Algebra II

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.

High School Pacing Guide SY 2022-2023

Module 1: Polynomials, Rational, and Radical Relationships

(40 Hours)

Overview

Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.

Standards

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

N-CN.A.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.

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

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

A-APR.B.2 4 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A-APR.B.3 5 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.

Unit Vocab

A Square Root of a Number (A square root of a number 𝑥 is a number whose square is 𝑥. In symbols, a square root of 𝑥 is a number 𝑎 such that 𝑎 2 = 𝑥.

Negative numbers do not have any square roots, zero has exactly one square root, and positive numbers have two square roots.)

The Square Root of a Number (Every positive real number 𝑥 has a unique positive square root called the square root or principle square root of 𝑥; it is denoted √𝑥. The square root of zero is zero.)

Pythagorean Triple (A Pythagorean triple is a triplet of positive integers (𝑎, 𝑏, 𝑐) such that 𝑎 2 + 𝑏 2 = 𝑐 2 . The triplet (3, 4, 5) is a

Pythagorean triple but (1, 1, √2) is not, even though the numbers are side lengths of an isosceles right triangle.)

End Behavior (Let 𝑓 be a function whose domain and range are subsets of the real numbers. The end behavior of a function 𝑓 is a description of what happens to the values of the function o as 𝑥 approaches positive infinity, and o as 𝑥 approaches negative infinity.)

Even Function (Let 𝑓 be a function whose domain and range is a subset of the real numbers. The function 𝑓 is called even if the equation, 𝑓(𝑥) = 𝑓(−𝑥), is true for every number 𝑥 in the domain. Even-degree polynomial functions are sometimes even functions, such as 𝑓(𝑥) = 𝑥 10, and sometimes not, such as 𝑔(𝑥) = 𝑥 2 − 𝑥.)

Odd Function (Let 𝑓 be a function whose domain and range is a subset of the real numbers. The function 𝑓 is called odd if the equation, 𝑓(−𝑥) = −𝑓(𝑥), is true for every number 𝑥 in the domain. Odd-degree polynomial functions are sometimes odd functions, such as 𝑓(𝑥) = 𝑥 11, and sometimes not, such as ℎ(𝑥) = 𝑥 3 − 𝑥 2 .)

Rational Expression (A rational expression is either a numerical expression or a variable symbol, or the result of placing two previously generated rational expressions into the blanks of the addition operator (__+__), the subtraction operator (__−__), the multiplication operator (__×__), or the division operator (__÷__).)

Parabola (A parabola with directrix line 𝐿 and focus point 𝐹 is the set of all points in the plane that are equidistant from the point 𝐹 and line 𝐿.)

Axis of Symmetry (The axis of symmetry of a parabola given by a focus point and a directrix is the perpendicular line to the directrix that passes through the focus.)

Vertex of a Parabola (The vertex of a parabola is the point where the axis of symmetry intersects the parabola.)

Dilation at the Origin (A dilation at the origin 𝐷𝑘 is a horizontal scaling by 𝑘 > 0 followed by a vertical scaling by the same factor 𝑘. In other words, this dilation of the graph of 𝑦 = 𝑓(𝑥) is the graph of the equation 𝑦 = 𝑘𝑓 ( 1 𝑘 𝑥). A dilation at the origin is a special type of a dilation.)

Module 1: All materials here

Topic A: Polynomials from Base 10 to Base X

Standards

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see as ( ) ( ) , thus recognizing it as a difference of squares that can be factored as ( )( ).

A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity ( ) ( ) ( ) can be used to generate Pythagorean triples

Overview

The focus in this topic is on polynomial arithmetic and how it is analogous to operations with integers. The module opens with a lively lesson that engages students in writing polynomial expressions for sequences by examining successive differences. Later in this topic, students gain fluency with polynomial operations and work with identities such as a2-b2=(a+b)(a-b). The topic closes with an emphasis on the use of factoring and the special role of zeros when solving polynomial equations.

Lessons

Topic B: Factoring - Use and Obstacles

Standards

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

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 – y 4 as (x2 ) 2 – (y2 ) 2 , thus recognizing it as a difference of squares that can be factored as (x2 – y 2 )(x2 + y2 ).

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

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

A-APR.D.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.

F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Overview

This topic focuses on factoring polynomials and the advantages of factored form of a polynomial to both solve equations and sketch graphs of polynomial functions

Lessons

Topic C: Solving and Applying Equations

Standards

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

A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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

REI.B.4 Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x 2 + y2 = 3.

