MATHEMATICS. Florida Standards

Transcription

1 Algebra 2 Algebra 2 Honors MATHEMATICS Florida Standards

2 Florida Standards Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity. 2. Reason abstractly and quantitatively. (MAFS.K12.MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols. 3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence. 4. Model with mathematics. (MAFS.K12.MP.4) Many everyday problems can be solved by modeling the situation with mathematics. 5. Use appropriate tools strategically. (MAFS.K12.MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding. 6. Attend to precision. (MAFS.K12.MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations. 7. Look for and make use of structure. (MAFS.K12.MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea. 8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently.

3 Pre Calculus Honors: Florida Standards Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course, organized into five units, are as follows: Polynomial, Rational, and Radical Relationships: This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Trigonometric Functions: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. Modeling with Functions: In this unit students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Inferences and Conclusions from Data: In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data including sample surveys, experiments, and simulations and the role that randomness and careful design play in the conclusions that can be drawn. Applications of Probability: Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.

4 Fluency Recommendations A/G- Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables). A-APR.1- Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in Algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent. A-SSE.1b- Fluency in transforming expressions and chunking (seeing parts of an expression as a single object) is essential in factoring, completing the square, and other mindful algebraic calculations. The following Mathematics and English Language Arts Florida Standards should be taught throughout the course: MAFS.912. 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 o their parts as a single entity. MAFS.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling. MAFS.912.F-IF.3.9: Compare properties of two functions each represented in a different way(algebraically, graphically, numerically in tables, or by verbal descriptions) LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing tasks, attending to special cases or exceptions defined in the text. LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in context and topics. LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information expressed visually or mathematically into words. LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners. LAFS.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of each source. LAFS.910.SL.1.3: Evaluate a speaker s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. LAFS.910.SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning. LAFS.910.WHST.1.1: Write arguments focused on discipline-specific content. LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.

5 Algebra 2 & Algebra 2 Honors: Florida State Standards At A Glance First Quarter Second Quarter Third Quarter Fourth Quarter DSA Unit 0 Prerequisite Functions (Linear & Quadratic) MAFS.912.A.-REI.1.1 MAFS.912.A-CED.1.4 MAFS.912.F-IF.2.6 MAFS.912.A-REI.2.4 MAFS.912.A-SSE.2.3a &b MAFS.912.N-CN.1.1 MAFS.912.N-CN.1.2 MAFS.912.N-CN.3.7 *MAFS.912.N-CN.3.8 MAFS.912.F-IF.3.8a DIA 1 (Unit 0) Unit 1- Non-Linear Functions Polynomial Functions MAFS.912.A-APR.1.1 MAFS.912.A-APR.2.3 MAFS.912.A-SSE.1.1 MAFS.912.A-SSE.1.2 MAFS.912.A-APR.3.4 MAFS.912.A-APR.2.2 *MAFS.912.N-CN.3.9 *MAFS.912.A-APR.3.5 DIA 2 (Unit 1 Polynomials) Radical Functions MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 MAFS.912.A-REI.1.2 MAFS.912.F-BF.2.4a *MAFS.912.F-BF.2.4b *MAFS.912.F-BF.2.4c *MAFS.912.F-BF.2.4d DIA 3 (Unit 1 Radicals) Exponential and Logarithmic Functions MAFS.912.A-SSE.2.3 MAFS.912.A-REI.1.1 MAFS.912.F-LE.1.4 MAFS.912.F-BF.1.2 MAFS.912.F-IF.1.3 MAFS.912.F-LE.1.2 MAFS.912.F-BF.1.1a MAFS.912.F-BF.2.a MAFS.912.A-SSE.2.4 Rational Functions MAFS.912.A-APR.4.6 MAFS.912.A.-REI.1.2 *MAFS.912.F-IF.3.7d SSA Unit 2- Modeling with Functions Graphing Functions MAFS.912.A-REI.4.11 MAFS.912.A-REI.3.7 MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.F-IF.3.7 MAFS.912.F-LE.2.5 Modeling Functions MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-CED.1.3 MAFS.912.G-GPE.1.2 MAFS.912.A.-REI.3.6 (3x3) MAFS.912.F-BF.1.1 MAFS.912.F-IF.3.8b DIA 4 (Unit 2) Unit 3 Trigonometric Functions MAFS.912.F-TF.1.1 MAFS.912.F-TF.1.2 MAFS.912.F-TF.3.8 MAFS.912.F-TF.2.5 DIA 5 (Unit 3) Unit 4-Statistitics MAFS.912.S-IC.1.1 MAFS.912.S-IC.1.2 MAFS.912.S-IC.2.3 MAFS.912.S-IC.2.4 MAFS.912.S-IC.2.5 MAFS.912.S-ID.1.4 MAFS.912.S-ID.2.6 MAFS.912.S-IC.2.6 DIA 6 (Unit 4) Unit 5- Probability MAFS.912.S-CP.1.1 MAFS.912.S-CP.1.2 MAFS.912.S-CP.1.3 MAFS.912.S-CP.1.4 MAFS.912.S-CP.1.5 MAFS.912.S-CP.2.6 MAFS.912.S-CP.2. 7 *MAFS.912.S-CP.2.8 *MAFS.912.S-CP.2.9 *MAFS.912.S-MD.2.6 *MAFS.912.S-MD.2. 7 DIA 7 (Unit 5) District EOC Exam * Standards highlighted are for HONORS only*

