Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Transcription

1 and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 1: Functions, Equations, Absolute Value and Inequalities 15 days of instruction plus assessment time- 3.5 weeks Teacher Note: This unit is designed as a review and extension of material covered in the PreAP Algebra I and PreAP Geometry Course. Discussion with previous teachers about how much of the curriculum was actually covered and concepts that were not well received by students is vital to planning the time for this unit. The teacher needs to go to the Quality website and download Quality Alg II Unit 2 Linear Equations and Inequalities for use in this unit and go to the NMSI website and download all of the lessons for this unit. A graphing calculator is needed for many of the lessons in this unit. NMSI s Laying the Foundation Lesson: Literal Equations (1 day) Teacher Note: For #1 the students need access to formulas. You could give them the Quality reference sheet so that the formulas are not holding them back from what we really want them to do and that is to be able to manipulate equations to solve for specific variables. This would be a great 1 st or 2 nd day activity. ** Checkpoint Unit 1 1,2,3 can be used with this lesson. AL COS Common Standard Common 23 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. A-CED4 NMSI s Laying the Foundation lesson: Introducing Interval Notation (.5 days) Teacher Note: Provides a review of inequalities and their graphs. Students should understand the concepts of and increasing, decreasing, and constant function. Introduces the concept of + and infinity. This lesson takes students from the inequality way of writing an interval to the interval notation and ties the graph on the number line to both. Answers to Quality core tests samples are both in inequality notation and interval notation. The committee believes that this is a standard that should have been left in and is a great review of Algebra I. AL COS Common Standard Common key features given a verbal description of the relationship. Key features include periodicity. Unit 1 Page 1

2 NMSI s Laying the Foundation lesson: Transformations of Functions Exploration (2.5 days) Teacher Note: Have students complete the first page for homework the night before. Students should be familiar with the following function notations: f(x) + c, f(x) c, f(-x), -f(x); and be able to do powers, square roots, and absolute value. Suggest students work in groups on part 2. Graphing Calculators are needed for this lesson and students will need graph paper. This may take a little longer if graphing calculators have not been used before. ** Checkpoint Unit 1 5,6,7,8 can be used with this lesson. AL COS Common Standard Common 34 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-BF3 NMSI s Laying the Foundation lesson: Even/Odd functions (1 day) Teacher Note: You want to talk with your Algebra I teacher and see if they did this lesson. If not, this is a great review of Algebra I to start the year. The teacher instructions suggest to use your information from Transformations of Functions Exploration to help you make connections. Graphing Calculators are needed for this lesson. The students must understand how to define an even/odd function f ( x) f ( x) and f ( x) f ( x) and connect that with the points on the graph. The committee believes that this is a standard that should have been left in and is a great review of Algebra I. ** Checkpoint Unit 1 9 can be used with this lesson. AL COS Common Standard Common key features given a verbal description of the relationship. Key features include periodicity. NMSI s Laying the Foundation lesson: Transforming Domain and Range (1 day) Teacher Note: The activity asks students to transform the absolute value function and then to discuss how the domain and range are affected. Problems 6-9 will be discovery for students. ** Checkpoint Unit 1 13,17,18,19 can be used with this lesson. AL COS Common Standard Common 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. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. F-BF3 F-IF5 Unit 1 Page 2

3 NMSI s Laying the Foundation lesson: Applying Piecewise Functions (1 day) Teacher Note: This lesson makes connections between distance-time graphs and speed graphs. The graphs are linear piecewise graphs. ** Checkpoint Unit 1 10,11,12,14,15,16,,23 and Illustrative Mathematics Pizza Promotion can be used with this lesson. AL COS Common Standard Common Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases (piecewise functions) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 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-IF7b F-IF8 F-IF5 F-IF6 NMSI s Laying the Foundation lesson: Walking Piecewise Graphs (1 day) Teacher Note: Another option for LTF lessons if you have the CBR. Students walk piecewise graphs. AL COS Common Standard Common 30 30a 24 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 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-IF7 F-IF7b F-IF6 Quality Alg II Unit 2 Linear Equations and Inequalities (C-2 C-7) (.5 day) Solving Inequalities Matching Game Teacher Note: This should be a review of how to solve different types of inequalities. This activity has all of the steps of solving an inequality and students must match and give justification for each step. AL COS Common Standard Common 20 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED1 Unit 1 Page 3

