Download DUE 6-13: Facilitators Guide Template - CC 6-12

Module Focus: Grade 7 – Module 2
Sequence of Sessions
Overarching Objectives of this November 2013 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 7 curriculum, A Story of Ratios.
Key Points
• The additive inverse is the opposite of a number because, when added to a number, the sum is zero.
• We can use arrows on a number line to show the sum, p + q, on a number line. The sum is the distance|q|from p to the right of p if
p is positive, and to the left of p if p is negative.
• Subtracting a number is the same as adding its opposite.
• The properties of operations justify the rules for multiplication and division of integers. (-1)(-1) = 1 can be justified using the
distributive property and additive inverse.
• The opposite of a sum is the sum of its opposites. Ex.: –(-3 + 4) = 3 + (-4).
• Distance between 2 rational numbers p and q on a number line is |p - q|
• Every rational number can be written as a decimal that either terminate s in zeros or repeat.
Session Outcomes
What do we want participants to be able to do as a result of this
session?
 Understand the sequence of mathematical concepts within this module.
How will we know that they are able to do this?
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
Time
Overview
20 mins
Introduction to instructional focus
of Grade 7 Module 2 of A Story of
Ratios.


Grade 7 – Module 2
Grade 7 – Module 2 PPT
Concept Development 108 mins
Examination of the development of
mathematical understanding
across the module using a focus on
Concept Development within the
lessons.



Grade 7 – Module 2
Grade 7 – Module 2 PPT
Grade 7 – Module 2
Lesson Notes
Module Review
Articulate the key points of this
session.
Introduction to the
Module
8 mins
Prepared Resources
Facilitator Preparation
Review Grade 7 Module 2
Overview, Topic Openers, and
Assessments.
Session Roadmap
Section: Introduction to the Module
Time: 20 minutes
[20 minutes] In this section, you will…
Introduce the mathematical models and instructional strategies to
support implementation 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 135
MINUTES IN LENGTH.
Welcome! In this module focus session, we will examine
Grade 7 – Module 2.
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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.
•Introduction to mathematical models and instructional
strategies to support implementation of A Story of Ratios.
GROUP
1
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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 second module in Grade 7 is called Rational
Numbers (click for red ring). The module is allotted 30
instructional days. It challenges students to build on
understandings from previous modules by:
1)Extending operations with positive rational numbers
to operations with positive and negative rational
numbers.
2)Advancing the understanding of “opposite direction
and value” and a number and its opposite on the number
line to: a + (-a) = 0, the additive inverse property.
3)Representing the subtraction of two rational numbers
as the distance between the numbers on the number
line.
4)Interpreting the absolute value of p – q as the distance
from p to q, regardless of direction.
5)Applying the properties of operations to justify
operations involving positive and negative rational
numbers.
6)Using the partitioning and sub-partitioning of the
number line and place value to understand the division
of rational numbers, and terminating and repeating
decimals.
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Locate “The Number System, 6-8” progressions
document in your supplemental materials. Turn to page
9 and take a few minutes to read pages 9-13.
(After 5 minutes) As you read this portion of the
document, what were some of your key take-aways?
(Click three times after participants share their
thoughts.) Answers will vary. Participants may also
state that counters or chips may be used to represent
integers, however; the number line model must also be
used so that students eventually understand the location
of rational numbers on a number line, and addition of
numbers using a number line model. This prepares
them for work with vectors in high school.
Participants may also reference the use of place value
and partitioning of the number line to represent the
division of rational numbers and the quotients as either
terminating or repeating decimals.
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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 23 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. You will want to read the
entire document at your leisure, following today’s
session; and perhaps many of you have already done so
as you either prepare to teach the content in this
module, or are currently doing so.
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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. (Additive
Identity, Additive Inverse, Break-Even Point, Distance
Formula, Debit, Loss, Multiplicative Identity, Profit,
Repeating Decimal, Terminating Decimal).
What representations will be used in this Module?
(Equations, Expressions, Integer Game, Number Line, Tape
Diagram)
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 23
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: 48 minutes
[48 minutes] In this section, you will…
Examine the conceptual understandings that are built in Grade 7
Materials used include:
Module 2, Topic A.
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- 9? (Opposite
quantities combine to make zero, adding rational
numbers and representing addition on the number line,
subtracting rational numbers, the distance between 2
numbers on the number line, applying the properties of
operations to add and subtract rational numbers).
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
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Let’s take a closer look at the development of key
understandings in Topic A.
GROUP
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Let’s take a moment to review the student outcomes for
the first lesson in this module.
I want to emphasize that the Integer Game plays a
critical role throughout the module in helping students
form a conceptual understanding of integer operations
(which extend to all rational numbers). We will now
take a closer look at the role of the Integer Game in
Lesson 1.
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In Lesson 1, prior to playing the Integer Game, students
practice counting up and counting down on a number
line, in accordance with card values. You will recall from
the progression document that it is imperative that
students understand the number line representation of
the addition of rational numbers. Their understanding
of vector addition will develop as they move through the
lessons in Topic A.
Refer to this example in your additional materials.
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The Integer Game is a card game. Students play the
game initially in Lesson 1 for addition, but variations of
the game are played (and referred to) throughout the
lessons in the module so that students relate the number
line representations to a real-world model.
The way in which the Integer Game is used in lessons is
included as a separate document in the Module. A
template for cards is included in that document as well.
While a deck of store-bought playing cards may be used
instead, it is suggested that the cards from the template
are photocopied and used for several reasons.
1)The Integer Game cards show one integer on each
card. They do not associate a certain symbol or color
with the numbers, but playing cards do, which may
interfere with students conceptualization of positive and
negative numbers. If you were to use playing cards, you
would have to assign a color to denote positive numbers,
and a color to denote negative numbers. And the
negative numbers would not physically appear on the
cards.
2)Playing cards do not include “1” cards nor “0” cards.
Most likely it would be difficult for ELL students to
understand that say, a joker represents zero, or an ace
represents one, if you were to use playing cards. And
again, those numbers would not physically appear on the
cards.
So it is highly recommended that the cards be cut out
from the template and placed in baggies or rubberbanded for student groups to use in the classroom.
However, as we are adults with a firm understanding of
integers and playing cards, today we will be using
playing cards.
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Since for today we will use playing cards, we will
consider a joker to equal zero and an ace to have a value
of one. We will also consider black cards to be positive
values and red cards to be negative values.
Read the directions for game play which are listed under
Lesson 1 - Exercise 3, on page 2 of your additional
materials. Note the object of the game is to pick up and
discard cards so that your hand’s total value stays as
close to zero as possible. Play the game in groups of 2 to
4 players; keep in mind students will play the game with
a number line along side of them to count up or down if
need be. I will give you 5 minutes to play the game.
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In Lesson 2, students advance their number line
modeling of addition. They use directional arrows
instead of the “jumps” or “loops” initially used. Take a
moment to follow the directions for Example 1 (in your
additional materials) and complete the example with
your table members. Discuss the ways in which students
must attend to precision (MP.6).
(After 3 minutes, Click to reveal the answer.) Next, take 2
minutes to complete Example 3.
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(After allowing 2 minutes for completion, click to reveal
the answers.)
What are your thoughts on the modeling of addition in
this way? How will it benefit students in high school?
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In Lesson 3, students make sense of integer addition on
the number line by relating it to the context of the gameplay. They show the sum of 2 integers as the distance of
the absolute value of the second addend (q) from the
first addend (p), to the right if q is positive, and to the left
if q is negative.
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Spend 3 minutes completing the Exit Ticket for Lesson 3.
(After 1 minute click to reveal the answer key.)
As you can see, students make sense of integer addition
on the number line by relating it to the context of the
game-play.
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In Lesson 5, students recognize that subtracting a
number is the same as adding its additive inverse. This
slide shows a portion of an example from the lesson,
where students record their conclusions and make an
analogy to the Integer Game to justify the rules for
subtraction. They recognize that subtracting a number
is the same as adding it opposite, and do so by adding to
the minuend the additive inverse of the subtrahend.
1
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This slide shows a portion of an example from the
lesson, where students record their conclusions and
make an analogy to the Integer Game to justify the rules
for subtraction.
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We haven’t spent a lot of time talking about an important
component of each lesson: The Closing Questions. This
“student debrief” section is important as it allows
teachers to gauge whether or not students understood
the lesson content. While the exit ticket serves as an
additional formative assessment, the closing questions
may bring to light students misconceptions, or foster
further questions from students which can be addressed
during this time. If time allows, you may wish to pose a
closing question and allow elbow partners to discuss it
before returning to a whole group discussion.
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Spend 3 minutes completing the Exit Ticket, located in
your additional materials.
(Click to reveal the answer.)
Which mathematical practice(s) are embodied in this
task? (MP.3-Construct viable arguments and critique the
reasoning of others, MP.4-Model with mathematics,
MP.6-Attention to precision)
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The Problem Set for this lesson is a good example of how
we can design homework for students that not only
builds fluency, but reaches those higher levels of Blooms
Taxonomy when, through carefully structured problems
and questioning, students discover patterns and
structure.
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Students recognize that the rules for adding and
subtracting integers apply to all rational numbers. They
have been using these rules with positive rational
numbers since earlier grades; partitioning the number
line and creating area models to represent the addition
and subtraction of mixed numbers and fractions. The
opposites of these numbers are now included in addition
and subtraction problems.
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Lesson 8-9 is a 2-day lesson that applies the properties
of operations to find the sums and differences of rational
numbers. Students improve their fluency with rational
numbers and use the properties of operations to
efficiently add and subtract rational numbers without
the use of a calculator.
For those of you who are familiar with the Grade 6
standards:
How does the third outcome build on students’
understandings from Grade 6?
(In Grade 6, students spent time developing an
understanding of the opposite of a number, a, to be –(a).
It is advisable to continue to point out that a negative
sign in front of a sum or number, means the opposite of
that sum or number. (Click twice for an illustration.)
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Take a couple of minutes to complete Example 2 for
Lesson 8.
(After 2 minutes:) Click to show a sample student
response.
In addition to the Number System 6-8 Progressions
document that we looked at earlier today, I would
encourage you to read the 3-5 Progression on Numbers
and Operations - Fractions, to better understand
students’ earlier foundational work and
conceptualization relating to number line
representations.
Section: Concept Development- Topic B
Time: 24 minutes
[24 minutes] In this section, you will…
Examine the conceptual understandings that are built in Grade 7
Module 2, Topic B.
Materials used include:
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
GROUP
1
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Let’s take a closer look at the development of key
understandings in Topic B.
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Turn to Topic Opener B in your materials. Read the Topic
Opener to yourself.
What material is included in Lessons 10 - 16? (Multiplication
and Division of Integers which then extends to all Rational
Numbers, Converting Between Fractions and Decimals,
Converting Rational Numbers to Decimals Using Long
Division, Applying the Properties of Operations to
Multiplication and Division of Rational Numbers).
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.
2
29
In Lesson 10, students revisit the Integer Game using an
abundance of duplicate cards, to develop an understanding of
integer multiplication. Their foundational understanding of
multiplication as repeated addition from earlier grades,
provides them with an understanding of why, for instance,
5 x (-2) equals -10, (as referenced in the progressions
document). It is because 5 x (-2) means:
(-2) + (-2) + (-2) + (-2) + (-2) = -10. In playing the Integer
Game in this lesson, students quickly realize that discarding
duplicates or triplicates of a negative number makes their
score increase just as it would if they picked up duplicates or
triplicates of the positive value instead. Hence, students’
intuition tells them that the product of two negative numbers
is a positive number.
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In Lesson 11, students formalize their conceptual
understanding of the multiplication of signed numbers; as
indicated in the student outcomes for this lesson.
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In Example 1 of Lesson 11, students record examples of
picking up and discarding several identical cards in the
Integer Game from Lesson 10, using the patterns to arrive at
the rules for the multiplication of signed numbers.
Take 5 minutes to complete problems #1-5 of the Problem Set
for Lesson 11. (Advance to the next slide while waiting for
participants to finish).
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(Click to reveal sample student responses.) What do you think
about this type of homework exercise? We want to make sure
we include plenty of opportunities for students to build
fluency and reach the higher levels of Bloom’s Taxonomy.
Notice the different types of questions, and that students must
demonstrate an understanding through their writing as well.
2
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Students know how to represent decimals as fractions and
vice-versa using place value. This foundation serves them
well in Lesson 14. Students explore why some rational
numbers are decimals that terminate in zeros, while others
repeat. The lesson begins with a calculator exploration.
Example 2 shown here, illustrates how students come to
recognize that fractions whose denominators are products of
2’s and/or 5’s are equivalent to decimals that terminate;
which stems from our base-ten number system and the fact
that 2 x 5=10. After this initial exploration, students use long
division to convert rational numbers in fractional form, to
decimals.
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Following Topics A and B is the Mid-Module Assessment.
There are 7 questions; but you will complete just one right
now – Question 6. It is listed in your additional materials.
Take 3 minutes to complete the question.
(After 3 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 (7.NS.A.1c). Which
Mathematical Practices are embodied in the task? (MP.3, MP.4,
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? (Experience with the
Integer Game, number line representations, absolute value as
distance, subtraction as adding the inverse, etc… .)
.
Section: Concept Development- Section C
Time: 36 minutes
[36 minutes] In this section, you will…
Examine the conceptual understandings that are built in Grade 7
Module 2, Topic C.
Materials used include:
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
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Let’s take a closer look at the development of key understandings in the final
topic of the module, Topic C.
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Turn to Topic Opener C in your materials. Read the Topic Opener to yourself.
What material is included in Lessons 17 - 23? (Comparing Tape Diagram
Solutions to Algebraic Solutions, Generating, Writing and Evaluating
Equivalent Expressions, Expressions, Investments and Performing
Operations with Rational Numbers, If-Then Moves using Integer Cards,
Solving Equations using Algebra). How do Topics A and B prepare students
for Topic C? (Applying operations involving rational numbers,
understanding the rules for adding, subtracting, multiplying and dividing
signed numbers, converting rational numbers to decimals, knowing the long
division algorithm to find a repeating decimal.)
Note that students will begin to focus on an understanding of Algebra in
these lessons, but that focus will become much more concentrated and indepth in Module 3.
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Students begin Topic C by comparing tape diagram models to equation
models in Lesson 17. This image is in your additional materials; and I would
like to begin by having you look at it for a moment. Then take 2 minutes to
discuss, as a table, how this approach to algebra is different than, perhaps
how you are accustomed to teaching/presenting it to students.
It is important to note the parallel models. Students will relate their past
experience with a visual model to this new, abstract algebraic model; that
connection will allow them to assimilate the algebraic steps to the numeric
steps that are familiar to them.
Students will “make zeros” and “make ones” (using the additive inverse and
multiplicative inverse) to work backwards. That is emphasized in the steps,
as we show the zero and the “1t”. Students will eventually drop these “baby
steps” as their understanding is solidified.
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Take the next 5 minutes to complete the Exit Ticket for Lesson 17; pair up
with your elbow partner to do so. One of you solve it using a tape diagram,
and the other, an equation model. Be sure to show “making zeros” and
“making ones”. Discuss your representations with your partner. Then, for
question two, switch roles so that you have a chance to create the other
representation for this question.
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I want to jump ahead to Lesson 21, to reference the introduction of “If-Then”
statements into the math curriculum. You will find these properties of
equalities in the Glossary of the Common Core State Standards for
Mathematics; they are listed in Table 4. While students may or may not
already know these to be true, it is the first time they are being formally
addressed. We introduce the if-then statements less formally in Lesson 21,
as we relate them to the Integer Game. They are addressed more formally in
lessons 22 and 23, as students use them to justify their algebraic steps.
This table from Lesson 21, Exercise 1 is listed in your additional materials,
along with the properties of equalities for all four operations. The properties
of equality may appear abstract to students without first relating them to the
context of a real-world scenario. So we have carefully selected this Module
and this lesson, with its reference to the Integer Game, as a way to introduce
them, so that students may first make sense of them. Then, in subsequent
Lessons, they reference them in their algebraic work.
We have just enough time for some short card-play. Get back in touch with
your elbow partner. One of you will construct hand one, and one of you will
construct hand two, according to the first row in the table. Locate the 3
integer cards that comprise your hand. Lay them out on the table in front of
you, as I ask the following questions.
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40
With your cards on the table in front of you (and the rest of the deck easily
accessible, I am going to ask you the following questions. I encourage you to
adjust your cards according to my line of questioning.
1.Do you both have the same card total? (Yes.) Are your hands equal in
value? (Yes; we each have cards that total -1.)
2.What happens IF you both add the same card-value to your hand? For
instance, what happens to your totals if you each add a “4”? (Each of our
totals goes up by 4.)
3.So THEN your hands are both still equal? (Yes; now each of our hands total
3.)
4.Will this be true if you both add the same of any card – or will it only work
if you each add a “4”? (It will work for any value; both of our totals would go
up by the same amount, so our totals would still be equal.)
….We could go on and model each of these with the Integer Game cards, but
we have just demonstrated a real-life analogy for the Addition Property of
Equality. The properties are listed in the student outcomes for this lesson
which are referenced beneath this chart in your additional materials. As
previously stated, they are part of the Common Core, and can be found in the
Glossary in Table 4.
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Before we take a peek at a couple of End-of-Module Assessment questions, I
wanted us to take note of the way in which the “If-Then” statements support
students’ algebraic work in Lessons 22 and 23, which comprise a 2-day
lesson. This slide shows an example of that; the illustration is in your
materials as well. This justification of our algebraic steps is new; and
students will need plenty of exposure and practice with it before it becomes
solidified. They know the “If-Then” statements to be true from their gameplay, now they need practice referencing them in their mathematical work to
justify mathematical statements.
Take a few minutes to put the “If-Then” statements into practice on the Exit
Ticket! (After 3 minutes advance to the next slide.)
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How did you do?
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Let’s take a look at two questions from the End-of-Module Assessment:
questions 2 and 3. Question 2 relates to the rational number and the algebra
standards (7.NS.3,7.EE.2, 7.EE.4a), and question 3 relates to operations with
rational numbers (7.NS.1). Count off at your table 1,2,1,2... , so that half of
you will complete question 2, and the other half will complete question 3.
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 3), and the 2’s should discuss
their question (question 2)*.
(*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 full-
circle 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: 8 minutes
[8 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
#
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2
45
Script/ Activity directions
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 2 minutes for participants to turn and talk. Bring the group to order
and advance to the next slide.
GROUP
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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.)
(Participants’ answers may include additional key points such as “ifthen” statements applying to true number sentences, and/or the use of
algebra to solve equations involving rational numbers.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided
●
Grade 7 Module 2 PPT
●
Grade 7 Module 2 Lesson Notes
Additional Suggested Resources
●
A Story of Ratios Curriculum Overview
●
CCSS Progressions Document: The Number System (6-8)
Active learning
Turn and talk