6 Florida Standards Conceptual Categories and Domains for Pre Calculus Honors Conceptual Category: Number and Quantity Domain: N-RN: The Real Number System Domain: N-CN: Complex Numbers Conceptual category: Algebra Domain: SSE: Seeing Structure in Expressions Domain: APR: Arithmetic with Polynomials and Rational Expressions Domain: CED: Creating Equations Conceptual category: Geometry Domain: G-GPE: Expressing Geometric Properties with Equations Conceptual category: Statistics and Probability Domain: S-IC: Making Inferences and Justifying Conclusions Domain: S-CP: Conditional Probability and the Rules of Probability Domain: S-MD: Using Probability to Make Decisions Domain: REI: Reasoning with Equations and Inequalities Conceptual category: Functions Domain: F-IF: Interpreting Functions Domain: F-BF: Building Functions Domain: F-LE: Linear, Quadratic, and Exponential Models Domain: F-TF: Trigonometric Functions Throughout the map, in the resource column, the Flipbook was used as a reference. The mapping committee recommended printing the flipbook as the pages referenced are the actual paper page numbers, not the file page numbers. The document also provided directions to print and create your flipbook:

7 Standard MAFS.912.A-REI.1.1 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. MAFS.912.A-CED.1.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. SMP #4 MAFS.912.F-IF.2.6 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. SMP # 4,#5 Course: Algebra 2 & Algebra 2 Honors Unit 0: Prerequisite Functions (Linear & Quadratics): Linear In what ways can the problem be solved, and why should one method be chosen over the other? How can algebra describe the relationship between sets of numbers? How can the relationship between quantities best be represented? I can: Remarks Resources apply order of operations and inverse operations to Provide examples for how the solve equations. same equation might be construct an argument to justify my solution process. solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation, the order of steps taken will not matter. solve formulas for a specified variable. Give students formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula. define interval, rate of change, and average rate of change. explain the connection between average rate of change and the slope formula. calculate the average rate of change of a function, represented either by function notation, a graph, or a table, over a specific input interval. compare the rates of change of two or more functions when they are represented with function notation, with a graph, or with a table. interpret the meaning of the average rate of change (using units) as it relates to a real-world problem. Book pgs Book pgs Book pgs 98-99