4 Quality Alg II Unit 2 Linear Equations and Inequalities (D-3-D-8, E-4- E-8) (3 days) Absolute Value and Inequalities Teacher Note: These activities show students how to solve different types of absolute equations and inequalities. AL COS Common Standard Common 21 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] Unit 1 Page 4 A-CED2 NMSI s Laying the Foundation lesson: Exploring Inequalities (.5 days) Teacher Note: Lesson links the solution of an inequality in 2 variables to an inequality in one variable. Students should have previous experience graphing linear, quadratic, and absolute value functions. ** Checkpoint Unit 1 #24 can be used with this lesson. AL COS Common Standard Common 27 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-REI11 NMSI s Laying the Foundation lesson: Systems of Linear Inequalities (1 day) Teacher Note: Prepare the model of the solid using cardstock prior to the lesson. Students should be familiar with graphing linear inequalities, finding areas and perimeters of polygons, and volumes of solids. The reference sheet could be used for the formulas of the different shapes. ** Checkpoint Unit 1 25,26,27,28 can be used with this lesson. AL COS Common Standard Common 27 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. 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. Quality Alg II Unit 2 Linear Equations and Inequalities (J-7 and L-2) (2days) A-REI11 A-CED3 Linear Programming Teacher Note: This should have been covered in Algebra I and should be a quick review. Talk with Algebra I teacher to determine how much time will need to be spent on this. This could be covered on the EOC test for. AL COS Common Standard Common Create equations in two or more variables to represent relationships between A-CED2 21 quantities; graph equations on coordinate axes with labels and scales. Represent constraints by equations or inequalities, and by systems of equations A-CED3

5 and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Explain why the x-coordinates of the points where the graphs of the equations y = 27 f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-REI11 Teacher led Instruction (1 day) Teacher Note: Use illuminations- Egg Launch contest (Quadratic). AL COS Common Standard Common 32 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-IF9 Unit 1 Page 5

6 and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 2: Arithmetic Series and Matrices 9 days of instruction plus assessment time-2 weeks Teacher Note: Teachers will need to go to Quality Website and download from Algebra II Unit 1 The purpose and Predictability of Patterns and Unit 3 What is a Matrix- Really Qulaity Alg II Unit 1 The Purpose and Predictability of Patterns- (D-6 F-10) (3 days) Arithmetic Sequences Series Teacher Notes: Students should have been introduced to arithmetic sequences in Algebra I. Communicate with the Algebra I teacher to understand depth of knowledge. It is very important to make the connection between linear functions that have constant slope and arithmetic sequences that have common differences. a a d n n 1 1 y The comparison of these two should be key to what you are doing. It is not a new formula, it is just b m( x 1) a line moved to the right one because you cannot start with the 0 term, so you move the line over to the right one (x-1) to start with the 1 st term. AL COS Common Standard Common Algebra I Construct linear and exponential functions, including arithmetic and sequences, F-LE2 38 given a graph, a description of a relationship, or two input-output pairs (include Partial Algebra I 35 reading these from a table). Write arithmetic sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F-BF2 Partial and PreAP Unit 2 Page 1

