Overview

This quarter includes the culmination of module 1, which in Topic D extends to a connection to geometry. 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.

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.

The student materials consist of the student pages for each lesson in Module 2.

The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.

In this module, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). Students 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 module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.

The student materials consist of the student pages for each lesson in Module 3.

The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.

Week 1: November 6

Standards Addressed in Week's Lessons

Topic C:

A-APR.D.6

A-REI.A.1

A-REI.A.2

Module

Topic

A-REI.B.4

A-REI.C.6

A-REI.C.7

G-GPE.A.2

Notes:

Lessons 33–35:

Students study the definition of a parabola as they first learn to derive the equation of a parabola given a focus and a directrix and later to create the equation of the parabola in vertex form from the coordinates of the vertex and the location of either the focus or directrix. Students build upon their understanding of rotations and translations from Geometry as they learn that any given parabola is congruent to the one given by the equation for similar parabolas.

Week 2: November 13

Standards Addressed in Week's Lessons

Topic D:

N-CN.A.1

N-CN.A.2

N-CN.C.7

Module

Topic

A-REI.A.2

A-REI.B.4

A-REI.C.7

Notes:

Lesson 36:

This lesson presents a third obstacle to using factors of polynomials to solve polynomial equations. Students begin by solving systems of linear and nonlinear equations to which no real solutions exist and then relate this to the possibility of quadratic equations with no real solutions.

Lesson 37:

This lesson introduces complex numbers through their relationship to geometric transformations. That is, students observe that scaling all numbers on a number line by a factor of −1 turns the number line out of its one-dimensionality and rotates it 180° through the plane.

Lesson 38:

Students discover that complex numbers have real uses; in fact, they can be used in finding real solutions of polynomial equations.

Lesson 39:

Students develop facility with properties and operations of complex numbers and then apply that facility to factor polynomials with complex zeros

Week 3: November 20

Module

Topic

Standards Addressed in Week's Lessons

Topic D:

N-CN.A.1

N-CN.A.2

N-CN.C.7

A-REI.A.2

A-REI.B.4

A-REI.C.7

Standards Addressed in EOM Assessment

A-APR.D.6

A-REI.A.1

A-REI.A.2

A-REI.B.4

A-REI.C.6

A-REI.C.7

G-GPE.A.2

N-CN.A.1

N-CN.A.2

N-CN.C.7

A-REI.A.2

A-REI.B.4

A-REI.C.7

Notes:

Lesson 40:

This lesson brings the module to a close with the result that every polynomial can be rewritten as the product of linear factors, which is not possible without complex numbers.

Week 4: November 27

Notes:

Lesson 1:

This is an exploratory lesson in which students model the circular motion of a Ferris wheel using a paper plate. The goal is to study the vertical component of the circular motion with respect to the degrees of rotation of the wheel from the initial position. This function is temporarily described as the height function of a passenger car on the Ferris wheel, and students produce a graph of the height function from their model.

Lesson 2:

This lesson introduces the co-height function, which describes the horizontal component of the circular motion of the Ferris wheel. Students again model the position of a car on a rotating Ferris wheel using a paper plate, this time with emphasis on the horizontal motion of the car.

Lesson 3:

This lesson provides historical background on the development of the sine and cosine functions in India around 500 C.E. In this lesson, students generate part of a sine table and use it to calculate the positions of the sun in the sky, assuming the historical model of the sun following a circular orbit around Earth.

Lesson 4:

Lesson 4 draws connections between the height function of a Ferris wheel and the sine and cosine functions used in triangle trigonometry in Geometry. This lesson extends the domain of the sine and cosine functions from the restricted domain (0,90) of degree measures of acute angles in triangles to the interval (0,360).

Week 5: December 4

Notes:

Lesson 5:

In Lesson 5, students need to come to know enough values of these functions to generate graphs of these functions and discern structure and properties about them (in much the same way that students were first introduced to exponential functions by studying their values at integer inputs).

Lessons 6 and 7:

Lessons 6 and 7 introduce the tangent and secant functions through their geometric descriptions on a circle and link those geometric descriptions to the appropriate ratios of sine and cosine. The remaining trigonometric functions, cotangent and cosecant, are also introduced.

Lesson 8:

In this lesson, students construct a graph of the sine and cosine functions as functions on the real line by measuring the horizontal and vertical components of a point on the unit circle, breaking a piece of spaghetti to the appropriate length, and gluing it to the graph.

Week 6: December 11

Module

Standards Addressed in Week's Lessons

Topic A:

F-IF.C.7e

F-TF.A.1

F-TF.A.2

Topic B:

F-IF.C.7e

F-TF.B.5

F-TF.C.8

S-ID.B.6a

Notes:

Lesson 9:

Lesson 9 introduces radian measure. We justify the switch to radians by drawing the graph of 𝑦 = sin(𝑥°) with the same scale on the horizontal and vertical axes, which is nearly impossible to draw.

Lesson 10:

The topic culminates with Lesson 10, which incorporates such identities as sin(𝜋 − 𝑥) = sin(𝑥) and cos(2𝜋 − 𝑥) = cos(𝑥) for all real numbers 𝑥 into an introduction to trigonometric identities that will be studied further in Topic B.

Lesson 11:

Lesson 11 continues the idea started in Lesson 9 in which students graphed 𝑦 = sin(𝑘𝑥°) for different values of 𝑘.