8 Standard MAFS.912.A-REI.2.4 Solve quadratic equations in one variable. SMP #4 MAFS.912.N-CN.1.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. SMP #6 MAFS.912.N-CN.1.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. SMP #7, #8 MAFS.912.N-CN.3.7 Solve quadratic equations with real coefficients that have complex solutions. SMP #1, #7 MAFS.912.N-CN.3.8 *Honors Only* Extend polynomial identities to the complex numbers. SMP #7, #8 Course: Algebra 2 & Algebra 2 Honors Unit 0: Prerequisite Functions (Linear & Quadratics): Quadratics Can the student model and solve quadratic equations using a variety of algebraic methods? Why structure expressions in different ways? I can: Remarks Resources solve quadratic equations by inspection (e.g., Some students may think that High for x² = 49), taking square roots, completing the rewriting equations into various forms School square, the quadratic formula and factoring, as (taking square roots, completing the square, using quadratic formula and appropriate to the initial form of the equation. Book pgs factoring) are isolated techniques use the method of completing the square to within a unit of quadratic equations. transform any quadratic equation in x into an Teachers should help students see equation of the form (x p)² = q that has the the value of these skills in the context same solutions, and derive the quadratic of solving higher degree equations equation from this form. and examining different families of 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. use the relation i² = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. solve quadratic equations with real coefficients that have complex solutions. express the solution of a quadratic equation as a complex number, a + bi. Write the factors of polynomials using complex numbers. use complex numbers to rewrite a sum of squares, a 2 + b 2 as the product of a complex number and its conjugate. show that factored quadratics have real coefficients when written in standard form. functions. Includes recognition of the cycles of i (i 2 =-1, i 3 =i 2 i = -i, etc.). In the cases of quadratic equations, when the use of quadratic formula is not critical, students sometime ignore the negative solutions. For example, for the equation x 2 = 9, students may mention 3 and forget about ( 3), or mention 3i and forget about (- 3i) for the equation x 2 = - 9. If this misconception persists, advise students to solve this type of quadratic equation either by factoring or by the quadratic formula. For example, rewrite x² + 4 as (x + 2i)(x 2i) High School Book pgs 27-30

9 Course: Algebra 2 & Algebra 2 Honors Unit 0: Prerequisite Functions (Linear & Quadratics): Quadratics Can the student model and solve quadratic equations using a variety of algebraic methods? Why structure expressions in different ways? Standard MAFS.912.F-IF.3.8 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. SMP #7, #8 MAFS.912.A-SSE.2.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. SMP #7 I can: Remarks Resources explain that there are three forms of quadratic functions: Use various representations of standard form, vertex form, and factored form. the same function to apply completing the square to rewrite a quadratic emphasize different Book pgs characteristics of that function function in vertex form. For example, the y-intercept of factor a quadratic expression to find the zeroes of the the function y = x 2-4x 12 is Graphs function it represents. easy to recognize as (0, -12). (2007) identify and factor perfect-square trinomials. However, rewriting the function define an exponential function. as y = (x 6)(x + 2) reveals Quadratics rewrite exponential functions using the properties of zeros at (6, 0) and at ( -2, 0). (2009) exponents. Building factor a quadratic expression to find the zeroes of the Connections function it represents. (NCTM) identify and factor perfect-square trinomials. define an exponential function. rewrite exponential functions using the properties of exponents. Furthermore, completing the square allows the equation to be written as y = (x 2) 2 16, which shows that the vertex (and minimum point) of the parabola is at (2, -16). Students often confuse the k value with the y-intercept. Be sure to explain that the y- intercept is not (0, k). Show them how to substitute 0 for x and solve for the y-intercept. Book pgs Book pgs 96-97

10 Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Polynomials How can properties of the real number system be useful when working with polynomials expressions? Standard Remarks I can: MAFS.912.A-APR.1.1 apply the definition of an integer to explain Consider mentioning that like Understand that polynomials form a closure. integers, polynomials are closed system analogous to the integers, apply the definition of a polynomial to explain under addition, subtraction, and namely, they are closed under the closure. multiplication, but not division. operations of addition, subtraction, add and subtract polynomials. and multiplication; add, subtract and multiply polynomials. When adding, subtracting, multiply polynomials. multiplying and dividing focus on polynomial expressions that simplify SMP #2, #7 to forms that are linear or quadratic in a positive integer power of x. Resources Book pg MAFS.912.A-APR.2.3 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. SMP #1, #8 identify the zeros of factored polynomials. identify the multiplicity of the zeros of a factored polynomial. explain how the multiplicity of the zeros provides a clue as to how the graph will behave as it approaches and leave the x- intercept. sketch a rough graph using the zeros and other easily identifiable points (y-intercept). Use graphing technology to expedite the exploration of a multitude of functions so students can more easily observe the relationships. Book pg. 50