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

Overview

Students solve polynomial, rational, and radical equations, and apply these types of equations to real-world situations. They examine the conditions under which an extraneous solution is introduced. They rewrite rational expressions in different forms and work with radical expressions as part of this process. Students work with systems of equations that include quadratic and linear equations and apply their work to understanding the definition of a parabola.

Lessons

Topic D: Complex Numbers Overcome Obstacles

Standards

N-CN.A.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.

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

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

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

A-REI.B.4 Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y 2 = 3.

Overview

Students extend their facility with solving polynomial equations to working with complex zeros. Complex numbers are introduced via their relationship with geometric transformations. The topic concludes with students realizing that every polynomial function can be written as a product of linear factors, which is not possible without complex numbers

Lessons

Module 2: Trigonometric Functions

(17 Hours)

Overview

Module 2 builds on students’ previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry. The heart of the module is the study of precise definitions of sine and cosine (as well as tangent and the co-functions) using transformational geometry from high school Geometry.

This precision leads to a discussion of a mathematically natural unit of rotational measure, a radian, and students begin to build fluency with the values of the trigonometric functions in terms of radians. Students graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions. The study of the properties culminates in the proof of the Pythagorean identity and other trigonometric identities.

Standards

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

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

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

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

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

F-TF.C.8 Prove the Pythagorean identity sin2 (𝜃) + cos2 (𝜃) = 1 and use it to find sin(𝜃), cos(𝜃), or tan(𝜃) given sin(𝜃), cos(𝜃), or tan(𝜃) and the quadrant of the angle

S-ID.B.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Unit Vocab

Radian (A radian angle is the angle subtended by an arc of a circle that is equal in length to the radius of the circle. A radian (1 rad) is a unit of rotational measure given by a rotation by a radian angle.)

Periodic Function (A function 𝑓 whose domain is a subset of the real numbers is said to be periodic with period 𝑃 > 0 if the domain of 𝑓 contains 𝑥 + 𝑃 whenever it contains 𝑥, and if 𝑓(𝑥 + 𝑃) = 𝑓(𝑥) for all real numbers 𝑥 in its domain.)

Sine (Let 𝜃 be any real number. In the Cartesian plane, rotate the initial ray by 𝜃 radians about the origin. Intersect the resulting terminal ray with the unit circle to get a point (𝑥𝜃, 𝑦𝜃). The value of sin(𝜃) is 𝑦𝜃.)

Cosine (Let 𝜃 be any real number. In the Cartesian plane, rotate the initial ray by 𝜃 radians about the origin. Intersect the resulting terminal ray with the unit circle to get a point (𝑥𝜃, 𝑦𝜃). The value of cos(𝜃) is 𝑥𝜃.)

Sinusoidal Function (A periodic function is sinusoidal if it can be written in the form 𝑓(𝑥) = 𝐴 sin(𝜔(𝑥 − ℎ)) + 𝑘 for real numbers 𝐴, 𝜔, ℎ, and 𝑘. In this form,

|𝐴| is called the amplitude of the function,
2𝜋/ |𝜔| is the period of the function,
|𝜔|/ 2𝜋 is the frequency of the function,
ℎ is called the phase shift, and the graph of 𝑦 = 𝑘 is called the midline.
Furthermore, we can see that the graph of the sinusoidal function 𝑓 is obtained by first vertically scaling the graph of the sine function by 𝐴, then horizontally scaling the resulting graph by 1 𝜔 , and, finally, by horizontally and vertically translating the resulting graph by ℎ and 𝑘 units, respectively.)

Period (The period 𝑃 is the distance between two consecutive maximal points, or two consecutive minimal points, on the graph of the sinusoidal function.)

Amplitude (The amplitude is the distance between a maximal point of the graph of the sinusoidal function and the midline.)

Frequency (The frequency of a periodic function is the unit rate of the constant rate defined by the number of cycles per unit length.)

Midline (The midline is the horizontal line that is halfway between the maximal line and the minimal line.)

Trigonometric Identity (A trigonometric identity is a statement that two trigonometric functions are equivalent.)

Tangent (Let 𝜃 be any real number such that 𝜃 ≠ 𝜋 2 + 𝑘𝜋 for all integers 𝑘. In the Cartesian plane, rotate the initial ray by 𝜃 radians about the origin. Intersect the resulting terminal ray with the unit circle to get a point (𝑥𝜃, 𝑦𝜃). The value of tan(𝜃) is 𝑦𝜃 𝑥𝜃 .)

