Module Focus: Pre-calculus and Advanced Math –
Module 3
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




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
●
This session is part of a sequence of Module Focus sessions examining the Pre-calculus and Advanced Math curriculum, A Story of Functions
Key Points
Students will work in the complex plane solving polynomial equations, graphing functions, converting equations for ellipses and
hyperbolas between complex and real form.
Students will form an understanding rational expressions and graphing rational functions.
Students will learn to compose functions and find inverse functions.
•
•
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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
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.


coherence in their classrooms and support their colleagues to do the
same.
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
17 min
Introduces key concepts and
topics in Module 3.
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3
Facilitator’s Guide
Review Pre-Calculus Module 3
Topic A: Polynomial
Functions and the
Fundamental
Theorem of Algebra
225
Explores polynomial functions and
examine the Fundamental
Theorem of Algebra.
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3
Facilitator’s Guide
Review Topic A
54 min
Explores rational functions and
expressions, showing that rational
expressions form a system
analogous to rational numbers.
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3
Facilitator’s Guide
Review Topic B
37 min
Explores inverse functions and
revisit the inverse relationship
between logarithmic and
exponential functions.
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3
Facilitator’s Guide
Review Topic C
7 min
Concludes your exploration of the
key topics and concepts of Precalculus Module 3.
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3
Facilitator’s Guide
Review Pre-Calculus Module 3
Topic B: Rational
Functions and
Composition of
Functions
Topic C: Inverse
Functions
Conclusion
Session Roadmap
Prepared Resources
Facilitator Preparation
Section: Introduction
Time: 17 minutes
In this section, you will introduce you to key topics and concepts in Materials used include:
Pre-calculus Module 3.
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3 Facilitator’s Guide
 Pre-calculus Module 3 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
Module 3.
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).
GROUP
Regardless of your role, what you all have in common is the need to
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 to:
• Examine 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 according to the Common Core State Standards.
The goal of today’s session is to take a look at the content in the lessons of
Module 3 and see how the concepts build as each lesson progresses. My
goal is to make the themes of the module clear.
2 min
4.
Here is our agenda for the day.
Say: “Overall, I’d like to spend our session discussing the overarching
themes of Module 3. 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.) Say: “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.)
3 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: “These are the standards we will address in the presentation. This is
not a comprehensive list of standards addressed in the module.”
Review Slide.
2 min
7.
Say: “The key concepts in each topic are…”
Review Slide.
Section: Topic A: Polynomial Functions and the
Fundamental Theorem of Algebra
Time: 225 minutes
In this section, you will explore polynomial functions and examine
the Fundamental Theorem of Algebra.
Materials used include:
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3 Facilitator’s Guide
 Pre-calculus Module 3 Participant Handout
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
3 min
Read the Topic A opener.
8.
Say: “Lesson 1 serves to review the Fundamental Theorem of Algebra (FTA)
that students saw at the end of Module 1 of Algebra II, before we move into
new algebraic terrain.
Let’s look at the lessons.”
8 min
9.
Say: “Let’s start right in with a question whose answer may be obvious, but
whose justification isn’t.”
Pose the question on the slide and encourage participants to provide a
PROOF for the obvious response of “two.”
The cleanest proof is to suppose that a+bi is a solution, then solve (a+bi)2 = 1
by equating the real and imaginary parts.
The quadratic formula can also provide proof.
GROUP
2 min
10.
Say: “Students saw the FTA in Algebra II, Module 1, Lesson 40.
Proving the fundamental theorem of algebra is beyond what even most
undergraduate mathematics students see. We present it to the students, in
its entirety, and use it, but we can only provide justification for the case
when n=2.”
2 min
11.
Say: “We rely on these facts every time we factor a polynomial or solve a
polynomial equation. Of course, actually solving the equation or factoring
the polynomial can be very difficult, but at least we know it can be done.”
8 min
12.
Say: “Justifying the FTA for n=2 is standard N.CN.9: (+) Know the
Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.”
Allow the participants to work either alone or in pairs or small groups to
establish the FTA in the case when n=2. Share approaches and results.