11 Course: Algebra 2 & Algebra Honors Unit 1: Non-Linear Functions Polynomials How can properties of the real number system be useful when working with polynomials expressions? Standard Remarks I can: MAFS.912.A-SSE.1.1 define expression, term, factor and coefficient Students should recognize that in the Interpret expressions that represent group the parts of an expression differently in expression 2x + 1, 2 is the a quantity in terms of its context. order to better interpret their meaning coefficient, 2 and x are factors, and a) Interpret parts of an expression, 1 is a constant, as well as 2x and 1 being terms of the binomial such as terms, factors and expression. coefficients b) Interpret complicated expressions by viewing one or more of their parts as a single entity. SMP #7 MAFS.912.A-SSE.1.2 Use the structure of an expression to identify ways to rewrite the polynomial function. (ex. Difference of Squares, Perfect Squares, etc.) SMP #7 MAFS.912.A-APR.3.4 Prove polynomial identities and use them to describe numerical relationships. SMP #8 MAFS.912.A-APR.2.2 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 if p(x). SMP #8 identify clues in the structure of expressions to rewrite the function: like terms, common factors, difference of squares, perfect squares, etc. explain why equivalent expressions are equivalent. apply models for factoring and multiplying polynomials to rewrite expressions. verify polynomial identities: difference of squares, difference of cubes, sum of cubes. factor polynomials completely. divide polynomials using long division and synthetic division. apply the Remainder Theorem, when appropriate, to check the answer. apply the Remainder Theorem to determine if a divisor is a factor of a polynomial. Have students create their own expressions that meet specific criteria (e.g., number of terms factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten in different forms. Additionally, pair/group students to share their expressions and rewrite one another s expressions. Illustrate how polynomial identities are used to determine numerical relationships such as: 25 2 =(20 + 5) 2 = Students can benefit from exploring the rational root theorem, which can be used to find all of the possible rational roots (i.e., zeros) of a polynomial with integer coefficients. When the goal is to identify all roots of a polynomial, including irrational or complex roots, it is useful to graph the polynomial function to determine the most likely candidates for the roots of the polynomial that are the x-intercepts of the graph. Resources Book pg (refer to pg. 34 and 35 as indicated) Book pg Book pg

12 Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Polynomials How can properties of the real number system be useful when working with polynomials expressions? Standard Remarks I can: MAFS.912.N-CN.3.9 *Honors Only* apply the Fundamental Theorem of Algebra to While this is only an Honors Know the Fundamental Theorem of demonstrate that the number of linear factors standard, Regular students would Algebra; show that is it true for all a polynomial has is equal to the degree of that also benefit from this knowledge. polynomials. polynomial. SMP #7 MAFS.912.A-APR.3.5 *Honors Only* 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. SMP #7 generate Pascal s Triangle to find the coefficients of a binomial expansion. Return to these learning targets after teaching combinations in Unit 4 Statistics and Probability: apply the combination formula. write the binomial expansion of (a + b) n by applying the Binomial Theorem (a + b) n = n C 0 a n b 0 + n C 1 a n-1 b 1 + n C 2 a n-2 b nc n a 0 b n Resources

