Module Focus: Algebra II – Module 2
Sequence of Sessions
Overarching Objectives of this October 2014 Network Team Institute

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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 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.
Related Learning Experiences
●
This session is part of a sequence of Module Focus sessions examining the Algebra II curriculum in A Story of Functions.
Key Points
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There is a rich historical context that motivates the study of trigonometry.
Students are given opportunities to explore the meaning of the trigonometric functions and their properties in order to develop a deeper
understanding.
Students make connections between the values of sine and cosine for particular amounts of rotation with the graphs of the functions and then to
trigonometric identities.
Trigonometric functions are useful for modeling periodic data.
A trigonometric identity is a statement that two functions are equivalent on a given domain and an identification of that domain.
Trigonometric identities can be proven graphically, numerically, and algebraically.
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
Introduction
33 min
Introduces Grade 11 Module 2


Algebra II Module 2 PPT
Facilitator Guide
175 min
Explore how trigonometric
functions are defined as functions
and the graphs of the sine and
cosine function are used to
discover basic trigonometric
identities.


Algebra II Module 2 PPT
Facilitator Guide
Mid-Module
Assessment
35 min
Allows Participants to complete a
Mid-Module Assessment and
follow up discussion.


Algebra II Module 2 PPT
Facilitator Guide
Review Mid-Module Assessment
Topic B:
Understanding
133 min
Explore how trigonometric
functions can model periodic


Algebra II Module 2 PPT
Facilitator Guide
Review Topic B
Topic A: The Story of
Trigonometry and its
Contexts
Prepared Resources
Facilitator Preparation
Review Grade 11 Module 2
Review Topic A
Trigonometric
Functions and
Putting Them to Use
End of Module
Assessment
behavior and the concept of
polynomial identities.
44 min
Allows Participants to complete an
End of Module Assessment and
follow up discussion.


