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● 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● 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?

How will we know that they are able to do this?

Understand the sequence of mathematical concepts within this module.

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.

Participants will be able to articulate the key points listed above.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to the Module

20 minsIntroduction 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

Review Grade 7 Module 2 Overview, Topic Openers, and Assessments.

Module Review 8 minsArticulate the key points of this session.

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:

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 1 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.

1 2 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.

1 3 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.

2 4 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.

8 5 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.

3 6 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.

3 7 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. )

1 8 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.

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

3 9 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

1 10 Let’s take a closer look at the development of key understandings in Topic A.

1 11 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.

2 12 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.

3 13 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 rubber-banded 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.

8 14 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.

4 15 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.

3 16 (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?

2 17 In Lesson 3, students make sense of integer addition on the number line by relating it to the context of the game-play. 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.

3 18 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.

2 19 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 20 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.

3 21 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.

3 22 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)

3 23 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.

1 24 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.

2 25 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.)

3 26 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:

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 27 Let’s take a closer look at the development of key understandings in Topic B.

3 28 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.

1 30 In Lesson 11, students formalize their conceptual understanding of the multiplication of signed numbers; as indicated in the student outcomes for this lesson.

7 31 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).

1 32 (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 33 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.

7 34 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

1 35 Let’s take a closer look at the development of key understandings in the final topic of the module, Topic C.

3 36 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 in-depth in Module 3.

4 37 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.

5 38 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.

3 39 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.

5 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.

6 41 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 game-play, 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.)

1 42 How did you do?

8 43 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…Faciliate as participants articulate the key points of this session and clarify as needed.

Materials used include:

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 44

2 45 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.

5 46 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 “if-then” 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 Active learning Turn and talk

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)