AFTER the participants have worked through and discussed, this, let them
know that this is done in Lesson 1, Example 1. Much of the lesson is a review
of solving polynomial equations using previous techniques.
3 min
13.
Say: “The square root of a complex number a + bi is a solution to the
equation
z2 = a+bi.
The FTA guarantees that this solution will always exist.
However – knowing it exists and finding it are two different things. In
Lesson 2, we look at how to find these square roots, both geometrically and
algebraically.”
8 min
14.
Say: “Before we can think about finding square roots of complex numbers,
we need to understand the effect of squaring a complex number.”
Pause here to allow participants to work this out quietly or to discuss this
questions with their neighbor.
Participants should conclude that when squaring a complex number, the
modulus is squared and the argument is doubled.
Thus, the square root of z should have a modulus equal to the square root of
|z| and argument half that of z.
3 min
15.
Ask Participants: “How many square roots does a complex number have?
Describe them.”
Answer: We’ve described one of them already – take the square root of the
modulus (which for this example is a bit under 2 units long) and divide the
argument theta in half. The other is the image of that number under rotation
by pi radians; it’s the negative of the first one, which fits with what we know
of square roots from prior experience.
3 min
16.
Read through the procedure on this slide before asking participants to work
through the example on the next slide.
8 min
17.
Say: “Work through Exercise 8a in Lesson 2 to find the square roots of z = 5
+ 12i. When you are finished, sketch z and its two square roots in the
complex plane.”
Answer: The square roots are 3+2i and -3-2i.
12 min
18.
Ask: “How many square roots of 1 are there?” Then ask, “How many cube
roots of 1 are there?”
Don’t wait for an answer to the second question; start the participants on the
Exploratory Challenge in Lesson 3 which answers this question.
Answer: There are 3 cube roots of 1; only one of them is real and the other
two are complex.
5 min
19.
Read the definition of the nth roots of unity, and ask the question.
Answer: For any positive integer n, there are n nth roots of unity. This is the
maximum allowed by the fundamental theorem of algebra. If they are all
distinct (not repeated), then there are n of them.
If participants are unsure of the answer, do not answer this question yet.
Return to it after doing the next set of exercises.
10 min
20.
Say: “Work through Exercises 1-4 in Lesson 3 find the 2nd, 4th, 6th and 5th
roots of unity. Give the answers in both rectangular (a+bi) and polar form
r(cos(theta) + i sin(theta)).”
5 min
21.
Ask participants to plot the nth roots of unity in the complex plane for n = 4
and 6, then to make a conjecture about how to describe the locations of the
nth roots of unity in the plane for a general n.
Answer: The nth roots of unity lie on the unit circle, evenly spaced every
2pi/n units around the circle starting at the point 1 on the positive real axis.
2 min
22.
Say: “The long, tedious calculation (omitted here), illustrates a need for a
process to quickly evaluate powers of binomials (u+v)n.”
6 min
23.
Say: “Pascal’s triangle is generated recursively; each row is constructed from
the elements in the row above. To find the entry in a specific position, you
add the two entries located above-right and above-left. For example, to
obtain the first 10 in Row 5, add the 4 and 6 that are above it to the right and
left in Row 4.”
If there is time: Have participants do Exercise 3 from Lesson 4: Add two
additional rows to the triangle shown.
3 min
24.
Say: “Pascal published the triangle in 1654, in a work titled (in French)
Treatise on Arithmetical Triangle, in which he collected many previously
known results about the triangle and used them to solve problems in the
brand-new field of probability theory.”
Then read the slide about the fact that the triangle was well-known outside
of Europe before Pascal.
3 min
25.
Say: “Students discover that these binomial coefficients are the entries in
Pascal’s triangle.”
Binomial coefficients have many uses in mathematics, but in these lessons
we are only interested in using them to compute binomial coefficients.
Combinatorial uses are not part of the module.
12 min
26.
Say: “Students discover that these coefficients in the expansion of (u+v)n are
also the entries in Pascal’s triangle.”
Share and discuss results of these exercises; how do they link the coefficients
of the expanded binomials to Pascal’s triangle? Explorations in the patterns
of the triangle continue in Lesson 5.
4 min
27.
Say: “We have now established the binomial theorem, which essentially says
that the coefficients of (u+v)n are the entries in row n of Pascal’s triangle.
12 min
28.
Say: “Lesson 4 established the Binomial Theorem, and Lesson 5 explores
patterns in Pascal’s triangle that can be justified using the Binomial
Theorem.”
Compare results. In Exercise 1, participants should find that the entries in
Row n of the triangle sum to 2n, and that we can establish that by applying
the binomial theorem to (1+1)n.
5 min
29.
Ask: “What mathematical idea does the problem in the Exit Ticket
anticipate?”
Answer: Calculus; the difference quotient is used to calculate derivatives.
12 min
30.
Say: “In Lesson 6, we begin to look geometrically at equations that lead to
ellipses and hyperbolas in the plane.”
Ask the questions posed by the slide. Answers:
a). A circle centered at the origin with radius 3
b). An ellipse centered at the origin that passes through points (5,0), (-5,0),
(0,3), (0,-3) .
3 min
31.
Say: “In these exercises, students convert equations of circles and ellipses
from complex form to real form.”
3 min
32.
Say: “There is a lot of new terminology associated with an ellipse; a Frayer
diagram or word wall will help students master it.”
12 min
33.
Say: “Students now know what the analytic equation for an ellipse looks like
in either the Cartesian plane or the complex plane. Exercises 1 and 2 allow
students to convert from the complex form to the real form of the equation
of an ellipse; Example 2 (and Exercise 3) shows students how to convert
from the real form to the complex form.”
3 min
34.
Mention that the equations for ellipses not centered at the origin are also
covered in the lesson.
4 min
35.
Omit the video if time is running short. Otherwise watch this 73-second clip.
3 min
36.
Read the slide, allowing time for participants to think about the image.
3 min
37.
Read the slide.
Say: “The questions in the problem set investigate what happens to an
ellipse centered at the origin as either a or b gets larger, or smaller.”
12 min
38.
Allow participants to work alone or in small groups.
3 min
39.
Say: “This companion to Lesson 7 introduces hyperbolas.”
Read the slide.
If the velocity is less than the escape velocity, the satellite travels in an
elliptical orbit (as shown in E1 and E2). If the velocity is exactly the escape
velocity, then the satellite travels along a parabolic path P. If the velocity is
higher than the escape velocity, then the satellite travels on the path of a
hyperbola, shown in H.
3 min
40.
Say: “The opening discussion walks students through graphing the equation
x2 – y2 = 1, introducing the asymptotes. The resulting graph is shown here,
with the foci marked in yellow.”
3 min
41.
Say: “What is the primary geometric property of a hyperbola? ”
Read the slide, noting how closely related the properties are for an ellipse
and for a hyperbola.
16 min
42.
Allow participants time to work through these exercises. Have them share
graphs after completing the problems.
3 min
43.
Say: “A blank copy of this graphic organizer is included in the teacher file so
that it can be completed by students during the class.”
5 min
44.
Ask the questions on the screen and summon responses from the
participants.
Section: Topic B: Rational Functions and Composition of
Functions
Time: 54 minutes
In this section, you will explore rational functions and expressions,
showing that rational expressions form a system analogous to
rational numbers.
Materials used include:
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3 Facilitator’s Guide
 Pre-calculus Module 3 Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
5 min
45.
Read the topic B opener.
Say: “In Topic B, we consider a new type of function – rational functions.
We study them first as rational expressions, showing that the rational
expressions form a system analogous to the rational numbers (A-APR.7),
before studying rational functions and their graphs. The topic ends with the
study of function composition, which leads to inverse functions in Topic C.”
12 min
46.
Have participants work in pairs on selected exercises from 1-10.
At the end of work time, have pairs share their graphs on the document
camera.
Ask each pair to quietly summarize the key features of a rational function,
and then share with the group.
3 min
47.
Read this slide to summarize the previous discussion and make sure that all
key features were mentioned.
3 min
48.