Algebra II Module 2 PPT
Facilitator Guide
Review End of Module
Assessment
Session Roadmap
Section: Introduction
Time: 33 minutes
In this section, you will be introduced to Grade 11 Module 2.
Materials used include:
 “Grade 11 Module 2” PPT
 “Grade 11 Module 2” Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
3 min
Introduce ourselves and talk about the session.
1.
Mostly we will be working on the student pages located at the front of your
binder. Avoid looking at the teacher pages for the most part. I mainly want
you to experience the module from the student perspective.
GROUP
5 min
2.
Start here. Then back up to opening slide.
Give participants 5 min to do the opening exercise.
After intro, come back to the opening exercise and share graphs. Why is
trigonometry based on movement around a circle? Why not a square?
3 min
3.
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
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
4.
We have three main objectives for this morning’s work. Our main task will
be experiencing lessons and assessments. As a secondary objective, you
should walk away from the study of Module 2 being able to articulate how
these lessons promote mastery of the standards and how they address the
major work of the grade. Lastly, you should be able to get a sense for the
coherent connections to the content of earlier grade levels.
2 min
5.
Here is our agenda for the day.
We will spend most of our time on G11 M2. As we go through the module, I
will talk about foundational skills developed in prior grades. We will
discuss some fluency drills and other scaffolds that can be used to address
possible gaps in content knowledge.
We will also discuss how this develops skills needed for grade 12
(precalculus).
3 min
6.
(Go through the bullets to give an overview of the progression or flow of
each topic and the module as a whole.)
5 min
7.
Display and have participants read the Module overview, scan the
standards and the new terminology.
This slide needs work….mention that the approach is exploration (very few
problem set lessons), talk about the definition of theta as a rotation.
-------------I’m going to try paraphrasing from the module overview, noting the key
points.
I’m not sure how to get the theta as rotation point across without speaking
in negatives – which I want to avoid. Can we emphasize this later on?
Yes, I think we mention it here and continue to emphasize throughout.
8 min
8.
(Review the bullet points with participants to remind them of the
background students are coming in to this module with.)
We will be discussing in more detail as we go through the module what
students’ previous experiences have been
2 min
9.
Section: Topic A: The Story of Trigonometry and its
Contexts
Time: 175 minutes
In this section, you will explore how trigonometric functions are
defined as functions and the graphs of the sine and cosine function
are used to discover basic trigonometric identities.
Materials used include:
 “Grade 11 Module 2” PPT
 “Grade 11 Module 2” Facilitators Guide
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
5 min
Read the topic A opener.
10.
Note that all the lessons are either exploratory or socratic.
8 min
11.
Supplies: Rulers, protractors, graph paper, card stock or manila
folders, metal brad/fastener.
Give participants time to work L1 Exploratory challenge 1. Share some
graphs. Model the classroom discussion. What sort of graphs would
you expect to see from students? Note varying levels of precision on
the graphs.
Give participants time to work L1 Exploratory challenge 2 and the
exploratory challenge in L2.
Discuss and share results.
Students come away with a context about how circular motion can be
described using both a vertical and horizontal component and the fact
that this motion leads to a new type of function that we have not seen
GROUP
before.
The ferris wheel will be referred to periodically throughout the module
as a familiar reference when thinking about circular motion. We will
explore how to create a function to model the motion of the ferris
wheel later in the module.
8 min
12.
Using this definition, why is the Ferris wheel height function an
example of a periodic function?
Is the co-height also a periodic function?
Why was it logical to call it the co-height?
10
min
13.
Provides historical context to tracking circular motion.
What would astronomers have been interested in?
• Pinpointing the location – distance from earth, rotation from the
horizontal
• height and “overness”
• Quadrants helped to track a location (think about how they are
numbered)
Why is counterclockwise considered positive? Why do clocks rotate in
the opposite direction?
Think back to opening exercise. If stars “moved” in a square motion,
we would have squine and cosquine instead.
Finally, at the end of this lesson, the term “sine” is mentioned.
10
min
14.
Supplies: protractor, paper, colored pencils
Give participants time to work example 1 and 2. Discuss.
Look at problem set problems.
Give them instructions on the graphic organizer and let them offer
suggestions or ideas.
To build fluency, sprints could be used throughout this module.
5 min
15.
Supplies: protractor, paper, colored pencils
Make sure students realize they don’t have to memorize the quadrants
in which sine and cosine are positive and negative if they understand
how the sine and cosine functions are defined.
Students could add to their graphic organizer throughout the module.
5 min
16.
17.
10
min
18.
We already extended the domain from (0,90) to (0,360). After this
lesson we will have examined all possible rotations and therefore
extended the domain to include all real numbers.
15
min
19.
Allow time to work opening exercise. It seems strange that we learned
about lines tangent to circles and this trigonometric function called
tangent.
Coincidence? Or is there a connection between the two?
Work on exercises 2 – 6 (on doc cam or give participants time).
5 min
20.
Before revealing, ask participants to share ways in which tangent is
interpreted in this lesson.
20
min
21.
Without the geometric context, secant, cosecant, and cotangent make
little sense to the students.
The second question is one my students always ask…if the names are
just random why not pair cosine with cosecant?
We really only need sine and cosine functions. The rest of trig can be
worked with these two. sec𝜃 could just as easily be written 1/cos𝜃. It
was settled upon for the purpose of printing. sec𝜃 was much easier to
produce on a printing press than 1/cos𝜃.
Work through opening exercise, example 1, and exercise 1. Discuss.
If ahead of schedule, work through more of the lesson.
25
min
22.
Supplies: sprints A and B
A sprint is another fluency exercise that can help to bridge gaps in
knowledge and increase automaticity of a skill. They are fast-paced and
don’t take up too much class time.
Students worked with radicals to some extent in G9 and G10. They also
covered them in Lesson 9 of this module. But this is still an area where
there could be potential gaps in content knowledge. This might be a
good place to do a rapid white board exchange as we saw earlier or a
sprint. We are going to do a sprint now.
Conduct sprint.
5 min
23.
Note that we are still working exclusively in degrees at this point.
We use the rotations that are multiples of 30, 45, 60 or 90 degrees.
Make a point that we have learned enough values of sine and cosine to
enable us to construct their graphs.
The first 5 key features are the same ways in which we described
polynomial graphs. Now we have periodic functions. We need ways to
describe the repetition of the graphs.
Note that these two terms (amplitude and period) are only being
INTRODUCED in this lesson. Mastery of the definitions or meaning is
not expected at this point.
5 min
24.
Students have worked with degrees for many years. They are
comfortable with degrees and probably see no reason to switch.
The graph might not be compelling to students. After all, we know that
we can rescale the horizontal axis. There is no reason we must use a
square grid. This is not the real reason that we are switching from
degrees to radians.
5 min
25.
Supply: graphing calculator
If participants have graphing calculators, ask them to find the limit first
with the calculator in degrees and then in radians. If they do not have
g.c., demonstrate using demos.
10
min
26.
Work exit ticket. Then, talk through the lesson summary. Point out the
approach taken in the lesson on converting between degrees and
radians (not relying on formulas but rather thinking of fractions of a
turn)
5 min
27.
If ahead of schedule, look at problem set # 6 – 10
5 min
28.
10
min
29.
Discuss the identities. Complete problem set #3, 4, 7, 8.
2 min
30.
5 min. Total time elapsed: 200 minutes + 20 minutes
discussion/questions/break = 220 minutes or 3 hours and 40 m
(Go through each point listed.)
Opening the module with circle-ometry and leading into trigonometry
makes the content more relatable to students.
Students often come away from trig with no concept of what it actually
is (i.e. thinking it is ok to write sin = 1 or that sinx/x = sin or that xsinx
= sin(x^2))
2 min
31.
Section: Mid-Module Assessment
Time: 35 min
In this section, you will allow Participants to complete a MidModule Assessment and follow up discussion.
Materials used include:
 “Grade 11 Module 2” PPT
 “Grade 11 Module 2” Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
25 min
32.
Have participants locate the assessment. Give them approximately 25 min
to take the assessment with their partner. After 20 minutes have passed
give a verbal warning for them to scan any remaining questions that they
have not yet attempted. If everyone finishes early, stop this part and start
the next portion of this session.
8 min
33.
Again, work with a partner to examine your work against the rubric and
exemplar. If you have any questions or concerns, jot them down on a post-it
note and we will address those before we move on.
After 10 minutes or so have passed, call the group together and address any
questions or concerns that participants noted on their post-it notes.
GROUP
2 min
34.
Section: Topic B: Understanding Trigonometric Functions Time: 133 min
and Putting Them to Use
In this section, you will explore how trigonometric functions can
model periodic behavior and the concept of polynomial identities.
Materials used include:
 “Grade 11 Module 2” PPT
 “Grade 11 Module 2” Facilitators Guide
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
5 min
Read topic opener for topic B
35.
GROUP
5 min
36.
Keep in mind that students studied these transformations extensively
in 9th grade. Where they may run into trouble is working with the
scaling on the horizontal axis.
15
min
37.
Why would the frequency be an important characteristic? Frequency is
not “part of the wave” like the amplitude, period, etc. In Physics it
refers to the number of waves per some time interval (often cycles per
second or Hertz).
When the period is decreased, the frequency increases (more waves
per interval)
Work exercise c and d. For (c), discuss the effect of the negative. Could
we rewrite the function using an identity from topic A ? [sin(-x) = sinx]
If ahead of schedule, work problem set extension problems # 4 and 5.
20
min
38.
Supply: graphing calculator
Give participants time to work through exercises 1 – 5 (exploratory
challenge)
Discuss the questions from the slide.
If ahead of schedule, work the exit ticket.
25
min
39.
Supply: graphing calculator
Students look at a variety of periodic phenomena and model the data
both with and without technology.
This graph shows the tides at Montauk, NY for the week of May 21–28.
Discuss this graph as a group.
What is the period? Why does it mean in terms of the tides?
What is the phase shift? Is that the only possible phase shift? Could we
write the function as a cosine function? Sure, but generally sine is used
to model data. Note that the graphing calculator only has sine
regression.
Write the equation. Note that students are most likely to make a
mistake with “w.”
Have them work through Exercises 2 – 6. Have the data saved on the
graphing calculator and transfer to their calculators to demonstrate
how this might save time in class.
Discuss the activity.
15
min
40.
Go through each point. In exploring the graph, students realize that
tangent is also periodic. What is the period of the tangent function?
Work on Exploratory challenge 2 (exercises 6 – 16).
Discuss.
5 min
41.
Click to advance.
Where do these 3 identities come from? The period of each function
Click to advance.
Where do these 3 identities come from? Sine is an odd function, cosine
is even, tan(-𝜃)=sin(-𝜃)/cos(-𝜃)
Click to advance.
10
min
42.
Supply: graphing calculator
Go through each point.
Look at exercise 3. One way to prove or determine whether an
equation is an identity is to graph each side. Demonstrate with the g.
calc. or with Desmos.
10
min
43.
Squaring both sides is not guaranteed to preserve equality. It is an
irreversible step that potentially alters a solution set.
Work through example 1 and if ahead of schedule problem set 2.
Students should never try to establish an identity by starting with that
identity. That is the mathematical equivalent of the statement “If I am
the Queen of England, then I am the Queen of England.” That statement
does not actually mean that I am the Queen of England.
15
min
44.
Students spend the first part of grade 12 module 1 looking at the topic
of linearity. They spend the first few lessons exploring the notion of
“Does linearity hold?”
Unfortunately it does not and we realize that sin(x+y) sin(x) + sin(y).
Work examples and discuss. Then work problem set #1 if time permits.
2 min
45.
5 min. Total time elapsed: 395 minutes +15 minutes
discussion/questions/break = 410 minutes or 6 hours and 50 m
(Go through each point listed.)
4 min
46.
Leave this slide blank.
Ask participants to list key points or ideas from module 1.
Ex. Take a moment now to re-read the standards that this module
covers…
Can you think back to moments in the lessons that get students to
arrive at those understandings? What things stand out to you now that
did not stand out early on?
2 min
47.
Section: End of Module Assessment
Time: 44 min
In this section, you will allow Participants to complete an End of
Module Assessment and follow up discussion.
Materials used include:
 “Grade 11 Module 2” PPT
 “Grade 11 Module 2” Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
25 min
Have participants locate the assessment. Give them approximately 25 min
to take the assessment with their partner. After 20 minutes have passed
give a verbal warning for them to scan any remaining questions that they
have not yet attempted. If everyone finishes early, stop this part and start
the next portion of this session.
48.
GROUP
8 min
49.
Again, work with a partner to examine your work against the rubric and
exemplar. If you have any questions or concerns, jot them down on a post-it
note and we will address those before we move on.
After 10 minutes or so have passed, call the group together and address any
questions or concerns that participants noted on their post-it notes.
6 min
50.
(Review each key point one at a time.)
5 min
51.
10 min
Total time elapsed: 475 minutes or 7 hours and 55 m
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? (pause
while participants reflect then click to advance to the next question). 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
●
Algebra II Module 2 PPT
●
Algebra II Module 2 Facilitator’s Guide
Additional Suggested Resources
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●
●
How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview
A Story of Functions CCLS Checklist
Active learning
Turn and talk