Week 7: December 18

Standards Addressed in Week's Lessons

Topic B:

F-IF.C.7e

F-TF.B.5

F-TF.C.8

S-ID.B.6a

Notes:

Lesson 12:

In Lesson 11, teams of students work to understand the effect of changing the parameters 𝐴, 𝜔, ℎ, and 𝑘 in the graph of the function 𝑦 = 𝐴(sin(𝜔(𝑥 − ℎ))) + 𝑘, so that in Lesson 12 students can fit sinusoidal functions to given scenarios, which aligns with F-IF.C.7e and F-TF.B.5.

Lesson 13:

In Lesson 13, students analyze given real-world data and fit the data with an appropriate sinusoidal function, providing authentic practice with MP.3 and MP.4 as they debate about appropriate choices of functions and parameters.

Lesson 14:

This lesson returns to the idea of graphing functions on the real line and producing graphs of 𝑦 = tan(𝑥). Students work in groups to produce the graph of one branch of the tangent function by plotting points on a specified interval.

Lesson 15:

To wrap up the module, students revisit the idea of mathematical proof in Lessons 15–17. Lesson 15 aligns with standard F-TF.C.8, proving the Pythagorean identity.

Week 8: January 8

Module

Topic

Notes:

Lesson 16:

Throughout Lessons 15, 16, and 17, the emphasis is on the proper statement of a trigonometric identity as the pairing of a statement that two functions are equivalent on a given domain and an identification of that domain.

Lesson 17:

In Lesson 17, students discover the formula for sin(𝛼 + 𝛽) using MP.8, in alignment with standard F-TF.B.9(+), but teachers may choose to present the optional rigorous proof of this formula that is provided in the lesson.

Standards Addressed in Week's Lessons

Topic B:

F-IF.C.7e

F-TF.B.5

F-TF.C.8

S-ID.B.6a

End-of-Module Assessment:

F-IF.C.7e

F-TF.A.1

F-TF.A.2

F-IF.C.7e

F-TF.B.5

F-TF.C.8

S-ID.B.6a

Week 9: January 16

Module

Topic

Notes:

Lesson 1;

Students review and practice applying the laws of exponents to expressions in which the exponents are integers. Students first tackle a challenge problem on paper folding that is related to exponential growth and then apply and practice applying the laws of exponents to rewriting algebraic expressions.

Lesson 2:

Lesson 2 sets the stage for the introduction of base-10 logarithms in Topic B of the module by reviewing how to express numbers using scientific notation, how to compute using scientific notation, and how to use the laws of exponents to simplify those computations in accordance with N-RN.A.2.

Lesson 3:

Lesson 3 begins with students examining the graph of 𝑦 = 2^𝑥 and estimating values as a means of extending their understanding of integer exponents to rational exponents.

Standards Addressed in Week's Lessons

Topic A:

N-RN.A.1

N-RN.A.2

N-Q.A.2

F-IF.6

F-BF.A.1a

F-LE.A.2

Week 10: January 22

Module

Notes:

Lesson 4:

This lesson continues the discussion of properties of exponents and radicals, and students continue to practice

MP.7 as they extend their understanding of exponents to all rational numbers and for all positive real bases

as specified in N-RN.A.1.

Lesson 5:

This lesson revisits the work of Lesson 3 and extends student understanding of the domain of the exponential function 𝑓(𝑥) = 𝑏^𝑥 , where 𝑏 is a positive real number, from the rational numbers to all real numbers through the process of considering what it means to raise a number to an irrational exponent.

Lesson 6:

This lesson is a modeling lesson in which students practice MP.4 when they find an exponential function to model the amount of water in a tank after 𝑡 seconds when the height of the water is constantly doubling or tripling and apply F-IF.B.6 as they explore the average rate of change of the height of the water over smaller and smaller intervals.

Lesson 7:

In this lesson, students use an algorithmic numerical approach to solve simple exponential equations that arise

from modeling the growth of bacteria and other populations (F-BF.A.1a).

Standards Addressed in Week's Lessons

Topic A:

N-RN.A.1

N-RN.A.2

N-Q.A.2

F-IF.6

F-BF.A.1a

F-LE.A.2

Topic B:

N-Q.A.2

A-CED.A.1

F-BF.A.1a

F-LE.A.4

Week 11: January 29

Standards Addressed in Week's Lessons

Topic B:

N-Q.A.2

A-CED.A.1

F-BF.A.1a

F-LE.A.4

Notes:

Lesson 8:

This lesson begins with the logarithmic function disguised as the more intuitive “WhatPower” function, whose behavior is studied as a means of introducing how the function works and what it does to expressions. Students find the power needed to raise a base 𝑏 in order to produce a given number.

Lesson 9:

Just as population growth is a natural example that gives context to exponential growth, Lesson 9 gives context to logarithmic calculation through the example of assigning unique identification numbers to a group of people. In this lesson, students consider the meaning of the logarithm in the context of calculating the number of digits needed to create student ID numbers, phone numbers, and social security numbers, in accordance with N-Q.A.2.

Lesson 10:

In this lesson, students discover the logarithmic properties by completing carefully structured logarithmic tables and answering sets of directed questions. Throughout these two lessons, students look for structure in the table and use that structure to extract logarithmic properties (MP.7).