Module Focus: Grade 6 – Module 3
Sequence of Sessions
Overarching Objectives of this November 2013 Network Team Institute
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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.
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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 6 curriculum, A Story of Ratios.
Key Points
• Directed measurement --- a rational number’s position on the number line is found using length and direction.
• The opposite of a number a, is –a. Both a and –a are located an equal distance from zero, in opposite directions.
• Rational numbers represent real-world situations. We can write and explain statements of order for rational numbers in realworld contexts.
• The absolute value of a number is its distance from zero; and can be used in the context of a situation to show magnitude. We can
use absolute value and the symmetry of the coordinate plane to solve problems related to distance.
Session Outcomes
What do we want participants to be able to do as a result of this
session?
How will we know that they are able to do this?
 Understand the sequence of mathematical concepts within this module.

Participants will be able to articulate the key points listed above.
 Articulate and model the instructional approaches that support
implementation of this module (both as classroom teachers and school
leaders), including an understanding of how this instruction exemplifies the
shifts called for by the CCLS.
 Articulate the 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.
 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.
Session Overview
Section
Introduction to the
Module
Concept Development
Time
Overview
Introduction to mathematical
models and instructional
strategies to support
implementation of A Story of
Ratios.
Examination of the development
of mathematical understanding
across the module using a focus on
Concept Development withint he
lessons.
Prepared Resources
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
Grade 6 – Module 3
Grade 6 – Module 3 PPT
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Grade 6 – Module 3
Grade 6 – Module 3 PPT
Grade 6 – Module 3
Lesson Notes
Facilitator Preparation
Review Grade 6 Module 3
Overview, Topic Openers, and
Assessments.
Ensure that participants have
access to grid paper, ruler, and
compass.
Module Review
Articulate the key points of this
session.
Session Roadmap
Section: Introduction to the Module
Time: 27 minutes
[27 minutes] In this section, you will…
Introduce the mathematical focus of the module and how it fits
within the sequence of A Story of Ratios.
Materials used include:
Time Slide # Slide #/ Pic of Slide
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Script/ Activity directions
NOTE THAT THIS SESSION IS DESIGNED TO BE 270 MINUTES IN LENGTH
(time includes a fifteen minute break).
Welcome! In this module focus session, we will examine Grade 6 – Module 3.
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Our objectives for this session are:
•Examination of the development of mathematical understanding across the
module using a focus on Concept Development within the lessons.
•Introduction to mathematical models and instructional strategies to
support implementation of A Story of Ratios.
GROUP
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We will begin by exploring the module overview to understand the purpose
of this module. Then we will dig in to the math of the module. We’ll lead you
through the teaching sequence, one concept at a time. Along the way, we’ll
also examine the other lesson components and how they function in
collaboration with the concept development. Finally, we’ll take a look back
at the module, reflecting on all the parts as one cohesive whole.
Let’s get started with the module overview.
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The third module in Grade 6 is called Rational Numbers (click for red ring).
The module is allotted 25 instructional days. It challenges students to build
on understandings from previous modules by:
1)Extending the number line to include the negative numbers.
2)Partitioning the number line into intervals to locate and represent rational
numbers and their opposites.
3)Extending the coordinate plane to include all 4 quadrants and using its
symmetry to problem-solve.
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Locate “The Number System, 6-8” progressions document in your
supplemental materials. Turn to page 7 and take a few minutes to read
pages 7-8, starting paragraph on p. 7 under the heading: “Apply and extend
previous understandings of numbers to the system of rational numbers”.
(After 4 minutes) As you read this portion of the document, what were some
of your key take-aways? (Click twice after participants share their
thoughts.) Answers will vary. Participants may also state that a number’s
distance from zero is its absolute value, and that absolute value shows
magnitude. They may also comment on the ordering and comparing of
rational numbers as they relate to the context of a situation.
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Turn to the Module Overview document. Our session today will provide an
overview of these topics, with a focus on the conceptual understandings and
an in-depth look at select lessons and the models and representations used
in those lessons.
Take a moment to look at the table of contents at the beginning of the
Module Overview. Notice the Module is broken into three topics which span
19 lessons. Following the Table of Contents is the narrative section. Focus
and Foundational standards, as well as the standards for Mathematical
Practice are listed in this overview document as well
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Let’s start by looking at the vocabulary, tools and representations that are
used in this Module. They are located near the end of the Module Overview
document. I will give you a minute to read the vocabulary, tools, and
representations lists.
Name some of the new vocabulary terms. (Absolute Value, Charge, Credit,
Debit, Deposit, Elevation, Integers, Magnitude, Negative Number, Opposites,
Positive Number, Withdrawal)
What representations will be used in this Module? (Horizontal and Vertical
number lines, Coordinate Plane)
How is this information useful? How is it useful for you in your role? (This
information is useful in planning for word walls, concept-related posters,
parent newsletters, interdisciplinary projects, and other ways in which math
vocabulary and representations are reinforced in classrooms across the
grade level, and at home. )
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The Module is broken into three Topics which span 19 lessons. Our session
today will provide an overview of these topics, with a focus on the
conceptual understandings along with an in-depth look at select lessons and
the terminology and representations used in those lessons.
Section: Concept Development- Topic A
Time: 38 minutes
[38 minutes] In this section, you will…
Materials used include:
Examine the conceptual understandings that are built in Grade 6
Module 3, Topic A
Grid Paper, Ruler, Compass
Time Slide Slide #/ Pic of Slide
#
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Script/ Activity directions
Turn to Topic Opener A in your materials. Read the Topic
Opener to yourself.
What conceptual understandings and mathematical
representations are included in Lessons 1- 6? (Positive and
negative numbers as opposites with zero being its own
opposite, Rational numbers as representations of real-world
quantities, Rational numbers as points on a number line).
What is your previous experience with this material? Take a
moment to discuss with table members the ways in which
students develop a deep understanding of positive and
negative numbers.
GROUP
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Let’s take a closer look at the development of key understandings in
Topic A.
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The first lesson in the Module is perhaps the most important lesson in
the Module. I have provided you with a full-length copy of this lesson.
Students extend the number line to include negative numbers, and in
doing so, understand that each number has an opposite (and that zero
is its own opposite). This fundamental understanding of opposite
direction and value on the number line is the foundation for students’
work throughout the Module. Take 5 minutes to read through the
lesson.
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So what are some key understandings in Lesson 1? Students learn
that (Click to advance 4 times). Let’s complete a student activity from
Lesson 1 that provides students with this understanding.
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Directions:
•Draw a horizontal line. Place a point on the line and label it 0.
•Use a compass to locate and label the next point 1, thus creating a
scale. (Continue to locate other whole numbers to the right of zero
using the same unit measure.)
•Using the same process, locate the opposite of each number on the
left side of zero. Label the first point to the left of zero, -1.
State the numbers on your number line in order from left to right. (-
5,-4,-3,-2,-1,0,1,2,3,4,5)
Materials: Grid Paper, Ruler, Compass
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Following Lesson 1 is a 2-day lesson where students relate integers to
the real world. Let’s take a moment to read the student outcomes.
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Notice the exercises in Lessons 2 and 3 require students to: (read the
outcomes listed on this slide aloud).
Follow up with these verbal directions to participants:
Complete Lesson 3’s Exit Ticket, which can be found in your
additional materials. As you answer the questions, make note of the
understandings that are necessary for students to complete this
exercise. (Advance to next slide to review sample student work for
the Exit Ticket.)
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Take a moment to look at the sample solutions (click to reveal). Are
there any questions?
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Students build on their understanding of opposites in Lesson 5 by
finding the opposite of a number’s opposite, and come to the
realization that it will in fact be the number itself. You will recall this
important conceptual understanding that was cited on page 7 of the
progression document. Let’s take a closer look at an example and
activity from Lesson 5.
Examination of key examples and exercises from each lesson will
include opportunities for participants to engage in the lessons as
students and debrief as teachers.
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(Allow participants 1 minute to complete Lesson 5, Example 1 in their
additional materials.)
The language related to Lesson 5 can be confusing for students and
adults, as we start referring to the opposite of an opposite. Students
will build a deep understanding through number line models and
real-world examples. (Click to show answers.)
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Notice the notation used to indicate an opposite, and the opposite of
an opposite. To help build a conceptual understanding, students
should relate the mathematical representation to a real world
example.
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In Lesson 6, students understand that the process used in Lesson 1
can be used to partition the number line to find any rational number
and its opposite. For positive rational numbers a and b,
•the unit fraction �/� is located on the number line by dividing the
segment between � and � into � segments of equal length. One of the
� segments has � as its left endpoint; the right endpoint of this
segment corresponds to the unit fraction �/� .
•�/� is located on the number line by joining � segments of length
�/� so that 1) the left endpoint of the first segment is �, and 2) the
right endpoint of each segment is the left endpoint of the next
segment. The right endpoint of the last segment corresponds to the
fraction �/�.
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Complete Lesson 6 – Exercise 1, located in your additional materials.
(Advance to the next slide for the answers.)
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Students will relate this partitioning to experiences in earlier grades
when representing whole numbers and fractions (2.MD.6 and 5.NF.7).
Now complete the Exit Ticket. (Allow 3 minutes then advance the
slide.)
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Compare your answers with your table members. Refer to the sample
student responses at the end of Lesson 6, following the Exit Ticket.
Section: Concept Development- Topic B
Time: 54 minutes
[54 minutes] In this section, you will…
Materials used include:
Examine the conceptual understandings that are built in Grade 6
Module 3, Topic B.
Time Slide # Slide #/ Pic of Slide
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Script/ Activity directions
Let’s take a closer look at the development of key understandings in Topic B.
GROUP
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Turn to Topic Opener B in your materials. Read the Topic Opener to yourself.
What material is included in Lessons 7 - 13? (Ordering and Comparing
Rational Numbers, Writing and Interpreting Inequality Statements, Absolute
Value as Magnitude and Distance).
What is your previous experience with this material? Take a moment to
discuss with table members the ways in which students develop a
conceptual understanding of the ordering of rational numbers and absolute
value.
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Lessons 7 and 8 comprise a 2-day lesson on Ordering Integers and Other
Rational Numbers, with the following student outcomes:
•Students write, interpret, and explain statements of order for rational
numbers in real world contexts.
•Students recognize that if a < b, then -a > -b, because a number and its
opposite are equal distances from zero; and moving along the horizontal
number line to the right means the numbers are increasing.
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We are going to play few rounds of a game that is played in Lesson 7.
Students use their teacher’s verbal clues to arrive at the values of 2 numbers,
and determine which is larger, based on their relative positioning on the
number line. You may use the number line in your (Lesson 7) additional
materials to help you as I read the clues. (Click to verify each answer, after
reading each clue and allowing for a participant to first state the answer.)
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Lesson 7-8 is a 2-day lesson on Ordering Integers and Other Rational
Numbers. Spend the next 6 minutes completing the Exercises 1-3 and the
Problem Set from Lesson 8, which are listed in your additional materials.
When your table is finished, discuss your answers.
(After 9 minutes) Question #4 in the problem set relates to a student
outcome for this lesson. Students should come to understand that if a < b,
then –a > -b.
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I am going to administer a fluency-building exercise from Lesson 10. You
will want to turn to the Lesson 10 section in your additional materials. You
will have 1 minute to complete side A. On your mark, get set, go! (After one
minute is up, review the answers, and acknowledge those who complete the
most questions correctly. Then conduct the exercise for side B in the same
fashion.)
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Lesson 11 starts with students finding two rational numbers that are the
same distance from zero. They conclude that the numbers must be
opposites, and therefore any pair of opposites have the same distance from
zero, or the same Absolute Value.
Complete the Lesson 11 Exercises listed in your additional materials.
Absolute value is a new and abstract concept for students. How will they
conceptualize it? When you are finished completing the Exercises, discuss
them with table members.
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Following Topics A and B is the Mid-Module Assessment. There are 5
questions; but you will complete just one right now – question 3. It is listed
in your additional materials. Take 5 minutes to complete the question.
(After 5 minutes:) Spend the next 2 minutes sharing your completed task
with table members. Following your share-out session, take a minute to
refer to the rubric and sample student work for this question, provided in
the Module’s materials
(After 3 minutes:)
Which standards relate to this question (6.NS.C.5, 6.NS.C.6, 6.NS.C.7)? Which
Mathematical Practices are embodied in the task? (MP.2, MP.3, MP.6) What
are the challenges students might face as they complete this task? How does
students’ work in Topics A and B’s lessons prepare them for this task?
.
Section: Concept Development- Topic C
Time: 1 hours, 54 minutes
[1 hour 54 minutes] In this section, you will…
Materials used include:
Examine the conceptual understandings that are built in Grade 6
Module 3, Topic C.
Time Slide # Slide #/ Pic of Slide
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Script/ Activity directions
Let’s take a closer look at the development of key understandings in Topic C.
GROUP
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Turn to Topic Opener C in your materials. Read the Topic Opener to yourself.
What material is included in Lessons 14 - 19? (Locating points on the
Coordinate Plane, Distance on the Coordinate Plane, Symmetry in the Plane,
Problem-Solving using the Coordinate Plane). How do Topics A and B
prepare students for Topic C? (Students’ understanding of absolute value as
distance, how to locate a number’s opposite, and the ordering of rational
numbers on both vertical and horizontal number lines will serve as a
foundation.)
Note that students’ prior understanding of the Coordinate Plane is limited to
the first quadrant, where both coordinates are positive. Students will use
Absolute Value and the plane’s symmetry to find distances on vertical and
horizontal lines. They will also make connections between the sign of the
coordinates and the quadrant in which the point lies.
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You’re the Expert!
Go to the appropriate table as designated by the number on your card. Each of
you will have fifteen minutes to discuss the lesson assigned with your group.
You will complete a lesson study, you will model an essential portion or
portions of the lesson to the entire group, and you will address concerns and
scaffolding that isn’t necessarily addressed within the teacher’s edition of the
lesson. During your presentation, please be sure to mention the lesson
progression you see within grade level and between grade levels. Note the
foundational skills and prerequisites you may deem necessary to address
prior to the administration of this lesson.
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Let’s take a look at two questions from the End-of-Module Assessment:
questions 1 and 5. Question 1 relates to Topic A, and question 5 relates to
Topic C. Count off at your table 1,2,1,2... , so that half of you will complete
question 1, and the other half will complete question 5.
Locate the questions in your additional materials and complete your assigned
question independently. When everyone at your table is finished, the 1’s
should discuss their question (question 1), and the 2’s should discuss their
question (question 5)*.
(*Remind participants that they may spend time outside of this session
completing the entire assessment and analyzing the sample student
responses and rubric.)
(After 6 minutes)
What is the progression from Topic A through Topic C? We have come fullcircle back to where we began today’s session: the progressions document. I
encourage you to read the document again outside of this session, as well as
the Module Overview, Topic Openers, and Assessments, so that you are
comfortable with the material presented in this Module.
Section: Module Review
Time: 11 minutes
[11 minutes] In this section, you will…
Materials used include:
Faciliate as participants articulate the key points of this session and
clarify as needed.
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
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Let’s take a few minutes to reflect on the purpose of today’s session.
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Take two minutes to turn and talk with others at your table. During this
session, what information was particularly helpful and/or insightful? What
new questions do you have?
Allow 3 minutes for participants to turn and talk. Bring the group to order and
advance to the next slide.
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Let’s review some key points of this session. I would like each tables’
members to take one minute to write down a key point from today’s session.
I will then call on each table to share out. (Click to advance and show key
points.)
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Active learning
Turn and talk
Turnkey Materials Provided
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Grade 6 Module 3 PPT
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Grade 6 Module 3 Lesson Notes
Additional Suggested Resources
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A Story of Ratios Curriculum Overview
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CCSS Progressions Document: The Number System (6-8)