Module Focus: Precalculus and Advanced Math –
Module 2
Sequence of Sessions
Overarching Objectives of this December 2014 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 Precalculus and Advanced Math curriculum, A Story of Functions
Key Points
Topic A
• Introduces matrices as a tool to organize networks.
• Add, subtract, and multiply square and rectangular matrices.
Topic B
• Properties of matrix operations.
• Linear transformations represented by matrices.
Topic C
• Solve systems of equations using inverse matrices.
Topic D
• Vectors, magnitude and direction.
• Vector applications.
Topic E
• Designing computer games (projecting 3-D images on to 2-D planes).
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.
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.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.
Session Overview
Section
Time
Overview
Prepared Resources
Facilitator Preparation
Introduces Pre-calculus Module 2.

Pre-calculus Module 2 PPT
Review Pre-calculus Module 2

Pre-calculus Module 2
Facilitator Guide
Topic A: Networks
and Matrices
Explores using matrices as a tool
to organize networks and add,
subtract, and multiply square and
rectangular matrices.

Pre-calculus Module 2 PPT
Review Topic A

Pre-calculus Module 2
Facilitator Guide
Topic B: Linear
Transformation in
Planes and Space
Explores properties of matrix
operations and linear
transformations represented by
matrices

Pre-calculus Module 2 PPT
Review Topic B

Pre-calculus Module 2
Facilitator Guide
Mid-Module
Assessment
Allows participants to complete a
Mid-Module Assessment and
engage in a follow-up discussion.

Pre-calculus Module 2 PPT
Review Mid- Module Assessment

Pre-calculus Module 2
Facilitator Guide
Introduction
15 min
Topic C: Systems of
Linear Equalities
Explores solving systems of
equations using inverse matrices.

Pre-calculus Module 2 PPT
Review Topic C

Pre-calculus Module 2
Facilitator Guide
Topic D: Vectors in
Plane and Space
Explores vectors and vector
applications.

Pre-calculus Module 2 PPT
Review Topic D

Pre-calculus Module 2
Facilitator Guide
Topic E: First Person
Videogames:
Projection Matrices
Explores designing computer
games and projecting 3-D images
onto 2-D planes.