13 Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Radicals How can properties of the real number system be useful when working with radical expressions? How does knowledge of integers help when working with rational and irrational numbers? Standard MAFS.912.N-RN.1.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. SMP # 3, 7, 8 MAFS.912.N-RN.1.2 Rewrite expressions involving rational exponents using the properties of exponents. SMP # 7 MAFS.912.A-REI.1.2 Solve simple radical equations in one variable, and give examples showing how extraneous solutions may arise. SMP # 1 I can: Remarks Resources apply properties of exponents to simplify Identify numerator as power algebraic expressions with rational exponents. and denominator as root for apply the definition of an nth root to rational exponents. Book pgs.10- demonstrate that ( n x) n = x for various values 13 of n (2,3, ) and explain why this is true. apply the properties of exponents to demonstrate that (x 1/n ) n = x for various values of n (2,3, ) and explain why this is true. apply the properties of exponents and definition of an nth root to explain that x 1/n = n x because ( n x) n = x and (x 1/n ) n = x. apply the properties of exponents to simplify algebraic expressions with integer and rational exponents. write radical expressions as expressions with rational exponents (and vice versa). solve a radical equation in one variable. explain which numbers are non-solutions of a radical equation. generate examples of radical equations w/ extraneous solutions. Students should note the impact of an exponent on the base whether it is the previous term or in parentheses. Solving a radical equation by squaring both sides requires checking the solution to avoid finding a solution when there isn t one. The square root of a negative value has no solution but squaring it makes it positive which does have a solution. This creates possible extraneous solutions. Book pgs Book pgs.66-68

14 Standard MAFS.912.F-BF.2.4a 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. SMP # 7 MAFS.912.F-BF.2.4b *Honors Only* Verify by composition that one function is the inverse of another SMP # 7 MAFS912.A.F-BF1.1c*Honors Only* Compose functions. Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Radicals How does knowledge of integers help when working with rational and irrational numbers? In what ways can the problem be solved, and why should one method be chosen over another? In what ways can functions be built? How can the relationship between quantities best be represented? I can: Remarks Resources write the inverse of a function by solving For example, f(x) = c for x such that c is considered the input and x is f(x) =2 x³ or f(x) = considered the output. (x+1)/(x 1) for x 1. Book pgs. write the inverse of a function in standard notation replacing x Remember restrictions with y and c with x in my inverse function. on the domain use the composition of functions to verify that g(x) and f(x) are inverses by showing that g(f(x)) = f(g(x)) = 1. compose two or more functions. explain a multi-step real world problem in terms of function composition and write an equation to describe the composition. Book pgs MAFS.912.F-BF.2.4c *Honors Only* Read values of an inverse function from a graph or a table, given that the function has an inverse. SMP # 7 MAFS.912.F-BF.2.4d *Honors Only* Produce an invertible function from a non-invertible function by restricting the domain SMP # 7 decide if a function has an inverse using the horizontal line test. use the definitions of function, inverse function and one-toone function to explain why the horizontal line test works. list values of an inverse given a table or graph of a function that has an inverse. identify and eliminate the part(s) of a graph that cause it to fail the vertical line test. state the domain of a relation that has been altered in order to pass the horizontal line test. write the inverse of the invertible function in function notation. Book pgs Book pgs

15 Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Exponential & Logarithmic Standard MAFS.912.A-SSE.2.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (c) use the properties of exponents to transform expressions for exponential functions. SMP #7 MAFS.912.A-REI.1.1 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. Why structure expressions in different ways? I can: Remarks Resources define an exponential function, f(x) = Offer multiple real-world examples of exponential High ab x. functions. To illustrate an exponential decay School rewrite exponential functions using the function, students need to recognize that in the equation for an automobile cost C(t) = properties of exponents. Book page explain and use A(t) = a(1 + r) t 20,000(.75)^t, the base is.75 and between 0 & and the value of $20,000 represents the initial cost of an automobile that depreciate 25% per year over the course of t years. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate apply order of operations and inverse operations to solve equations. construct an argument to justify my solution process. is 15%. Provide examples for how the same equation might be solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation, the order of steps taken will not matter. High School Book pgs 64-65