7 Quality Alg II Unit 3 What is a Matrix- Really? (Entire Unit) (6 days) Teacher Notes: Students have probably not been introduced to matrices until now. Adding, Subtracting and multiplying by a scalar is simple and could be done in one day (B-5 D-6). Multiplying matrices will take two days (E-1 F-7) one day for practice and another for application. This unit does a good job with application. Solving Matrices Equations requires the discussion of the identity matrix and determinant. This unit goes into great detail about students discovering the pattern for the inverse matrix by using the identity matrix. There may not be time for this and you may need a direct approach to finding determinants and inverse matrices. You might want to skip H and K if you are worried about time. The key to this concept is the application of a matrix to solve a problem. This is where the technology comes in to save on time. Students need to know how to solve a 3 variable equation by hand, but do not need to spend a great deal of time on this. The connection of how to solve a 3 variable equation using matrices is very important. Quality put the solving 3 variable equations in the graphing calculator part of their test. There is not enough graphing calculator practice problems and I would find another source, textbook or google it, to obtain some more. There is no mention of row reduction or row echelon form in either CCSS or QC. (3 or 4 days) AL COS Common Standard Common PreCalc 26 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. Add, subtract, and multiply matrices of appropriate dimensions. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. This is not specifically designated in Common. They suggest the use of technology. Represent a system of linear equations as a single matrix equation in a vector variable. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). N-VM6 N-VM7 N-VM8 N-VM9 N-VM10 A-REI8 A-REI9 and PreAP Unit 2 Page 2

8 and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 3: Quadratic and Root Functions 17 days of instruction plus Checkpoint time- 3.5 weeks NMSI s Laying the Foundation lesson: Transformations of Conic Sections (1 day) 2 Teacher Note: Graphing and translating x, x, and 2 2 x r. The students should have graphed They should graph 2 y x in Algebra 1 and y 2 2 x r in Geometry. This lesson should be a good review of these and introduce y x as the function that is made when you reflect 2 y x about the line y x. Only Responsible for the parabolas, square root and circles (would be okay if did not do v-vii on circles). Hyperbolas are taken care of in PreCalc. AL COS Common Standard Common Precalc Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, AL 28 and degenerate conics, from second-degree equations. Example: Graph x 2 6x + Standard y 2 12y + 41 = 0 or y 2? 4x + 2y + 5 = a. Formulate equations of conic sections from their determining characteristics. key features given a verbal description of the relationship. Key features include periodicity. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Write a function that describes a relationship between two quantities.* [F-BF1] a. Combine standard function types using arithmetic operations. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-IF5 F-IF7 F-IF7b F-BF1 F-BF1b F-BF3 and PreAP Unit 3 Page 1

9 NMSI s Laying the Foundation lesson: Composition of Functions Graphically (.5 day) Teacher Note: This is a neat lesson to teach composition of functions. ** Checkpoints Unit 1 #4 can be used with this lesson. AL COS Common Standard Common This is a foundational lesson. NMSI s Laying the Foundation lesson: Composition of Functions (.5 day) Teacher Note: Introduction of the f g ( x) symbol for composition and begins to compose functions that are and are not inverses. #2 deals with composing 2 y x and y x. **Checkpoints Unit 3 #3 can be used with this lesson. AL COS Common Standard Common 35a 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. F-BF4a NMSI s Laying the Foundation lesson: Composition of Functions Exploration (1 day) 1 f Teacher Note: Introduces notation for inverse and drives home the fact that the composition of inverse functions results in y x. AL COS Common Standard Common Solve an equation of the form f(x) = c for a simple function f that has an inverse, F-BF4a 35 and write an expression for the inverse. NMSI s Laying the Foundation lesson: Solving equations Graphically-Is there a solution? (1 day) Teacher Note: This lesson asks students to find solutions of linear, absolute value, and quadratic functions. The radical function is then introduced and the idea of extraneous answers is discussed **Checkpoint Unit 3 #6 and Illustrative Mathematics How does the Solution change? can be used with this lesson AL COS Common Standard Common Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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. key features given a verbal description of the relationship. Key features include periodicity. and PreAP Unit 3 Page 2 A-REI2 A-REI11

10 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. 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. Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. a. Formulate equations of conic sections from their determining characteristics. F-IF5 F-IF7 F-IF7b F-BF1 F-BF1b F-BF3 AL Standard Teacher Led Discussion: Complex Roots and Quadratics (3 days) Teacher Note: There is not a specific lesson that discusses complex roots. Your textbook should do a good job of explaining the complex roots and how the Quadratic Formula s discriminant helps us decide what type of roots a quadratic has. This might also be a good place to graph quadratic inequalities. ** Checkpoint Unit 3 5,7 can be used with this lesson. AL COS Common Standard Common Know there is a complex number i such that i 1, and every complex number has the form a + bi with a and b real. 2 Use the relation i 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Solve quadratic equations with real coefficients that have complex solutions. Extend polynomial identities to the complex numbers. Example: Rewrite x 2 4 as x 2i x 2i Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Use the structure of an expression to identify ways to rewrite it x y as x y, thus recognizing it as a difference of squares Example: See that can be factored as x y x y. 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. key features given a verbal description of the relationship. Key features include and PreAP Unit 3 Page 3 2 N-CN1 N-CN2 N-NC3 N-CN7 N-CN8 N-CN9 A-SSE2 A-REI11

11 periodicity. Relate the domain of a function to its graph and, where applicable, to the 29 quantitative relationship it describes Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. 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. Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. a. Formulate equations of conic sections from their determining characteristics. F-IF5 F-IF7 F-IF7b F-BF1 F-BF1b F-BF3 AL Standard NMSI s Laying the Foundation lesson: Quadratic Functions- Adaptation of AP Calculus 1997 AB 2 (1 or 2 days) and PreAP Unit 3 Page 4 2 Teacher Note: Students are asked to graph y x and a transformation and discuss the different aspects of the new graph. The students then find average rate of change, instantaneous rate of change and area under the curve using rectangles. ** Checkpoint Unit 3 2,8, and Illustrative Mathematics Building a general quadratic function can be used with this lesson. AL COS Common Standard Common Create equations and inequalities in one variable and use them to solve problems. A-CED1 20 Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations in two or more variables to represent relationships between A-CED2 21 quantities; graph equations on coordinate axes with labels and scales key features given a verbal description of the relationship. Key features include periodicity.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F-IF5 F-IF7 F-IF7b Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) F-BF3

12 34 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. NMSI s Laying the Foundation lesson: Accumulation with a Quadratic Function (1 day) Teacher Note: Table of data about rate of water flow is input into calculator. Use regression capabilities to find quadratic fit and use function to estimate another data point. Draw rectangles and estimate area under curve to estimate total amount of water after a certain amount of time to help family make a decision. #7-11 is an extension of this by using the sum/sequence capabilities of calculator to get the area more precise. ** Checkpoint Unit 3 23 can be used with this lesson. AL COS Common Standard Common 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. A-CED1 A-CED2 F-BF1 F-BF1b NMSI s Laying the Foundation lesson: Takin Care of Business (1 day) Teacher Note: This lesson is a revenue, cost, profit problem that will use a quadratic model. AL COS Common Standard Common 31 28a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 20. Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. a. Formulate equations of conic sections from their determining characteristics. key features given a verbal description of the relationship. Key features include periodicity.* F-IF8 NMSI s Laying the Foundation lesson: Quadratic Optimization (1.5 days) Teacher Note: This is similar to the Algebra I Lesson by the same title. This is a great time to pull in the symmetry of parabolas and mention that if they know the roots, the vertex must be in the middle of the roots! This would allow them to not even need a graphing calculator to get the maxima. ** Checkpoint Unit 3 1,4,10,11,12 can be used with this lesson. AL COS Common Standard Common Create equations and inequalities in one variable and use them to solve problems. 20 Include equations arising from linear and quadratic functions, and simple rational and exponential functions. and PreAP Unit 3 Page 5 A-CED1