Module 2: Trigonometric Functions - All materials here

Topic A: The story of Trigonometry and it's contents

Standards

F-IF.C.7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

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

Overview

In this topic, students develop an understanding of the six basic trigonometric functions as functions of the amount of rotation of a point on the unit circle (F-TF.A.1), and then translate that understanding to the trigonometric functions as functions on the real number line (F-TF.A.2). Students study graphs of the functions to discover properties of the trigonometric functions (F-IF.C.7e).

Lessons

Topic B: Understanding Trigonometric Functions

Standards

Students will use trigonometric functions to model periodic phenomena (F-TF.B.5) by fitting sinusoidal functions to data (S-ID.B.6a). Students use the properties of the graphs of sinusoidal functions to help fit functions to data to solve problems in the context of the data (F-IF.C.7e). We end the module with the study of trigonometric identities and how to prove them (F-TF.C.8)

Overview

F-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

F-TF.C.8 Prove the Pythagorean identity sin2 (𝜃) + cos2 (𝜃) = 1 and use it to find sin(𝜃), cos(𝜃), or tan(𝜃) given sin(𝜃), cos(𝜃), or tan(𝜃) and the quadrant of the angle. S-ID.B.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear quadratic, and exponential models.

Lessons

Module 3: Exponential and Logarithm Functions

(33 Hours)

Overview

The process of choosing and using mathematics/statistics to analyze empirical situations, to understand them better, and make decisions, is the heart of this module.

Standards

N-RN.A.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.

N-RN.A.22 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

N-Q.A.23 Define appropriate quantities for the purpose of descriptive modeling.

A-SSE.B.34 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. .

A-SSE.B.45 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★

A-CED.A.16 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-REI.D.11 Explain why the 𝑥𝑥-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

F-IF.A.38 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by 𝑓𝑓(0) = 𝑓𝑓(1) = 1, 𝑓𝑓(𝑛𝑛 + 1) = 𝑓𝑓(𝑛𝑛) + 𝑓𝑓(𝑛𝑛 − 1) for 𝑛𝑛 ≥ 1.

Unit Vocab

Arithmetic Series (An arithmetic series is a series whose terms form an arithmetic sequence.)

Geometric Series (A geometric series is a series whose terms form a geometric sequence.)

Invertible Function (Let 𝑓𝑓 be a function whose domain is the set 𝑋𝑋, and whose image is the set 𝑌𝑌. Then 𝑓𝑓 is invertible if there exists a function 𝑔𝑔 with domain 𝑌𝑌 and image 𝑋𝑋 such that 𝑓𝑓 and 𝑔𝑔 satisfy the property: For all 𝑥𝑥 ∈ 𝑋𝑋 and 𝑦𝑦 ∈ 𝑌𝑌, 𝑓𝑓(𝑥𝑥) = 𝑦𝑦 if and only if 𝑔𝑔(𝑦𝑦) = 𝑥𝑥. The function 𝑔𝑔 is called the inverse of 𝑓𝑓, and is denoted 𝑓𝑓−1. The way to interpret the property is to look at all pairs (𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌: If the pair (𝑥𝑥, 𝑦𝑦) makes 𝑓𝑓(𝑥𝑥) = 𝑦𝑦 a true equation, then 𝑔𝑔(𝑦𝑦) = 𝑥𝑥 is a true equation. If it makes 𝑓𝑓(𝑥𝑥) = 𝑦𝑦 a false equation, then 𝑔𝑔(𝑦𝑦) = 𝑥𝑥 is false. If that happens for each pair in 𝑋𝑋 × 𝑌𝑌, then 𝑓𝑓 and 𝑔𝑔 are invertible and are inverses of each other.)

Logarithm (If three numbers, 𝐿𝐿, 𝑏𝑏, and 𝑥𝑥 are related by 𝑥𝑥 = 𝑏𝑏𝐿𝐿, then 𝐿𝐿 is the logarithm base 𝑏𝑏 of 𝑥𝑥, and we write 𝐿𝐿 = log𝑏𝑏(𝑥𝑥). That is, the value of the expression log𝑏𝑏(𝑥𝑥) is the power of 𝑏𝑏 needed to be equivalent to 𝑥𝑥. Valid values of 𝑏𝑏 as a base for a logarithm are 0 < 𝑏𝑏 < 1 and 𝑏𝑏 > 1.)

