Module Focus: Precalculus and Advanced Math –
Module 4
Sequence of Sessions
Overarching Objectives of this March 2015 Network Team Institute

Participants will be able to identify, practice, and use best instructional moves and scaffolds for chosen common core standards.
High-Level Purpose of this Session
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


Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for teaching
these modules.
Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within this
module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.
Standards alignment the major work of the grade in order to fully implement the curriculum.
Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while
maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
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This session is part of a sequence of Module Focus sessions examining the Precalculus and Advanced Math curriculum, A Story of Functions
Key Points
Students will develop an understanding of trigonometric identities and properties of trigonometric functions.
Students will enhance their understanding of The Law of Sines and Law of Cosines further from what they learned in Algebra II.
Students will explore and understand inverse trigonometric functions.
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•
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Session Outcomes
What do we want participants to be able to do as a result of this
session?

Participants will draw connections between the progression documents
and the careful sequence of mathematical concepts that develop within
this module, thereby enabling participants to enact cross- grade
coherence in their classrooms and support their colleagues to do the
same.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.


Participants will be able to articulate how the topics and lessons promote
mastery of the focus standards and how the module addresses the major
work of the grade in order to fully implement the curriculum.
Participants will be prepared to implement the modules and to make
appropriate instructional choices to meet the needs of their students
while maintaining the balance of rigor that is built into the curriculum.
Session Overview
Section
Time
Overview
Introduction
16 min
Introduces key concepts of Module
4.
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4
Facilitator’s Guide
Review Pre-calculus Module 4
39 min
Explores the addition and
subtraction formulas of
trigonometric functions.
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4
Facilitator’s Guide
Review Topic A
53 min
Explores the Law of Sines and Law
of Cosines and explore the proof of
these law analytically.
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4
Facilitator’s Guide
Review Topic B
54 min
Explores developing and using
inverse trigonometric functions.
7 min
Concludes and summarizes key
concepts of Module 4.
Topic A:
Trigonometric
Functions
Topic B:
Trigonometry and
Triangles
Topic C: Inverse
Trigonometric
Functions
Conclusion
Prepared Resources


Pre-calculus Module 4 PPT
Pre-calculus Module 4
Facilitator’s Guide
Review Topic C


Pre-calculus Module 4 PPT
Pre-calculus Module 4
Facilitator’s Guide
Review Pre-calculus Module 4
Session Roadmap
Section: Introduction
Facilitator Preparation
Time: 16 minutes
In this section, you will be introduced to the key concepts and topic Materials used include:
in the Trigonometry Module of the Pre-Calculus curriculum.
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4 Facilitator’s Guide
 Pre-calculus Module 4 Participant Handout
 Document Camera
 Pencil
 Graphing calculator
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
3 min.
Welcome to this Grade 12 segment of the NTI. Today we will take a look at
the trigonometry in Module 4, which draws upon the students’ prior
experiences with trigonometry in Geometry and Algebra II.
1.
Needed tools/materials
Document Camera
Pencil
Graphing Calculator
3 min.
2.
In order for us to better address your individual needs, it is helpful to know
a little bit about you collectively.
Pick one of these categories that you most identify with. As we go through
these, feel free to look around the room and identify other folks in your
same role that you may want to exchange ideas with over lunch or at breaks.
By a show of hands who in the room is a classroom teacher?
Math trainer?
Principal or school-level leader
District-level leader?
And who among you feel like none of these categories really fit for you.
(Perhaps ask a few of these folks what their role is).
Regardless of your role, what you all have in common is the need to
GROUP
understand this curriculum well enough to make good decisions about
implementing it. A good part of that will happen through experiencing
pieces of this curriculum and then hearing the commentary that comes from
the classroom teachers and others in the group.
2 min.
3.
Our objectives for this session are:
• Examination of the development of mathematical
understanding across the module using a focus on Concept
Development within the lessons.
• Work through examples that demonstrate themes and
changes present in the Common Core State Standards.
The goal of today’s session is to take a look at the content in the lessons of
Module 4 and see how the concepts build as each lesson progresses. My
hope is that the themes of the module are clear and that what is changing
under the CCSS is apparent.
2 min.
4.
Here is our agenda for the day.
Overall, I’d like to spend our session discussing the overarching themes of
Module 4. The idea is to leave with an understanding of where the major
shifts in Precalculus are and use examples to make sense of those changes.
(Click to advance animation.) Let’s begin with an orientation to the
materials for those that are new to the materials (Skip if participants are
already familiar with the materials).
2 min.
5.
Say: Take a few minutes to read the module overview. Notice the focus
standards and those that are considered to be foundational and make some
notes on those.
Review Slide.
2 min.
6.
Say: “Take a few minutes to read the module overview. Notice the focus
standards and those that are considered to be foundational and make some
notes on those.”
Say: “Some of these standards were touched on in Module 2 of Algebra II but
will be addressed fully in this Module.”
Review Slide.
2 min.
7.
Say: The key concepts in each topic are…
Review Slide.
Section: Topic A: Trigonometric Functions
Time: 39 minutes
In this section, you will explore the addition and subtraction formulas Materials used include:
of trigonometric functions.
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4 Facilitator’s Guide
 Pre-calculus Module 4 Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
3 min.
8.
Read the Topic A opener.
Say: “The early lessons in the module contain a review of some properties
that we established by inspection in Module 2 of Algebra II, namely the
periodicity and symmetry of the trigonometric functions, their values at
multiples of pi/6, pi/4 and pi/2. Students have also had a first glance at
the addition and subtraction formulas for sine and cosine in Algebra II. In
Precalculus, we look more deeply into these properties and results.”
Let’s look at the lessons.
4 min.
9.
Say: “In Algebra II, the sine and cosine functions were introduced as
functions of the horizontal and vertical position of a rider on a Ferris
wheel.
In Lessons 1 & 2 of Module 4 in Precalculus, we revisit the idea of circular
motion by assigning a coordinate system to a carousel and using the sine
function to represent the motion of a rider in the front/back direction and
the cosine function to represent motion in the left/right direction.”
Allow participants to think about or discuss the answer to these
questions before continuing.
8 min.
10.
Allow participants to work together to derive this formula using the
images given.
For pairs that finish early, allow them to continue to work through
Exercises 1 & 2.
3 min.
11.
Say: “The work in Example 1 and Exercises 1 and 2 leads to the three
formulas for the sums and differences of cosine and the difference
formula for sine. Example 2 establishes the sum formula for the tangent
function. Students are asked to establish the sum formula for sine on the
Exit Ticket.”
8 min.
12.
Say: “It is mathematically critical that when students are asked to verify
an identity that they do not start with that identity and work down to a
statement that is always true, such as 1=1. The logic used in such a
process is faulty, and does not adequately prove the identity is valid.
(Essentially, the logic boils down to “if my statement is correct, then my
statement is correct.”)
Encourage students to approach a verification or proof problem in one of
two ways:
1. Start with one side and transform it to the other.
2. Start with an equation that we already know to be true (such as
the Pythagorean identity, or the addition formula for sine) and
apply rules of algebra and logic to transform the equation into the
identity we want.”
7 min.
13.
Say: “Lesson 3 established and used the addition and subtraction
formulas for sine, cosine and tangent. Lesson 4 uses these formulas to
derive the double-angle formulas and then the half-angle formulas for
sine, cosine and tangent.”
Say: “Allow students to struggle with these exercises before suggesting
the approach to take. You can encourage advanced students to find other
ways of expressing these formulas in terms of only sine or only cosine.”
Say: “Remember that it is not valid for student to start with a statement in
order to verify or establish that same statement. They need to start with
an equation that is known to be valid.”
2 min.
14.
Say: “This is a summary of all of the trigonometric identities that have
been established in the lesson. Remember that an identity is both a
statement that two functions are equivalent AND a statement of the
domain on which they are equivalent.”
4 min.
15.
Ask the questions on the screen and summon responses from the
participants.
Section: Topic B: Trigonometry and Triangles
Time: 53 minutes
In this section, you will explore the Law of Sines and Law of Cosines
and explore the proof of these law analytically.
Materials used include:
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4 Facilitator’s Guide
 Pre-calculus Module 4 Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
2 min.
16.
Read the topic B opener.
Say: “Topic B is primarily about the Law of Sines and Law of Cosines. These
were established in Module 2 of Algebra II, but were not used extensively.
Here, we approach the proof of these Laws analytically, as opposed to the
geometric proof presented in Algebra II. A fundamental difference is that
the teacher provided the proof in Algebra II, but the students are
responsible for providing the proof in Precalculus.”
2 min.
17.
Read the slide.
8 min.
18.
Have the participants work through Problems 1 - 6 in the Exploratory
Challenge (the rest will be done after some discussion). Questions for
discussion are on the next slide, and should be addressed before proceeding
to develop the formula.
3 min.
19.
Allow participants to answer these questions before proceeding. On the
next slide, students work through the rest of the Exploratory Challenge to
develop the formula.
8 min.
20.
Exercise 7 is a bridge between the case in the earlier exercises when the
triangle in question was isosceles and the general case in Exercise 8.
8 min.
21.
Allow participants to work through Exercises 9 & 10 individually or in pairs.
Share results.
2 min.
22.
Exercise 10 f uses a calculator to approximate the area of a regular polygon
inscribed in the unit circle. As the number of sides of the polygon
approaches infinity, the area of the polygon should approach the area of the
unit circle, which is pi.
2 min.
23.
Say: “Lesson 8 develops the Law of Sines and Lesson 9 develops the Law of
Cosines. We will omit the details in this presentation, and just use the
results.”
6 min.
24.
Say: “The opening exercise asks students to consider which method to use
to find the missing measurement in a given triangle. Work through that
now for a quick reminder.”
Before continuing, ask participants to summarize when to use the Law of
Sines and when to use the Law of Cosines.
Have participants share their reasoning for choosing which method to use
in each case before continuing. The next task is to apply these formulas to
solve problems.
8 min.
25.
Have participants work on as many of these exercises as time will allow.
4 min.
26.
Ask the questions on the screen and summon responses from the
participants.
Section: Topic C: Inverse Trigonometric Functions
Time: XX minutes
In this section, you will explore developing and using inverse
trigonometric functions.
Materials used include:

Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
2 min.
27.
Read the Topic C opener.
Say: “Topic C is primarily about developing and using inverse trigonometric
functions. Before we can define them, however, we need to recall the
graphs of trigonometric functions in order to identify domains on which
they are strictly increasing or strictly decreasing. Only on those domains
can we define the inverse trigonometric functions.”
4 min.
28.
Say: “Allow participants time to think independently or discuss quietly with
a partner before sharing answers.”
6 min.
29.
After the participants have worked through these exercises, say: “The
opening exercises lead the students to the conclusion that the sine, cosine
and tangent functions are not invertible, and that in order to make them
invertible we must choose a suitable restricted domain.”
2 min.
30.
Say: “What if we apply our procedure for finding inverse functions to the
sine function, once we’ve suitably restricted the domain.”
“Before we can consider how to describe an inverse of the sine function
would be, we need to think about some values of this function.”
3 min.
31.
Answers: The restricted sine function used to make the inverse sine
function has domain –pi/2 ≤ x ≤ pi/2 and range -1 ≤ y ≤ 1. The domain of
the inverse sine function is the range the restricted sine function. This is -1
≤ x ≤ 1.
The range of the inverse sine function is the domain of the restricted sine
function. This is –pi/2 ≤ y ≤ pi/2.
3 min.
32.
Answers: Both sin(pi/6) and sin(5pi/6) are equal to ½. However, the value
of the sine inverse function at ½ is pi/6 because the range of the inverse
sine function is –pi/2 ≤ y ≤ pi/2.
Note that the value of the inverse sine function at -1/2 is –pi/6 (not 11pi/6).
8 min.
33.
Say: “These exercises allow students to develop the inverse cosine and
inverse tangent functions on their own, and to get practice with evaluating
the inverse trigonometric functions.”
4 min.
34.
When we are solving the equation cos(x) = ½, we are looking for all values
of x in a specified interval that satisfy the equation. There are likely to be
multiple solutions.
When evaluating the inverse cosine of ½, we are looking for the ONE value
of x between 0 and pi so that the cosine of that value is ½.
2 min.
35.
Say: “These last two lessons of the module align with the modeling practice
standard as well as
(+) F-TF.7: Use inverse functions to solve equations that arise in modeling
contexts.”
8 min.
36.
Allow participants to work though Example 1. Guide them if necessary, but
encourage them to work it out in pairs or small groups.
Early finishers can continue with the exercise that follows.
8 min.
37.
Allow participants to work though Exercise 3. Guide them if necessary, but
encourage them to work it out in pairs or small groups.
Be sure to allow time to discuss solutions and summarize approaches to the
problem.
4 min.
38.
Ask the questions on the screen and summon responses from the
participants.
Section: Conclusion
Time: 7 minutes
In this section, you will conclude your exploration of topics in
Precalculus Module 4.
Materials used include:
 Pre-calculus Module 4 PPT
 Pre-calculus Module 4 Facilitator’s Guide

Time Slide # Slide #/ Pic of Slide
2 min.
39.
5 min.
40.
Pre-calculus Module 4 Participant Handout
Script/ Activity directions
GROUP
Take a few minutes to reflect on this session. You can jot your thoughts on
your copy of the PowerPoint. What are your biggest takeaways?
Now, consider specifically how you can support successful implementation
of these materials at your schools given your role as a teacher, school
leader, administrator or other representative.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Active learning
Turn and talk
Turnkey Materials Provided
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Pre-calculus Module 4 PPT
Pre-calculus Module 4 Facilitator’s Guide
Pre-calculus Module 4 Participant Handout
Additional Suggested Resources
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How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview