Module Focus: Grade 9 – Module 5
Sequence of Sessions
Overarching Objectives of this May 2014 Network Team Institute

Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate
how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom
teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding
how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the
mid-module assessment and end-of-module assessment.
High-Level Purpose of this Session
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Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the first half of the lessons.
Standards alignment and focus: 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.
Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.
Related Learning Experiences
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This session is part of a sequence of Module Focus sessions examining the Grade 10 curriculum, A Story of Functions.
Key Points Module 5 Lessons
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Reinforce the notion that we are modeling with functions –
•
Push students to recognize when they are or are not given enough information to be sure that the function will be an
accurate representation of the data or information described.
•
Use precision when speaking and encourage students to do the same by questioning them for accuracy and precision.
•
Remain aware of the discrete vs. continuous domain in the context of the situation.
Approach Lesson Summaries as opportunities for students to consolidate their knowledge – ask & scaffold, don’t tell.
Key Points End-of-Module Assessment

Shorter modules, like G9-M5 do not have Mid-Module Assessments, only an End-of-Module Assessment designed to assess all
standards of the module.

Recall, as much as possible, assessment items are designed to assess the standards while emulating PARCC Type 2 and Type 3 tasks.

Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the
assignment of grades.
Key Points Topic A

The lessons in Topic A serve to consolidate student knowledge about linear, exponential and quadratic functions.
Session Outcomes
What do we want participants to be able to do as a result of this
session?

Participants will develop a deeper understanding of the sequence of
mathematical concepts within the specified modules and will be able to
articulate how these modules contribute to the accomplishment of the
major work of the grade.