Series (Let 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, 𝑎𝑎4, … be a sequence of numbers. A sum of the form 𝑎𝑎1 + 𝑎𝑎2 + 𝑎𝑎3 + ⋯ + 𝑎𝑎𝑛𝑛 for some positive integer 𝑛𝑛 is called a series, or finite series, and is denoted 𝑆𝑆𝑛𝑛. The 𝑎𝑎𝑖𝑖’s are called the terms of the series. The number that the series adds to is called the sum of the series. Sometimes 𝑆𝑆𝑛𝑛 is called the 𝑛𝑛th partial sum.)

𝒆 (Euler’s number, 𝑒𝑒, is an irrational number that is approximately equal to 𝑒𝑒 ≈ 2.7182818284590.)

𝚺 (The Greek letter sigma, Σ , is used to represent the sum. There is no rigid way to use Σ to represent a summation, but all notations generally follow the same rules.

Topic A: Real Numbers

Standards

N-RN.A.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.

5. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

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

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

F-BF.A.1a Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context

F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)

Overview

In Topic A, students both review what they already know about exponential expressions and functions with integer exponents, and extend the meaning of an exponential expression to allow for first rational and then irrational exponents (N-RN.1). Students practice applying properties of exponents to expressions with non-integer exponents (N-RN.2). Students use the average rate of change and repeated reasoning to discover Euler’s number, e (F-IF.6).

Lessons

Topic B: Logarithms

Standards

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

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

F-BF.A.1a Write a function that describes a relationship between two quantities.★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-LE.A.4 For exponential models, express as a logarithm the solution to 𝑎𝑎𝑏𝑏𝑐𝑐𝑐𝑐 = 𝑑𝑑 where 𝑎𝑎, 𝑐𝑐, and 𝑑𝑑 are numbers and the base 𝑏𝑏 is 2, 10, or 𝑒𝑒; evaluate the logarithm using technology.

Overview

At the beginning of Topic B, students apply the properties of exponents to solve exponential equations numerically (F-BF.1a) as a way to motivate the need for logarithms, which are first introduced by the more intuitive name “WhatPower”. In the intermediate lessons, students discover the logarithmic properties by creating and examining logarithmic tables and answering sets of directed questions. In the final lessons in Topic B, they solve logarithmic equations by applying the inverse relationship between exponents and logarithms (F-LE.4).

Lessons

Topic C: Functions and their Graphs

Standards

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

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

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

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

F-BF.A.1a Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

Overview

In Topic C, students graph logarithmic functions, identifying key features (F-IF.4, F-IF.7) and discover how the logarithmic properties are evidenced in the graphs of corresponding logarithmic functions. The inverse relationship between an exponential function and its corresponding logarithmic function is made explicit (F-BF.3).

In the final lesson in Topic C, students synthesize what they know about linear, quadratic, sinusoidal, and exponential functions to determine which function is most appropriate to use to model a variety of real-world scenarios (F-BF.1a).

Lessons

Topic C: Using Logarithms in Modeling Situations

Standards

A-SSE.B.3c 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.

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A-REI.D.11 Explain why the 𝑥𝑥-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

F-IF.B.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by 𝑓𝑓(0) = 𝑓𝑓(1) = 1, 𝑓𝑓(𝑛𝑛 + 1) = 𝑓𝑓(𝑛𝑛) + 𝑓𝑓(𝑛𝑛 − 1) for 𝑛𝑛 ≥ 1.

F-IF.B.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.

Overview

Topic D opens with a hands-on simulation and modeling activity in which students gather data and apply the analysis of Lesson 22 in Topic C to model it with an exponential function (A-CED.2, F-LE.5). Students use logarithms to solve exponential equations analytically, and express the solution as a logarithm (F-LE.4).

Students study the relationship between exponential growth and decay and geometric series (F-IF.3), and students must use properties of exponents to interpret expressions for exponential functions (F-IF.8b). Armed with a more thorough understanding of exponential functions and equations, students revisit the topic of Newton’s Law of Cooling that was introduced in Algebra I (F-BF.1b).

Lessons

Topic C: Geometric Series and Finance

Standards

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

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

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

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

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as 𝑦𝑦 = (1.02)𝑡𝑡, 𝑦𝑦 = (0.97)𝑡𝑡, 𝑦𝑦 = (1.01)12𝑡𝑡 , 𝑦𝑦 = (1.2) 𝑡𝑡 10, and classify them as representing exponential growth or decay.

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

F-BF.A.1b Write a function that describes a relationship between two quantities.★

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

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

Overview

Topic E is a culminating series of lessons driven by MP.4, Modeling with Mathematics. Students apply what they have learned about mathematical models and exponential growth to financial literacy while developing and practicing the formula for the sum of a finite geometric series (A-SSE.4). Throughout this set of lessons, students study the mathematics behind car loans, credit card payments, savings plans and mortgages, developing the needed formulas from summing a finite geometric series in each case. Key features of tables and graphs are used to answer questions about finances (F-IF.7e).