13 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. a. Formulate equations of conic sections from their determining characteristics. key features given a verbal description of the relationship. Key features include periodicity.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. A-CED2 A-CED4 F-IF5 F-IF7 F-IF7b F-BF1 F-BF1b Teacher Led Discussion--Radical equations (5 days) Teacher Note: There is not a specific NMSI s Laying the Foundation lesson that discusses Radical Equations. Your textbook should do a good job of having these type of problems. G.1.b, G.1.c, G.1.d, G.1.e should have been covered in Algebra I and Geometry. A quick review of those and then a day of G.1.f and G.1.g and this should be covered. ** Checkpoint Unit 3 13,14,15,16,17,19,21 and Illustrative Mathematics 1. Checking a calculation of a decimal exponent and 2. Rational or Irrational? can be used with this lesson. and PreAP Unit 3 Page 6

14 and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 4: Polynomials and Complex Roots 9 days of instruction plus assessment time- 2 weeks NMSI s Laying the Foundation lesson: Graphical Transformations (1 day) Teacher Note: The students are given a function that looks like a polynomial. They are asked to transform the graph and how the domain, maximum, x-intercept and y-intercept change. The students are then asked to find the average rate of change and area bounded by the graph. ** Checkpoint- Illustrative Mathematics Interpreting the Graph AL COS Common Standard Common key features given a verbal description of the relationship. Key features include periodicity.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* 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.* Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-IF5 F-IF6 F-BF3 NMSI s Laying the Foundation lesson: Investigating Functions (2 days) Teacher Note: Connects the characteristics of polynomial functions with transformations. Focuses on how the roots, maximum and minimum are affected by transformations. Introduces students to f x and f ( x ) ** Checkpoint Unit 4 #1 can be used with this lesson. AL COS Common Standard Common Identify zeros of polynomials when suitable factorizations are available, and use the A-APR3 17 zeros to construct a rough graph of the function defined by the polynomial. key features given a verbal description of the relationship. Graph polynomial functions, identifying zeros when suitable factorizations are 30b available, and showing end behavior. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) 34 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 and PreAP Unit 4 Page 1 F-IF7c F-BF3

15 graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Teacher Led Discussion --Polynomials end behavior and roots (4 days) Teacher Note: This part of the course is where students discover how polynomials act due to the highest power and the leading coefficient. Long or synthetic division is used to break down polynomials in order to find factors/roots in order to make a rough sketch of the polynomial. Use the rational root theorem to help you decide what numbers might be roots to use in the synthetic/long division. Look at graphs of polynomials and if there are not as many roots as the highest degree, then there must be pairs of imaginary roots. The discussion of double roots will probably be a part of this discussion. The connection about how many maxima and minimum and the highest power is also an outcome of this discussion. Textbooks should handle this nicely. There will be a polynomial discovery lesson for the graphing calculator in the dropbox. ** Checkpoint Illustrative Mathematics Computations With Complex Numbers can be used with this lesson. AL COS Common Standard Common 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. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 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. 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). 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. Prove polynomial identities and use them to describe numerical relationships. Example: The polynomial identity ( x 2 y 2 ) 2 ( x 2 y 2 ) 2 2xy 2 can be used to generate Pythagorean triples. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. N-CN1 N-CN9 A-APR1 A-APR2 A-APR3 A-APR4 A-CED2 F-IF7 F-IF8 F-BF1 F-BF1b and PreAP Unit 4 Page 2

16 NMSI s Laying the Foundation lesson: Adaptation of AP Calculus 1997 AB1 (2 days) Teacher Note: Graphing Calculator is required. Students must graph functions, calculate values, using the table, and finding zeros of a function. This is a velocity and position function problem. Average rate of change and particle motion are key concepts of this lesson. Students explore motion on a horizontal line utilizing both graphical and analytical skills. **Checkpoint Unit 4 #2 can be used with this lesson. AL COS Common Standard Common b Interpret expressions that represent a quantity in terms of its context.* a. Interpret parts of an expression such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. 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. 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.* key features given a verbal description of the relationship. 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.* Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. A-SSE1 A-SSE1a A-SSE1b A-APR3 A-REI11 F-IF6 F-IF7c and PreAP Unit 4 Page 3