End of Module
Assessment and
Conclusion
Allows participants to complete an 
Pre-calculus Module 2 PPT
Review End of Module
End of Assessment and engage in a 
Pre-calculus Module 2
Assessment
follow-up discussion.
Facilitator Guide
Session Roadmap
Section: Introduction
Time: XX minutes
In this section, you will be introduced to Pre-calculus Module 2.
Materials used include:
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
3 min
1.
Welcome to this Grade 12 segment of the NTI. Today we will take a look at
Module 2 and the foundational work done in Grades 9 – 11 that prepares
students for it.
Needed tools/materials
Document Camera
Straight edge/ruler
Graph paper
White paper
Colored pens or pencils
2 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).
Also ask how many have worked with Module 1 materials?
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:
• Examination of the development of mathematical
understanding across the module using a focus on Concept
Development within the lessons.
• 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 2 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.
1 min
4.
Here is our agenda for the day.
Overall, I’d like to spend our session discussing the overarching themes of
Module 2. The idea is to leave with an understanding of where the major
shifts in Pre-Calculus 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).
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.
6.
Say: The key concepts in each topic are…
Review Slide.
Section: Topic A: Networks and Matrices
Time: XX minutes
In this section, you will explore using matrices as a tool to organize
networks and add, subtract, and multiply square and rectangular
matrices.
Materials used include:
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide Slide #/ Pic of Slide
#
7.
Script/ Activity directions
Read the topic A opener.
Say: Topic A is very short and introduces students to a second
application of matrices – networks. Through this work, students also
multiply by scalars, and perform matrix operations on square and
rectangular matrices.
Let’s look at the lessons.
GROUP
8.
Say: Lesson 1 starts by defining a network diagram. (read definition)
<Click>
This is an example of a network diagram. Notice vertices are connected
by arcs in some fashion.
<Click>
This can be called a network diagram, but more specifically is a
directed graph. A directed graph is a type of network diagram that
shows the direction of the path. Directed graphs must make sense in a
context. Most of the graphs in this Topic are actually directed graphs
(digraphs) – but we refer to them all as network diagrams.
9.
Say: In exploratory challenge #1 students look at a very basic network
diagram that represent travel between 4 cities.
<Click>
Ask: How many ways can we travel from City 1 to City 4? (3 ways)
<Click>
Ask: How many ways can we reasonably travel from City 4 to City 1? (1
way)
10.
Say: In exploratory challenge, the network gets a little more
complicated.
<Click>
Ask: What might the loop at City 1 represent? (tour bus)
<Click>
Ask: How many ways can you travel from City 1 to City 4 if you want to
stop in City 2 and make no other stops?
Discuss with participants.
<Click>
Say: There are 3 routes from City 1 to City 2. There are 2 routes from
City 2 to City 4. (Look at the different options together, maybe
assigning routes letters and looking at combinations of routes
possible). There are 6 routes from City 1 to City 4 through City 2.
11.
Ask: How many possible ways are there to travel from City 1 to City 4
without repeating a city.
Allow participants time to work.
Ask for answers.
<Click>
Say: I got 17 possible ways. Use the next bullets to explain your
calculation.
12.
Say: Let’s create a table that shows the possible routes. We will list the
destination cities across the top and the cities of origin on the bottom.
Let’s do the first row together: How many routes are there that start in
city 1 and end in city 1? 1 – fill that into our table….continue with row
1.
<Click>
Say: Now you complete the rest of the table.
<Click>
13.
Say: What if we took the row and column that represented the cities
out and just looked at the interior of the table? What is this beginning
to resemble? (a matrix)
<Click>
Now take out the lines creating the table. Getting closer? (yes)
<Click>
Now let’s put brackets around these values. <Click>
I can represent this network with a matrix.
14.
Ask: What is the value or 4(2,3)? Discuss that the 2 stands for the city
of origin (the row) and 3 the destination city (the column).
<Click>
The answer is 2 and that is the number of direct routes from City 2 to
City 3.
<Click>
Ask: What is the value of r(2,3) times r(2,1)? Explain.
<Click>(4, the number of 1-stop routes from City 2 to City 1 stopping in
City 3)
15.
Ask: Write an expression for the total number of one-stop routes from
City 4 to City 1 and determine the number of routes stopping in one
city.
<Click>
R(4,2)times r(2,1) plus r(4,3) times r(3,1)
<Click>
6
<Click>
Do you notice a pattern? (The city of origin is the number first and the
connecting city is next times the connecting city is first then the
destination city last)
We conclude lesson 1 by having students analyze a social network.
16.
Say: Your turn! Let’s apply what we have just learned in Lessons 2 and
3. Work these exercises and challenges listed in small groups.
17.
Presenter: Review lesson summary.
18.
Presenter: Review lesson summary.
19.
Ask the questions on the screen and summon responses from the
participants.
Section: Topic B: Linear Transformations of Planes and
Space
Time: XX min
In this section, you will explore properties of matrix operations and Materials used include:
linear transformations represented by matrices.
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide Slide #/ Pic of Slide
#
20.
Script/ Activity directions
Read the topic B opener.
Say: Topic B is all about linear transformations of planes and space,
tying our work of M2 to M1. Students transform points in 2 and 3
dimensions, while working with matrices. The look at compositions of
transformations in the plane and continue their work with matrices
look at properties that apply to matrix addition and multiplication
comparing those to real number properties. The last lesson ties all of
these concepts together as students use matrices to encrypt a code and
then break a code.
GROUP
21.
Ask: Do you remember the conditions for linearity from Module 1?
<Click>
What is the form of the matrix that models complex number
multiplication?
<Click>
How did we write a point using matrices in 2-dimensions?
<Click>
How do you think we would write a point in 3-dimensions using
matrices?
<Click>
22.
Say: Let’s read this problem and analyze Wesley’s answer.
Read with participants – discuss.
<Click> to reveal answers.
23.
Say: Draw the vector from the origin to point x.
Now draw the vector from the origin to point y.
How can we calculate and show x + y?
(either draw x and attach y or draw y and attach x)
<Click>
Discuss the parallelogram formed and the meaning of the diagonal and
the coordinates of z versus calculating x + y.
24.
Say: Now lets add x and y in 3-dimensions. What is z? (4, 4, 2)
25.
Say: Lesson 6 consists of a series of discovery challenges leading to
student understanding.
<Click>
We start by looking at linear transformations represented by matrices
in both 2 and 3-dimensions.
<Click>
Students see that the identity matrix serves the same role as the
number 1 in the real number system (the multiplicative identity).
<Click>
And the zero matrix serves the same role as 0 (the additive identity).
<Click>
Students then discover that vector addition is commutative, associative,
and distributive.
26.
Say: We use specially designed GeoGebra aplications to explore linear
transformation on cubes in Lesson 7.
<Click>
Let’s do Exploratory Challenges 1 and 2 together.
<Click>
Take participants through the exploratory challenge demonstrating
GeoGebra. There are also alternatives for classroom without access to
technology.
27.
Presenter: Review Lesson Summary.
28.
Say: In lessons 8 & 9 we look at composition of transformations.
Read problems and allow participants time to determine answer.
<Click>
Reveal and discuss answer.
29.
Presenter: Read problems and allow participants time to determine
answer.
<Click>
Reveal and discuss answer.
30.
Presenter: Read problems and allow participants time to determine
answer.
<Click>
Reveal and discuss answer.
31.
Presenter: Read problems and allow participants time to determine
answer.
<Click>
Reveal and discuss answer.
32.
Presenter: Read problems and allow participants time to determine
answer.
<Click>
Reveal and discuss answer.
33.
Presenter: Read problems and allow participants time to determine
answer.
<Click>
Reveal and discuss answer.
34.
Presenter: Read problems and allow participants time to determine
answer.
<Click>
Reveal and discuss answer.
35.
Presenter: Your turn! Complete the listed exercises and challenge in a
small group.
Allow participants time to work and then discuss results.
36.
Presenter: Read the exercise and have participants find AB and BA.
<Click>
<Click>
What have we just shown?
<Click>
Matrix multiplication is NOT commutative.
37.
Say: Your turn!
Allow students time to complete Exercise 4.
38.
Ask: What did you discover about the matrices?
<Click> to reveal answer
<Click> to reavel next quations and answers.
Presenter: Make sure participants understand that while in special
cases matrix multiplication appears to be commutative, this is not a
property that holds for all matrix multiplication, so matrix
multiplication is NOT commutative.
39.
Say: Your turn!
Allow participants time to complete example and discuss.
40.
Say: Lesson 13 shows students another application for matrices –
encryption.
Complete example 1 and exercise 1 in small groups.
Discuss.
41.
Ask the questions on the screen and summon responses from the
participants.
Section: Mid- Module Assessment
Time: XX minutes
In this section, you will complete a Mid-Module Assessment and
follow up discussion.
Materials used include:
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
42.
Section: Topic C: Systems of Linear Equations
Time: XX minutes
In this section, you will explore solving systems of equations using
inverse matrices
Materials used include:
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
GROUP
Time Slide # Slide #/ Pic of Slide
43.
Script/ Activity directions
Read the topic C opener.
Say: In Topic C, students solve systems of equations by first writing the
systems as matrices, then using inverse matrices to solve.
44.
Say: Write this system as a linear transformation.
<Click> to reveal answer
Does L have an inverse?
<Click> to reveal inverse
Solve the system using the inverse matrix.
<Click> to reveal answer.
45.
Say: Your turn! Complete the following in small groups.
Discuss
GROUP
46.
Presenter: Review question on slide.
Section: Topic D: Vectors in Plane and Space
Time: XX minutes
In this section, you will explore vectors and vector applications. Materials used include:
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide Slide #/ Pic of Slide
#
47.
Script/ Activity directions
Read the topic D opener.
Say: In Topic D, students study vector as transformations in 2 and 3dimensions. This leads to scalar multiplication, vector addition, and
vector subtraction. Students also find a vector’s magnitude and direction.
In the later lessons, students write vectors as parametric equations
showing how our work with functions and linear transformations
translate to vectors.
GROUP
48.
Ask: Which arrows represent the same vector? Explain.
(w, u, and a because they indicate a translation of 1 unit right and 3 units
up <1, 3>.
V and b indicate a translation of 3 units right and 1 unit up <3, 1>.
C and d represent a translation of 1 unit left and 3 units down <-1, -3>.)
Ask: Why do c and u not represent the same vector?
(same magnitude but opposite direction)
49.
Say: Draw vector v – first earthquake.
Draw vector t – second earthquake.
Ask: Which earthquake shifted all points further?
Compare vectors and see that v moves further – trace with transparency
and see that one is longer than other.
50.
Presenter: Read statement on magnitude.
<Click>
Ask: How can we find the magnitude?
(Pythagorean Theorem)
Say: Write the general formula for the magnitude of a vector.
<Click> to reveal answer
Ask: What is the magnitude of vectors v and t?
<Click>
Magnitude of v = root (125)
Magnitude of t = root (117)
The first earthquake shifted point further.
51.
Ask: What is the total shift after both earthquakes?
What are do we need to find?
<Click> to reveal answer
How do we calculate v + t?
<Click> to reveal answer
<Click>
<Click>
How would we show that graphically?
<Click> to show point
<Click> to show resulting vector
52.
Say: In lesson 18, we look at vector and their translations maps in 2 and
3-dimensions.
<Click>
This figure shows a vector translated in 2-D space and shows if it was a
tangent point how the transformation would occur and how this relates
to the center of the circle.
<Click>
This figure shows a circle translated in 3-D. Notice we are moving in 3
directions.
53.
Say: In lesson 18, we also find the magnitude of a vector in 3-dimensional
space.
Ask: How would you represent w^2?
<Click> to reveal answer
How about v^2?
<Click> to reveal answer
Can you put those together in some way to find the magnitude of vector
v?
<Click>
<Click>
<Click>
<Click> to reveal the magnitude of v
54.
55.
56.
57.
58.
Stone arch bridge construction relies on the fact that the arch distributes
the force of gravity acting on the stone, known as the thrust, through the
curve of the arch and into the column of stones under the arch, which we
will call the base columns. If the resultant forces are not contained in the
stones or the ground, the structure will collapse.
Stone arch bridge construction relies on the fact that the arch distributes
the force of gravity acting on the stone, known as the thrust, through the
curve of the arch and into the column of stones under the arch, which we
will call the base columns. If the resultant forces are not contained in the
stones or the ground, the structure will collapse.
59.
Let’s do the Lesson 20 Exploratory Challenge together.
60.
Say: In lesson 21, students begin to write vectors as parametric
equations. This allows us to tie our previous work with functions and
linearity to vectors showing coherence.
61.
Say: Let’s do exercise 1 – 3 together, then you will do Lesson 21 Exercise
4 and 6 and Lesson 22 Exercise 1.
62.
Say: Lesson 23 begins with a video so students can see that we use
vectors all the time in the real world.
Show video.
Ask: Which of these phenomena can be represented as a vector?
(Wind, position of a ball that has been thrown, velocity of a ball that has
been thrown, water current.)
Discuss
63.
Say: Lets do Lesson23 example 1 together and then you can do Lesson 23
exercise 1.
64.
Presenter: Ask question.
Section: Topic E: First-person Videogames: Projection
Matrices
Time: XX minutes
In this section, you will explores designing computer games and Materials used include:
projecting 3-D images onto 2-D planes.
 Pre-calculus Module 2 PPT
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
GROUP
65.
Read the topic E opener.
Say: In Topic E, explore how to project 3-dimensional objects in a 2dimensional plane as on TVs. They play with computer software and use
matrices and vectors to move objects to create their own video game.
66.
Say: In lesson 25, students begin to look at perspective, vanishing points,
and what that means in terms of transformations. Students play with a
program called ALICE 3.1 which helps them see how 3-dimensional
transformations are portrayed in 2-dimensions.
67.
Say: We work with technology, but there is some of this lesson that is
done by hand. Let’s do the exploratory challenge together.
In this drawing task the “eye” or “camera” is the point, and the shaded
figure is the “TV screen”. The cube is in 3-D.
Use lines drawn from each vertex of the cube to the point and draw the
image of the cube on the TV screen.
68.
Say: In lesson 26, we look at the math behind the projections that we
studied in Lesson 25.
Project the point onto the plane y = 1. What is the image?
Say: We must make the y coordinate 1 using the parametric equation 4t
for y.
<Click>
The image is
<Click>
Now project the point onto z = 3.
What does this mean mathematically?
<Click>
What are the coordinates of the image?
<Click> to reveal answer
69.
Say: Next students look at rotations which is more difficult. Let’s look at
the example together.
70.
Say: Lesson 27 is a fun lesson where students think they are playing –
but they are actually putting everything they have learned on matrices
and vectors into practice. ALICE is used and very specific directions are
given so that students can actually program a scenario, then they design
their own.
Let’s look at Lesson 27 together.
71.
Presenter: Ask question.
Section: End of Module Assessment and Conclusion
Time: XX minutes
In this section, you will complete an End of Module Assessment and Materials used include:
participate in a follow-up discussion as well as conclude your
 Pre-calculus Module 2 PPT
explorations of Module 2.
 Pre-calculus Module 2 Facilitator’s Guide
Time Slide # Slide #/ Pic of Slide
72.
Script/ Activity directions
Presenter – review slide.
GROUP
2 min
73.
Go over themes.
3 min
74.
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
Turnkey Materials Provided
●
●
Pre-calculus Module 2 PPT
Pre-calculus Module 2 Facilitator’s Guide
Active learning
Turn and talk
Additional Suggested Resources
●
●
●
How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview
A Story of Functions CCLS Checklist