16 Standard MAFS.912.F-LE.1.4 For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. MAFS.912.F-BF.1.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Exponential & Logarithmic How the relationship between quantities best be represented? I can: define logarithmic function in relation to the exponential function. write an exponential equation ab ct = d in logarithmic form log a d = ct and solve it for t. a explain using the properties of exponentials and logarithms why ab ct = d and log a d = ct are equivalent. a apply powers of 2 or 10 to estimate the value of log 2 (x) or log 10 (x). use a calculator to evaluate a logarithm with a base of 10 or e. apply the change of base formula to evaluate the logarithm with a base of 2 using a calculator. distinguish between explicit and recursive formulas for sequences. define arithmetic and geometric sequences. determine the common difference and the common ratio. explain how to find the next term in a sequence. write an explicit formula for an arithmetic or geometric sequence. explain why the recursive formula for a geometric sequence uses multiplication and why the explicit formula uses exponentiation. translate between the recursive and explicit forms of geometric and arithmetic sequences. decide when a real world problem models an arithmetic or geometric sequence and write an equation to model the situation. Remarks Provide examples of arithmetic and geometric sequences in graphic, verbal, or tabular forms, and have students generate formulas and equations that describe the patterns. Apply exponential functions to real-world situations. Have students draw the graphs of exponential and other polynomial functions on a graphing calculator and examine the fact that the exponential curve will eventually get higher that the polynomial function s graph. Students may believe that the best (or only) way to generalize a table of data is by using a recursive formula. Students may also believe that arithmetic and geometric sequences are the same. Students need experiences with both types of sequences to be able to recognize the difference and more readily develop formulas to describe them. Resources Book page 122 Book page 108

17 Standard MAFS.912.F-IF.1.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. MAFS.912.F-LE.1.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Exponential & Logarithmic In what ways can functions be built? I can: convert a list of numbers (a sequence) into a function by making the whole number the inputs and the elements of the sequence the outputs. explain that a recursive formula tells me how a sequence starts and tells me how to use the previous value(s) to generate the next element of the sequence. explain that an explicit formula allows me to find any element of a sequence without knowing the element before it. distinguish between explicit and recursive formulas for sequences. determine if a function is linear or exponential given a sequence, a graph, a verbal description, or a table. construct a linear function from an arithmetic sequence, graph, table of values, or a description of the relationship. describe the algebraic process used to construct the exponential function that passes through two given points. Remarks Students may believe that all relationships having an input and an output are functions, and therefore, misuse the function terminology. Students may also believe that the notation f(x) means to multiply some value f times another value x. The notation alone can be confusing and needs careful development. For example, the FibonaFSi sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. Students may believe that all functions have a first common difference and need to explore to realize that, for example, a quadratic function will have equal second common differences in a table. Resources Book page 93 FSSS Flip Book page 118 MAFS.912.F-BF.1.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. define explicit and recursive expressions of a function. identify the quantities being compared in a real-world problem. write an explicit and/or recursive expressions of a function to describe a real-world problem. Students naturally tend to look down a table to find the pattern but need to realize that finding the 100 th term requires knowing the 99 th term unless an explicit formula is developed. FSSS Flip Book page 105

18 MAFS.912.SSE.2.4 Derive the formula for the sum of a finite geometric series (when the common ration is not 1), and use the formula to solve problems define a finite geometric series and common ratio. derive the formula for the sum of a finite geometric series. express the sum of a finite geometric series. calculate the sum of a finite geometric series. recognize real-world scenarios that are modeled by geometric sequences. apply and use the formula for the sum of a finite geometric series to solve real-world problems. Some students cannot distinguish between arithmetic and geometric sequences, or between sequences and series. To avoid this confusion, students need to experience both types of sequences and series. Students commonly do not understand what it means to find the sum of a series. FSSS Flip Book page 42 Students often do not recognize that there are multiple ways of finding sums of series. Although it is not always practical, students could use a conceptual method to find the sums rather than using a formula.