Participants will be able to articulate and model the instructional
approaches that support implementation of specified modules (both as
classroom teachers and school leaders), including an understanding of
how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of
the specified module and content of grades above and below,
understanding how the mathematical concepts that develop in the
modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that
prepare students to express reasoning and/or conduct modeling required on
the mid-module assessment and end-of-module assessment.
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
Introduction
36 min
Introduces Grade 9 Module 5
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Topic A, Topic B, and
End-of-Module
Assessment
255-295
min
Explores Grade 9 Module 5
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Facilitator Preparation
Grade 9 Module 5 PPT
Grade 9 Module 5
Facilitator Guide
Review Grade 9 Module 5
Grade 9 Module 5 PPT
Grade 9 Module 5
Facilitator Guide
Review Topic A
Review Topic B
Session Roadmap
Section: Introduction
Time: 36 minutes
In this section, you will be introduced to the concepts and key
points of Grade 9 Module 5.
Materials used include:
● Grade 9 Module 5 PPT
● Grade 9 Module 5 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
3 min
(Note that it will be important to monitor how much time is being spent on
each slide of exercises. As the presenter, work through the session in
advance making note of how long you have to present this session and how
long you are estimating for each slide; then make note of exercises you may
want to spend less time on so you can get through the module on pace.)
1.
GROUP
5 min
2.
Give participants 5 min to do the opening exercise.
1 min
3.
During this session you will be actively engaged in unpacking the content of
Grade 9 Module 5. You will be asked to interact with the materials from
both the student’s and teacher’s perspective at various times during the
session to deeply understand the content of the module.
We will revisit the opening exercise shortly. First let’s get to know each
other a bit.
2 min
4.
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
5.
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 5 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
6.
Here is our agenda for the day. If needed, we will start with orienting
ourselves to what the materials consist of. This module is a synthesis of
students’ capacity to use what they know about linear, exponential and
quadratic functions to model data, descriptions, or graphs with an explicit
formulas of the function they judge to be the best choice. There is no new
content here, rather there are a variety of application problems for students
to develop their capacity to model with functions. Thus we will spend our
time examining and experiencing a sampling of these examples, exercises
and discussions from the lessons in this module, beginning in Topic A, then
moving to Topic B and the end of module assessment.
(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).
4 min
7.
(Not accounted for in the timing – these slides are optional if participants
are new to the materials.)
Each module will be delivered in 3 main files per module. The teacher
materials, the student materials and a pack of copy ready materials.
Teacher materials include a module overview, and topic overviews, along
with daily lessons and a mid- and end-of-module assessment. (Note that
shorter modules of 20 days or less do not include a mid-module
assessment.)
Student materials are simply a package of daily lessons. Each daily lesson
includes any materials the student needs for the classroom exercises and
examples as well as a problem set that the teacher can select from for
homework assignments.
The copy ready materials are a single file that one can easily pull from to
make the necessary copies for the day of items like exit tickets, or fluency
worksheets that wouldn’t be fitting to give the students ahead of time, as
well as the assessments.
4 min
8.
(Not accounted for in the timing – these slides are optional if participants
are new to the materials.)
There are 4 general types of lessons in the 6-12 curriculum. There is no set
formula for how many of each lesson type we included, we always use
whichever type we feel is most appropriate for the content of the lesson.
The types are merely a way of communicating to the teacher, what to expect
from this lesson – nothing more. There are not rules or restrictions about
what we put in a lesson based on the types, we’re just communicating a
basic idea about the structure of the lesson.
Problem Set Lesson – Teacher and students work through a sequence of 4 to
7 examples and exercises to develop or reinforce a concept. Mostly teacher
directed. Students work on exercises individually or in pairs in short time
periods. The majority of time is spent alternating between the teacher
working through examples with the students and the students completing
exercises.
Exploration Lesson – Students are given 20 – 30 minutes to work
independently or in small groups on one or more exploratory challenges
followed by a debrief. This is typically a challenging problem or question
that requires students to collaborate (in pairs or groups) but can be done
individually. The lesson would normally conclude with a class discussion on
the problem to draw conclusions and consolidate understandings.
Socratic Lesson – Teacher leads students in a conversation with the aim of
developing a specific concept or proof. This lesson type is useful when
conveying ideas that students cannot learn/discover on their own. The
teacher asks guiding questions to make their point and engage students.
Modeling Cycle Lesson --Students are involved in practicing all or part of the
modeling cycle (see p. 62 of the CCLS, or 72 of the CCSSM). The problem
students are working on is either a real-world or mathematical problem that
could be described as an ill-defined task, that is, students will have to make
some assumptions and document those assumptions as they work on the
problem. Students are likely to work in groups on these types of problems,
but teachers may want students to work for a period of time individually
before collaborating with others.
5 min
9.
(Not accounted for in the timing – these slides are optional if participants
are new to the materials.)
Follow along with a lesson from the materials in your packet.
The teacher materials of each lesson all begin with the designation of the
lesson type, lesson name, and then 1 or more student outcomes. Lesson
notes are provided when appropriate, just after the student outcomes.
Classwork includes general guidance for leading students through the
various examples, exercises, or explorations of the day, along with
important discussion questions, each of which are designated by a solid
square bullet. Anticipated student responses are included when relevant –
these responses are below the questions; they use an empty square bullet
and are italicized. Snapshots of the student materials are provided
throughout the lesson along with solutions or expected responses. The snap
shots appear in a box and are bold in font. Most lessons include a closing of
some kind – typically a short discussion. Virtually every lesson includes a
lesson ticket and a problem set.
What you won’t see is a standard associated with each lesson. Standards are
identified at the topic level, and often times are covered in more than one
topic or even more than one module… the curriculum is designed to make
coherent connections between standards, rather than following the notion
that the standards are a checklist of items to cover.
Student materials for each lesson are broken into two sections, the
classwork, which allows space for the student to work right there in the
materials, and the problem set which does not include space – those are
intended to be done on a separate sheet so they can be turned in. Some
lessons also include a lesson summary that may serve to remind students of
a definition or concept from the lesson.
3 min
10.
We’re ready to begin examination of Topic A. First let’s read the standards
we’ll be assessing in this module. (Give participants time to read the
standards listed on pages 3-4 of the teacher materials.)
Now let’s go through an overview of the flow of the module as it addresses
those standards.
2 min
11.
(Go through the bullets to give an overview of the progression or flow of
each topic and the module as a whole.)
3 min
12.
(Review the bullet points with participants to remind them of the
background students are coming in to this module with.)
Section: Topic A, Topic B, and End-of-Module Assessment Time: 255-295 minutes
In this section, you will explore the key points of Grade 9 Module 5
in-depth.
Time Slide # Slide #/ Pic of Slide
Materials used include:
● Grade 9 Module 5 PPT
● Grade 9 Module 5 Facilitator Guide
Script/ Activity directions
GROUP
20-30
min
13.
(Direct participants to use their student materials to do the work; reflecting
back on the accompanying teacher materials after they’ve finished the
selected problems.)
Examine the graph given in Example 1 and complete Exercises 1. (Allow
time for participants to work.)
What might students say if you ask them, what is the advantage or purpose
of writing an equation to go with the graph? (Key responses: the capacity
to more precisely find the function value for x-values where the graph does
not fall nicely on a grid mark; the capacity to find function values beyond the
range of what is visible on the graph. Encourage participants to challenge
students as to their confidence that the function equation is an exact model
of the function or merely a good model, and as such, how confident are they
that their calculations are exact? The answer will depend on what they
know about the situation being depicted by the graph.)
Complete Exercises 2-6. (Allow time.)
When you are finished compare your results with a neighbor.
How well do you think your students will do with these exercises? Are there
any that you think they will struggle with? (Discuss as needed.)
What questions might you ask to sum up their in-class experience? I might
ask students at the end of this lesson what surprised them about their work
today.
Complete Problem Set exercise 1
Are we sure that this u-shaped curve can be represented exactly by a
quadratic function?
How do we know? (Using the points that appear on the graph to be (0,0) (2,
50) (4, 80) and (5, 90) there is evidence of constant second differences over
equal intervals, suggesting a quadratic function.)
Complete Problem Set exercise 2
(Discuss the reasonableness of the graph given the description of the
context.) Note that to confirm data is following a square root function, one
would look for constant second differences of x-values for equally spaced
intervals of y-values. (Repeatedly encourage teachers to be clear with
students when they do or do not have enough information to say that an
explicit formula is an exact representation of the relationship vs. a good
model of the relationship.)
10-20
min
14.
Complete the Opening exercise of Lesson 2.
(The aim is for participants to recognize that over equal x-intervals, the first
table of values shows a constant set of differences in the y-values; the
second table shows a constant set of second differences of the y-values, and
the third data set shows a constant ration between consecutive y-values.)
Complete Exercise 2 (Allow time and then ask for choral response for what
they answered for each table of values.)
Complete Problem Set problem 4. (There are likely to be some mistakes on
this one as it takes careful consideration; consider having participants
compare their answer with a neighbor and then go over as a group.)
15 min
15.
Complete Exercises 1-4, then compare with a neighbor.
(Host a discussion of how they think their students will do on this task and
where they think students might struggle.)
6 min
16.
So you can see how the lessons in Topic A are already consolidating student
knowledge about linear, exponential and quadratic functions as well as
developing fluency in pulling out that knowledge in relationship to only a
description about a functional relationship, or only a graph, or only data.
Any comments or questions before we move in to Topic B?
3 min
17.
Great, now we are ready for an examination of Topic B: Completing the
Modeling Cycle. Before we begin look at the Opening of Lesson 4 and notice
the modeling cycle graphic provided to students. This is a direct excerpt
from the CCSM standards. I encourage you to find one or more times
throughout this module to reflect on this cycle with students. Perhaps even
asking them to describe what the process of modeling has been like / is like
and then compare and contrast their reflections to this cycle and see what
students’ reaction is to this graphic.
30 min
18.
Take time now to work Example 1 of Lesson 4. (Allow time for participants
to work.)
You may find this example to be fairly challenging.
I find that a valuable approach is to example the relationship between a few
key points on the graph of the suspected parent function, the square root
function, and this graph. From the square root function we typically
consider the points (0,0), (1, 1) (4, 2) and (16, 4). On this graph we can
identify the points (0,0) (2, 8) and (8, 16). Let’s consider a correspondence
between the points, not making use of the (1,1) but making a
correspondence between the remaining 3 points on the graph of the square
root function and these 3 points on this graph.
The points (0,0) and (0,0) correspond exactly, suggesting to me that we
won’t need to consider a translation, and observing that this graph is also in
the first quadrant I don’t think we need to consider a reflection. So that
leaves me to consider only scalings. Given that appear to be no translations,
if there were a vertical scaling the y-coordinates of one would be a constant
multiple of the y-coordinates in the other. Is this the case? (Yes, each yvalue is 4 times the value of the y-values from the parent function.) If there
were a horizontal scale, I would expect the x-values to be a constant
multiple, is this the case? (Yes, each x-value is one half the x-value from the
parent function.) So how would these two combined scalings be
represented in the equation of the function? (f(x) = 4 * square root of 2x).
Glance over Exercise 1 and then complete Exercise 2. (Allow time and then
review as a group.)
Look at the questions from the exit ticket for this lesson; they are shown on
the slide or you can also find them on page 67 of your student materials.
After having completed lessons 1-4, how do you think your students will
answer these questions? (Discuss.)
30 min
19.
Read the Opening Exercise from Lesson 5.
Are there functions that could model this data well? (Allow time to work
then share responses.)
This next discussion question is from the teacher materials page 75. If a
student has up to 30 minutes of actual gym time during a soccer practice,
how many exercises in the circuit would she be able to complete? (Allow
time to work then share responses; if needed suggest, as the teacher notes
do, to create a 3rd sequence by adding another column to the table and filling
in the ‘total circuit time’.)
So we can create a new function by adding two other functions?
What could we use as an explicit formula for the function that provides the
total time to complete x exercises?
Reflect again on the question – do we believe in patterns? Just because our
function produces exactly this finite set of data points, does that mean that
the function we created IS an exact model of the situation?
Complete problem 2 from the Problem Set and compare answers with a
neighbor. (Allow time.)
Complete problem 3 from the Problem Set and compare answers with a
neighbor. (Allow time.)
Complete problem 4 from the Problem Set and compare answers with a
neighbor. (Allow time.)
(Host discussion as needed.)
10 min
20.
Do the exercise itself then critique the problem.
(This is the opening exercise for this session. If needed, give participants
more time to do the exercise and critique it. Ask for input; point out that the
question implies that one can determine what THE representation of THE
function is. In reality, all we have here is a table of values without context or
any additional information. Just as we learned in G9-M3, without more
information, we have no idea if the functions we create would be a good
model for the data other than the fact that it ought to do a good job of
modeling the five points shown.)
20-30
min
21.
(At your discretion have participants work or merely glance through Lesson
6 Example 1 and Exercises 1-3. Have participants compare or discuss an
appropriate choice of model for each situation; host discussion as needed.)
Notice how explicit page 85 of the teacher materials makes the process of
looking for constant second differences.
Now work the Exit Ticket of Lesson 6 found in the teacher materials on page
88. (Allow time.)
How likely do you think your students are to recognize this as quadratic
without the analysis of second differences?
Lesson 7 is designed to use the regression analysis of a TI-84 to find
equations that model data.
Look over Lesson 7 Exercise 2. The emphasis in these exercises is not only
the use of the technology, but with technology we are increasingly
empowered to do modeling with more real-world data, and thus a focus is
on interpreting the analysis of the data correctly within the context of the
problem.
Consider the exit ticket for this lesson on page 100 of the teacher notes. Part
c of this exit ticket relies on students experience in Module 2 of this year.
30-40
min
22.
(Consider allowing participants to review Lesson 8 Examples 1-2 and
Problem Set #1 to see if they are interested in working through any of those
as a group. Then have participants do Lesson 8 Problem Set #2 and
compare with a neighbor, discussing as a group as needed.)
Now turn to page 117 in the teacher materials and complete the Exit Ticket
for Lesson 9. Discuss your answers to each question with a neighbor.
Finally, let’s examine Problem Set #2 from Lesson 9. (Demonstrate the
solution to the problem and discuss as needed.)
20 min
23.
Review each key point one at a time.
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
24.
Finally let’s examine the end-of-module assessment for G9-M4.
25 min
25.
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.
20 min
26.
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 6 minutes or so have passed, call the group together and address any
questions or concerns that participants noted on their post-it notes.
4 min
27.
(Review each key point one at a time.)
10 min
28.
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
Turnkey Materials Provided
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Grade 9 Module 5 PPT
Grade 9 Module 5 Facilitator Guide
Reflect on a prompt
Active learning
Turn and talk
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