17 and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 5: Exponential and Logarithmic Functions 15 days of instruction plus Checkpoint time- 3.5 weeks Teacher Note: The Quality Unit 1 must be downloaded from the Quality website. Additional use of textbook and other teacher sources are needed for this unit. NMSI s Laying the Foundation lesson: And So They Grow (2 days) Teacher Note: Students will use dice and a graphing calculator to simulate the growth of a population. Since we cannot physically measure how a population increases and decreases, the students will use a simulation to model the situation. Students should have some knowledge of the behavior of exponential functions and know how to use a graphing calculator to create an exponential regression function. This is a great reintroduction to exponential growth. ** Checkpoint Illustrative Mathematics Extending the Definitions of Exponents can be used with this lesson. AL COS Common Standard Common 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. key features given a verbal description of the relationship. Key features include: periodicity. 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. Teacher Led Discussion --Graphing Exponential and Logarithmic functions (1 Day) and PreAP Unit 5 Page 1 A-CED1 A-CED2 Teacher Note: This would be a good time to make the connection between the graphs of exponential graphs that they x have already seen and y e. They may have never seen e before, so you might want to explain that it is a special base and the approximate value is The formal explanation for e should take place in Precalculus, but you x could do it here if you choose. Include a discussion of translations of y e and its symmetry and asymptotic behavior. Introduce logs as the inverse undo function of exponential graphs and show graphs. Introduce y= ln x as the inverse of x y e. **Checkpoint Unit 5 1,2,3,4,5,7,8,9,13 and Illustrative Mathematics Comparing Exponential can be used with this lesson. AL COS Common Standard Common key features given a verbal description of the relationship. Key features include: F-IF6

18 34 36 periodicity. 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. 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. [F-BF3] F-LE4 Teacher Led Discussion -Exponential and Log Laws (2 Days) Teacher Note: The students have been exposed to the exponential laws, but with variables as bases. You will need to review these with bases that are numbers and the base of e. Introduce the log laws as the inverse or going backwards of the exponential laws. Use your textbook to practice these. ** Checkpoints Unit 5 6,10,11,12 can be used with this lesson. AL COS Common Standard Common 36 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. and PreAP Unit 5 Page 2 F-LE4 NMSI s Laying the Foundation lesson: Solving Systems of Exponential, Logarithmic, and Linear Equations (1 day) Teacher Note: Students will solve systems of equations using properties of exponents and logarithms. Students should be familiar with properties of exponents and logarithms, solving exponential, logarithmic and quadratic equations, solving systems of equations, and graphing functions then finding the intersection of the functions. Graphing calculator required. #1 and #2 can be solved very similarly to solving linear systems after a few laws are applied. #3 and #4 requires substitution and factoring, then applying log laws to solve. #5 requires rewriting in terms of ln and then solving. The concept of ln both sides in order to solve exponential is designated for PreCalculus, but would be very helpful for this exercise. ** Checkpoints Unit 5 #14 can be used with this lesson. AL COS Common Standard Common Use the structure of an expression to identify ways to rewrite it. 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. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. A-SSE2 A-REI11 F-IF5 Graph functions expressed symbolically and show key features of the graph, by F-IF7

19 30 hand in simple cases and using technology for more complicated cases. Combine standard function types using arithmetic operations. 33 For exponential models, express as a logarithm the solution to ab ct = d where a, c, 36 and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F-BF1b F-LE4 NMSI s Laying the Foundation lesson: Exponential and Natural Log Functions (2 days) Teacher Note: This lesson serves as a good review of many topics covered with exponential and natural logarithmic functions, such as sketching graphs using transformations, using properties of exponents and logarithms, solving both types of equations, growth and decay problems, and linearization of data. AL COS Common Standard Common 30 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. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. A-CED3 F-IF7 F-IF7e NMSI s Laying the Foundation lesson: Motion Problems Using Exponential and Natural Logarithmic Functions (1 day) Teacher Note: Students will apply exponential and logarithmic functions to the concepts of motion of an object along a horizontal line. Students should be able to factor polynomial expressions and be familiar with natural logarithms and exponential functions. Students should be able to solve both equations and inequalities involving both. This lesson is designed to be completed without a graphing calculator. Be sure to check out the two web sites listed under material and resources. AL COS Common Standard Common Use the structure of an expression to identify ways to rewrite it. key features given a verbal description of the relationship. Key features include: periodicity. 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. A-SSE2 F-LE4 and PreAP Unit 5 Page 3