19 Course: Algebra 2 & Algebra 2 Honors Unit 1: Non-Linear Functions Rational How can the properties of the real number system be useful when working with rational expressions? How can the relationship between quantities best be represented? Standard MAFS.912.A-APR.4.6 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. SMP # 5, 7 MAFS.912.A-REI.1.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. SMP # 1, 3 MAFS.912.F-IF.3.7d *Honors Only* b. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. SMP #7, 8 I can: Remarks Resources define rational expression. determine the best method of simplifying the given 9 rational expression (inspection e.g. x 2 + x Book pgs.53- = 5x 1, 54 9x long division, computer algebra system). write a rational expression a(x)/b(x) where a(x) is the dividend and b(x) is the divisor in the form q(x) + r(x)/b(x). solve a rational equation in one variable. explain which numbers are non-solutions of a rational equation. generate examples of rational equations w/ extraneous solutions. identify the y-intercept and x-intercepts of a rational function. state or describe the end behavior of a rational function when looking at or interpreting the graph of the function. sketch a graph of a rational function based on its domain, x-intercepts, y-intercept, and asymptotes. use technology to graph rational functions and to find precise values for the x-intercept(s) and the maximums and minimums (turning points). Rational expressions follow the same rules for operations as for rational numbers. Watch for students to cancel terms instead of factors. Remind them to divide the numerator and denominator by a common factor rather than cancel. Remind students to set the denominator equal to zero and solve for the excluded values. Book pgs Book pgs

20 Course: Algebra 2 & Algebra 2 Honors Unit 2: Modeling with Functions Graphing Can the student model and graph linear, quadratic and non-linear equations? Standard MAFS.912.A-REI.4.11 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. SMP #5 MAFS.912.A-REI.3.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. SMP #8 I can: Remarks Resources explain that a point of intersection on the graph of a system of equations represents a solution to both equations. use a graphing calculator to determine the approximate solutions to a system of equations. determine the approximate solution of a system of equations in which on equation is linear and one equation is quadratic by graphing and estimating the point(s) of intersection Students may believe that the graph of a function is simply a line or curve connecting the dots, without recognizing that the graph represents all solutions to the equation. Provide students opportunities to practice linear vs. non-linear systems; consistent vs. inconsistent systems; pure computational vs. real-world contextual problems (e.g., chemistry and physics applications encountered in science classes). Book pg. 83 (refer to page 82 as indicated) CCSS Flip Book pgs 79-80

21 MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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 explain why f(x) + k translates the original graph of f(x) up k units and why f(x)-k translates the original graph of f(x) down k units. explain why f(x + k) translates the original graph of f(x) left k units and why f(xk) translates the original graph of f(x) right k units. explain why kf(x) vertically stretches or shrinks the graph of f(x) by a factor of k and predict whether a given value of k will cause a stretch or shrink. explain why f(kx) horizontally stretches or shrinks the graph of f(x) by a factor of 1/k and predict whether a given value of k will cause a stretch or a shrink. describe the transformation that changed a graph of f(x) into a different graph when given pictures of the pre-image and image. determine the value of k given the graph of a transformed function. graph the listed transformations when given a graph of f(x) and a value of k ( f(x) k, f(x k), kf(x), and f(kx). use a graphing calculator to generate examples of functions with different k values. analyze similarities and differences between functions with different k value Expose students to absolute value and piecewise defined functions as you teach the transformations. In this standard, k is referred to as a constant. Traditionally, a is the factor that causes a vertical stretch, h is the horizontal translation, and k is the vertical translation CCSS Flip Book pgs

22 Standard MAFS.912.F-IF. 2.4 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. SMP #1, #7, #8 MAFS.912.F-IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Course: Algebra 2 & Algebra 2 Honors Unit 2: Modeling with Functions Graphing Can the student model and graph quadratic equations? I can: Remarks Resources identify the x-intercepts, increasing/decreasing intervals. define and identify the relative maximum/minimum. CCSS Flip Book use the problem situation to explain the end behavior of the function. pg use the problem situation to explain why a function is periodic. explain how the domain of a function is represented in its graph. identify the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domains. Students may believe that the slope of a linear function is merely a number used to sketch the graph of the line. Apply the real world meaning for slopes as a rate of change. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the CCSS Flip Book page 96

