150 hats and 100 jackets, will the conditions be satisfied? What is the maximum number of jackets they can buy and still meet the conditions? - Chapter 3, Section 4 TE p. 160 #13, 17-24 - Chapter 4, Section 9 TE p. 263 #47, 57-60 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. MP.1 Make sense of problems and persevere in solving them. - Chapter 1, Section 4 The Pythagorean Theorem expresses the relation between the legs a and b of a right triangle and its hypotenuse c with the equation a 2 + b 2 = c 2. Why might the theorem need to be solved for c? Solve the equation for c and write a problem situation where this form of the equation might be useful. Motion can be described by the formula below, where t = time elapsed, u=initial velocity, a = acceleration, and s = distance traveled s = ut + ½at 2 Why might the equation need to be rewritten in terms of a? Rewrite the equation in terms of a. - Chapter 1, Section 4 TE p. 31, #55-60
Common Core State s-mathematics Page # : 15 Domain: REI- Reasoning with Equations and Inequalities Cluster: Understand solving equations as a process of reasoning and explain the reasoning A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. - Chapter 6, Section 5 (Radical equations) - Chapter 8, Section 6 (Rational equations) - Chapter 6, Section 5 TE. p. 395, #26-28, 35-44 - Chapter 8, Section 6 TE p. 546, #8-16, 43-52
Modeling Common Core State s-mathematics Page # : 16 Domain: REI- Reasoning with Equations and Inequalities Cluster: Represent and solve equations and inequalities graphically 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. MP.6 Attend to precision. - Chapter 3, Section 1 (Linear system from table) - Chapter 5, Section 3 (Polynomial functions) - Chapter 7, Section 5 (Logarithmic functions) - Chapter 8, Section 6 (Rational functions) Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions. Example: Given the following equations determine the x value that results in an equal output for both functions. f(x) = 3x-2 g(x) = (x+3)2-1 - Chapter 3, Section 1 TE p. 138, #14-16 - Chapter 3 Section 6 TE p. 180, # 32-37 - Chapter 5, Section 3 TE p. 301, #37-38 - Chapter 7, Section 5 TE p. 473, #23-30 - Chapter 8, Section 6 TE p. 546, #21-23, 43-52
Modeling Common Core State s-mathematics Page # : 17-18 Domain: IF- Interpreting Functions 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. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning. - Chapter 4, Section 2 Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. Examples: A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h = 16t 2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. What is a reasonable domain restriction for t in this context? Determine the height of the rocket two seconds after it was launched. Determine the maximum height obtained by the rocket. Determine the time when the rocket is 100 feet above the ground. Determine the time at which the rocket hits the ground. How would you refine your answer to the first question based on your response to the second and fifth questions. It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the storm was over, with a total rainfall of 3 inches. It didn t rain for the rest of the day. Sketch a possible graph for the number of inches of rain as a function of time, from midnight to midday. - Chapter 4, Section 2 TE. p. 206, #32, #43 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. MP.6 Attend to precision. - Chapter 4, Section 3 Given a function, determine its domain. Describe the connections between the domain and the graph of the function. Know that the domain taken out of context is a theoretical domain and that the practical domain of a function is found based on a contextual situation given, and is the input values that make sense to the constraints of the problem context. Students may explain orally, or in written format, the existing relationships. - Chapter 4, Section 3 TE. p. 213, # 25-28
Modeling Common Core State s-mathematics Page # : 17-18 Domain: IF- Interpreting Functions Cluster: Interpret functions that arise in applications in terms of the context F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. - Chapter 4, Section 1 Students should be able to describe patterns of changes from tables and/or graphs of linear and exponential functions. Sample vocabulary may include, increasing/decreasing at a constant rate or increasing or decreasing at an increasing or decreasing rate. Students should be comfortable in their understanding of rates of change to apply their knowledge linear and non-linear graphical display. - Chapter 4, Section 1 TE p. 200 #47 Progress Monitoring Assessments p. 24, #13