20 NMSI s Laying the Foundation lesson: Curing the Sniffles (1 day) Teacher Note: Students will determine the amount of medicine that remains in the body at any given time. A simulation will be used to model the situation. Students should have some experience with recursively defined sequences. This requires that students use the sequential mode on their graphing calculator. ** Checkpoints Unit 5 #16 can be used with this lesson. AL COS Common Standard Common Algebra I 35 key features given a verbal description of the relationship Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F-BF2 Quality Unit 1 Patterns (G-10 J-4) (3 days) Teacher Note: The students should have been introduced to geometric series in Algebra I, but you may need to remind/reteach this concept. The pages listed have vocabulary and activities starting with the introduction to a geometric sequence to summing of a finite and infinite geometric series. You will need to choose what you need of this unit depending on the knowledge of your students. AL COS Common Standard Common 14 Algebra I #38 Algebra I #34 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 Construct exponential functions and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. A-SSE4 F-LE2 F-BF2 and PreAP Unit 5 Page 4

21 and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 6: Rational Functions 8 days of instruction plus assessment time- 2 weeks NMSI s Laying the Foundation lesson: Rational Functions-Short Run Behavior (1 day) Teacher Note: Students will investigate the behavior of rational functions near the vertical asymptotes. Graphing calculator required. AL COS Common Standard Common key features given a verbal description of the relationship. Key features include periodicity.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* 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.* Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Interpret expressions that represent a quantity in terms of its context.* a. Interpret parts of an expression such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. F-IF5 F-IF6 F-IF8 A-SSE1 A-SSE1a A-SSE1b NMSI s Laying the Foundation lesson: Rational Functions-Long Run Behavior (1 day) Teacher Note: Students will find the end behavior asymptote of a rational function. Graphing calculator required. AL COS Common Standard Common key features given a verbal description of the relationship. Key features include periodicity.* Relate the domain of a function to its graph and, where applicable, to the F-IF5 29 quantitative relationship it describes.* Calculate and interpret the average rate of change of a function (presented F-IF6 symbolically or as a table) over a specified interval. Estimate the rate of change 24 from a graph.* Write a function defined by an expression in different but equivalent forms to F-IF8 and PreAP Unit 6 Page 1

22 31 reveal and explain different properties of the function. Interpret expressions that represent a quantity in terms of its context.* 12 a. Interpret parts of an expression such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. A-SSE1 A-SSE1a A-SSE1b NMSI s Laying the Foundation lesson: Rational Functions-Transformations of Rational Functions (1 day) Teacher Note: Students will apply transformations to the graphs of rational functions, describe the transformations, and graph the transformed functions. They will need to do long division in order to accomplish writing the rational functions as a sum of two parts. Students need graph paper for this lesson. ** Checkpoint Unit 6 3,5,6 can be used with this lesson. AL COS Common Standard Common 19 Algebra I 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 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. 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. and PreAP Unit 6 Page 2 A-APR6 A-APR7 F-BF3 NMSI s Laying the Foundation lesson: Rational Function Exploration (1 day) Teacher Note: Students will investigate end behavior asymptotes of rational functions. Students should be able to graph using a table of values, graph a function, and use function notation. Graphing calculator required. AL COS Common Standard Common 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. 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. F-BF3 A-CED1 Illuminations lesson: Light It Up illuminations.nctm.org AL COS Common Standard Common 33 Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. F-BF1 F-BF1b