23 MAFS.912.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. SMP #6, #7 graph quadratic functions showing intercepts, maxima, and minima. use technology to graph a quadratic function and to find precise values for the intercepts, maximum, or minimum. graph square root, cube root and piecewise-defined functions, including step functions and absolute value functions. explain that there are three forms of quadratic functions: standard form, vertex form, and factored form. explain that the graph of all three forms of quadratic functions is a parbola. find the x-intercepts of a quadratic to find the axis of symmetry. identify the line of symmetry. sketch a graph of a parabola written in vertex form. apply completing the square to rewrite a quadratic function in vertex form. positive integers would be an appropriate domain for the function. Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domain Use various representations of the same function to emphasize different characteristics of that function. For example, the y-intercept of the function y = x2-4x 12 is easy to recognize as (0, - 12). However, rewriting the function as y = (x 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0). CCSS Flip Book pgs Graphs (2007) Quadratics (2009) Building Connections (NCTM) MAFS.912.F-LE.2.5 Interpret the parameters in a linear or exponential function in terms of a context identify the names and definitions of the parameters a, b, c in the exponential function. explain the meaning of the constant a of an exponential function when the exponential function models a real-world relationship. explain the meaning of the y-intercept and other points on an exponential function when the exponential function a real-world relationship. explain the meaning of the constant b of an exponential functions when the exponential function models a real-world relationship. explain the meaning of the constant c of an exponential function when the Students may believe that an exponential function can appear to be abstract until applying it to a real-world situation. CCSS Flip Book page 123

24 exponential function models a real-world relationship. compose an original problem situation and construct an exponential function to model it.

25 Course: Algebra 2 & Algebra 2 Honors Unit 2: Modeling with Functions Modeling How can algebra describe the relationship between sets of numbers? Standard MAFS.912.A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. SMP #4 I can: Remarks Resources identify the variables and quantities represented in a Begin with simple real-world problem. equations and CCSS Flip determine the best model for the real-world problem. inequalities and build up Book pgs 55- write the equation or inequality that best models the to more complex 57 problem. equations. solve the equation or inequality. interpret the solution in the context of the problem. MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales SMP #4 MAFS.912.A-CED.1.3 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. SMP #4 identify the variables and quantities represented in a real-world problem. determine the best model for the real-world problem. write the equation that best models the problem. determine appropriate scale and label the axes. graph equations on coordinate axes with appropriate labels and scales. identify the variables and quantities represented in a real-world problem. determine the best models for the real-world problem. write the system of equations and/or inequalities that best models the problem. graph the system on coordinate axes with appropriate labels and scales. interpret solutions in the context of the situation modeled and decide if they are reasonable While this standard is initially introduced in the context of linear functions, it applies to all units and should be addressed continually. The standard specifically addresses solving a system by graphing. CCSS Flip Book pgs CCSS Flip Book pgs 60-61

26 Course: Algebra 2 & Algebra 2 Honors Unit 2: Modeling with Functions Modeling How can algebra be useful when expressing geometric properties? In what ways can the problem be solved, and why should one method be chosen over another? How can the relationship between quantities best be represented? Standard MAFS.912.G-GPE.1.2 Derive the equation of a parabola given a focus and directrix. SMP #2, #3, #7 I can: Remarks Resources define a parabola. Define a parabola as a determine the distance from a point on the parabola to set of points satisfying CCSS Flip the directrix. the condition that their Book pg 181 determine the distance from a point on the parabola to distance from a fixed the focus using the distance formula (Pythagorean point (focus) equals their Theorem). distance from a fixed equate the two distance expressions for a parabola to line (directrix). Start with write its equation. a horizontal directrix and identify the focus and directrix of a parabola when given a focus on the y-axis, its equation. and use the distance formula to obtain an equation of the resulting parabola in terms of y and x2. Next use a vertical directrix and a focus on the x-axis to obtain an equation of a parabola in terms of x and y2. MAFS.912.A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (3 x 3 only) SMP #7 define system of linear equations and solution of a system. explain why some linear systems have no solutions and identify linear systems that have infinitely many solutions. solve a system of linear equations algebraically (by substitution or elimination) to find an exact solution. graph a linear equation on a coordinate plane. determine the approximate solution to a system of linear equations by graphing both equations and estimating the point of intersection CCSS Flip Book pgs 77-78