Modeling Common Core State s-mathematics Page # : 19-22 Domain: IF- Interpreting Functions 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. MP.6 Attend to precision. - Chapter 2, Section 7 Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Examples: Describe key characteristics of the graph of f(x) = x 3 + 5. - Chapter 2, Section 7 TE p. 111, #8-28 F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. MP.6 Attend to precision. - Chapter 2, Section 4 Concept Byte with Lesson 2-4 (Piecewise functions) - Chapter 6, Section 8 (square root and cubed root functions) - Chapter 2, Section 7 (absolute value functions) Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Examples: Describe key characteristics of the graph of f(x) = x 3 + 5. - Chapter 2, Section 4 Concept Byte TE p. 91, #1-4 - Chapter 6, Section 8 TE p. 418, #7-10; 25-27, 38-40, 43-48 - Chapter 2, Section 7 TE p. 111, #8-22, 36-44
Modeling Common Core State s-mathematics Page # : 19-22 Domain: IF- Interpreting Functions Cluster: Analyze functions using different representations F-IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. MP.6 Attend to precision. - Chapter 5, Section 1 and Section 2 Graph the function f(x) = 2x by creating a table of values. Identify the key characteristics of the graph. Graph f(x) = 2 tan x 1. Describe its domain, range, intercepts, and asymptotes. Draw the graph of f(x) = sin x and f(x) = cos x. What are the similarities and differences between the two graphs? - Chapter 5, Section 1 TE p. 285 #32-37 - Chapter 5, Section 2 TE p. 293 #13-18 * EMPHASIZE TO INCLUDE GRAPHING WITH ALL KEY CHARACTERISTICS F-IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. MP.6 Attend to precision. - Chapter 7, Section 1 - Chapter 13, Section 1 - Chapter 7, Section 1 TE p. 439, #1-17 - Chapter 13 Section 4 TE p. 856 # 15-36 - Chapter 13, Section 1 TE p. 833, #19-20
Modeling Common Core State s-mathematics Page # : 19-22 Domain: IF- Interpreting Functions 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. - Chapter 4, Section 6 Students should manipulate a quadratic function to identify its different forms (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. - Chapter 4, Section 6 TE p. 238, #46-51 (standard form to vertex form) F-IF.8a 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. - Chapter 4, Section 6 (completing the square) - Chapter 4, Section 4 (factoring) - Chapter 4, Section 6 TE p. 238, #75 - Chapter 4, Section 4 TE p. 222, #72
Modeling Common Core State s-mathematics Page # : 19-22 Domain: IF- Interpreting Functions Cluster: Analyze functions using different representations F-IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. - Chapter 7, Section 5 - Chapter 7, Section 5 TE p. 473, #31-45 - Progress Monitoring Assessments p. 10, #8 F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). MP.6 Attend to precision. - Chapter 4, Section 2 - Chapter 4, Section 2 TE p. 207, #48
Modeling Common Core State s-mathematics Page # : 23 Domain: BF- Building Functions Cluster: Build a function that models a relationship between two quantities F-BF.1 Write a function that describes a relationship between two quantities. F-BF.1b 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. MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning. - Chapter 7, Section 2 - Chapter 6, Section 6 Students will analyze a given problem to determine the function expressed by identifying patterns in the function s rate of change. They will specify intervals of increase, decrease, constancy, and, if possible, relate them to the function s description in words or graphically. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model functions. Examples: You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Express the amount remaining to be paid off as a function of the number of months, using a recursion equation. A cup of coffee is initially at a temperature of 93º F. The difference between its temperature and the room temperature of 68º F decreases by 9% each minute. Write a function describing the temperature of the coffee as a function of time. The radius of a circular oil slick after t hours is given in feet by r=10t 2-0.5 t, for 0 t 10. Find the area of the oil slick as a function of time. - Chapter 7, Section 2 TE. p. 448, #22-30 - Chapter 4, Section 2 TE. p. 402, #21-26, 47-58