Lessons

Module 4: Inferences and Conclusions from Data

(30 Hours)

Overview

Students build a formal understanding of probability, considering complex events such as unions, intersections, and complements as well as the concept of independence and conditional probability. The idea of using a smooth curve to model a data distribution is introduced along with using tables and technology to find areas under a normal curve. Students make inferences and justify conclusions from sample surveys, experiments, and observational studies. Data is used from random samples to estimate a population mean or proportion. Students calculate margin of error and interpret it in context. Given data from a statistical experiment, students use simulation to create a randomization distribution and use it to determine if there is a significant difference between two treatments.

Standards

S-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

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

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

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

S-IC.B.6 Evaluate reports based on data.

S-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B

Unit Vocab

Complement of an Event (The complement of an event, A, denoted by A^C, is the event that A does not occur.)

Conditional Probability (The probability of an event given that some other event occurs. The conditional probability of A given B is denoted by P(A│B).)

Experiment (An experiment is a study in which subjects are assigned to treatments for the purpose of seeing what effect the treatments have on some response.)

Hypothetical 1000 Table (A hypothetical 1000 table is a two-way table that is constructed using given probability information. It represents a hypothetical population of 1000 individuals that is consistent with the given probability distribution and also allows calculation of other probabilities of interest.)

Independent Events (Two events A and B are independent if P(A│B)=P(A). This implies that knowing that B has occurred does not change the probability that A has occurred.)

Intersection of Two Events (The intersection of two events, A and B, denoted by A∩B, is the event that A and B both occur.)

Lurking Variable (A lurking variable is one that causes two variables to have a high relationship even though there is no real direct relationship between the two variables.)

Margin of Error (The margin of error is the maximum likely error when data from a sample are used to estimate a population characteristic, such as a population proportion or a population mean.)

Normal Distribution (A normal distribution is a distribution that is bell-shaped and symmetric.)

Observational Study (An observational study is one in which the values of one or more variables are observed with no attempt to affect the outcomes.)

Random Assignment (Random assignment is the process of using a chance mechanism to assign individuals to treatments in an experiment.)

Random Selection (Random selection is the process of selecting individuals for a sample using a chance mechanism that ensures that every individual in the population has the same chance of being selected.)

Sample Survey (A sample survey is an observational study in which people respond to one or more questions.)

Treatment (A treatment is something administered in an experimental study.)

Union of Two Events (The union of two events, A and B, denoted by A∪B, is the event that either A or B or both occur.)

Module 4: Inferences about Data - All Materials Here

Topic A: Probability

Standards

S-IC.A.2 Decide if a specified model is consistent with results from a given data generating process, e.g. using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? S-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S-CP.A.2 Understand that two events 𝐴 and 𝐵 are independent if the probability of 𝐴 and 𝐵 occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S-CP.A.3 Understand the conditional probability of 𝐴 given 𝐵 as 𝑃(𝐴 and 𝐵)/𝑃(𝐵), and interpret independence of 𝐴 and 𝐵 as saying that the conditional probability of 𝐴 given 𝐵 is the same as the probability of 𝐴, and the conditional probability of 𝐵 given 𝐴 is the same as the probability of 𝐵.

S-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare results.

Overview

Chance experiments, sample space, and events such as unions, intersections, and complements are explained. Probabilities of unions and intersections are calculated using data in two-way tables and Venn diagrams are used to represent a sample space and reinforce the probability formulas. Conditional probability is also introduced.

Lessons

Topic B: Modeling Data Distribution

Standards

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

Overview

Smooth curves are used to model given data distributions, eventually using the normal distribution to model data distributions that are bell shaped and symmetric. Tables and technology are used to calculate normal probabilities.

Lessons

Topic C: Drawing Conclusions using Data from a Sample

Standards

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

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

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

S-IC.B.6 Evaluate reports based on data.

Overview

The purposes of and differences among observational studies, surveys, experiments are explored along with how randomization relates to each. Data from random samples are used to estimate a population mean or population proportion and use simulation to understand margin of error.

Lessons

Topic D: Drawing Conclusions using Data from an experiment

Standards

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

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

S-IC.B.6 Evaluate reports based on data.

Overview

Conclusions are drawn based on data from a statistical experiment. Simulation is used to explore whether an observed difference in group means is significant. Published reports based on statistical experiments that compare two treatments are critiqued and evaluated.

Lessons