Say: “In Lesson 16, we explore one of the fundamental ideas of Topic B –
Function composition. We need the idea of function composition to explore
another major theme in the upcoming lessons – function inverses, including
the inverse relationship between logarithmic and exponential functions in
Topic C, and inverse trigonometric functions in Module 4.”
Say: “Lesson 16 opens with this example. The teacher could call out times of
descent from the first table, and have students call out with the
corresponding pressure on the diver from the second table.”
5 min
49.
Have participants work through Example 1.
Say: “After this example, the teacher leads the class in a discussion leading
to the definition of a composite function before proceeding to Example 2,
which is designed to clarify standard composition notation.”
2 min
50.
Read the slide to clarify the definition of composite functions before
continuing to Example 2.
6 min
51.
Say: “In Example 2, students grapple with the unfamiliar composition
notation in a non-numerical context, associating people to their birthdays
and people to their father. Removing the idea of composition from a
numerical context makes it easier to understand. More traditional exercises
follow.”
Discuss the results.
6 min
52.
Say: “Lesson 17 will focus on solving problems using function composition,
but this problem from the Exit Ticket of Lesson 16 assesses whether not the
idea of composition has been understood.”
2 min
53.
Say: “In Lesson 16, students practiced with the ideas and notation of
function composition. In Lesson 17, they use function composition to solve
problems.”
8 min
54.
Have participants work through Example 2 in Lesson 17, and share the
result.
4 min
55.
Ask the questions on the screen and summon responses from the
participants.
Section: Topic C: Inverse Functions
Time: 37 minutes
In this section, you will explore inverse functions and revisit the Materials used include:
inverse relationship between logarithmic and exponential
 Pre-calculus Module 3 PPT
functions.
 Pre-calculus Module 3 Facilitator’s Guide
 Pre-calculus Module 3 Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
3 min
56.
Read the Topic C opener.
Say: “Topic C is a short topic that presents the ideas of inverse functions and
revisits the inverse relationship between logarithmic and exponential
functions first seen in Module 3 of Algebra II. These ideas, particularly the
concept of restricting the domain to create a function with an inverse, are
needed again in Module 4 when students are introduced to inverse
trigonometric functions.”
6 min
57.
Say: “The idea and process of inverting a function is presented through a
sequence of examples. Work through Exercises 1-10.”
Discuss and share results.
2 min
58.
Read through the definition on the slide.
6 min
59.
Say: “In these examples, students see the reflective property for inverse
functions before being told about it.”
2 min
60.
Say: “Students observe the reflection property of a function and its inverse
through doing the exercises. More traditional exercises follow, with
functions given symbolically, and students practice the process of ‘swap x
and y and solve for y’.”
5 min
61.
Say: “The problem set covers writing the inverse of a function represented
numerically, symbolically or graphically. In Exercises 3a-d, students
practice the reflection property described on the previous slide. Work
through those problems now.”
6 min
62.
Ask: “Along with gaining more practice with reading function inverses from
a graph or table, or finding the symbolic representation of a function
inverse, Lesson 19 introduces the key step of restricting the domain of a
function to make it invertible. This curriculum does not introduce a
horizontal line test because the Common Core standards do not formally
define relations, which are needed to make a coherent statement of the
vertical and horizontal line test.”
3 min
63.
Say: “Lesson 20, Inverses of Logarithmic and Exponential Functions, reestablishes the inverse relationship between logarithmic and exponential
functions that was first established in Module 3 of Algebra II. That
knowledge is used in Lesson 21 as we solve problems using the inverse
relationship between logarithmic and exponential functions.”
4 min
64.
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 the key
topics and concepts of Precalculus Module 3.
Materials used include:
 Pre-calculus Module 3 PPT
 Pre-calculus Module 3 Facilitator’s Guide
 Pre-calculus Module 3 Participant Handout
Time Slide # Slide #/ Pic of Slide
2 min
65.
5 min
66.
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
●
●
●
Pre-calculus Module 3 PPT
Pre-calculus Module 3 Facilitator’s Guide
Pre-calculus Module 3 Participant Handout
Additional Suggested Resources
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How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview