Module Focus: Grade 6 – Module 3 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 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 real-world 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.

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

Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios.

Grade 6 – Module 3 Grade 6 – Module 3 PPT

Concept Development Examination of the development of mathematical understanding across the module using a focus on Concept Development withint he lessons.

Grade 6 – Module 3 Grade 6 – Module 3 PPT Grade 6 – Module 3

Lesson Notes

Review Grade 6 Module 3 Overview, Topic Openers, and Assessments.

Ensure that participants have access to grid paper, ruler, and

compass.

Module ReviewArticulate 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 Script/ Activity directions GROUP

1 1 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.

1 2 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.

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 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.

6 5 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.

10 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 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

5 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. (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. )

1 8 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…

Examine the conceptual understandings that are built in Grade 6 Module 3, Topic A

Materials used include:

Grid Paper, Ruler, Compass

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

8 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- 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.

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

6 11 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.

1 12 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.

5 13 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

1 14 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.

5 15 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.)

2 16 Take a moment to look at the sample solutions (click to reveal). Are there any questions?

1 17 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.

3 18 (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.)

1 19 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.

2 20 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 /𝟏 𝒃 is located on the number line by dividing the segment between 𝟎 and 𝟏 into 𝒃 segments of equal length. One of the 𝒃 segments has 𝟎 as its left endpoint; the right endpoint of this segment corresponds to the unit fraction / 𝟏 𝒃 .• /𝒂 𝒃 is located on the number line by joining 𝒂 segments of length

/𝟏 𝒃 so that 1) the left endpoint of the first segment is 𝟎, 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 /𝒂 𝒃.

4 21 Complete Lesson 6 – Exercise 1, located in your additional materials. (Advance to the next slide for the answers.)

4 22 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.)

2 23 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…

Examine the conceptual understandings that are built in Grade 6 Module 3, Topic B.

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

1 24 Let’s take a closer look at the development of key understandings in Topic B.

6 25 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.

1 26 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.

5 27 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.)

10 28 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.

6 29 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.)

10 30 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.

15 31 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…

Examine the conceptual understandings that are built in Grade 6 Module 3, Topic C.

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

1 32 Let’s take a closer look at the development of key understandings in Topic C.

8 33 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.

90 34 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.

15 35 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 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: 11 minutes

[11 minutes] In this section, you will…

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

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

1 36 Let’s take a few minutes to reflect on the purpose of today’s session.

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 6 Module 3 PPT

● Grade 6 Module 3 Lesson Notes

Additional Suggested Resources

● A Story of Ratios Curriculum Overview

● CCSS Progressions Document: The Number System (6-8)