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Module Focus: Grade 6 – Module 5 Sequence of Sessions

Overarching Objectives of this May 2014 Network Team Institute Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate

how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

High-Level Purpose of this Session● 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● Students utilize and build upon their previous understanding of composition and decomposition from Grades 1 – 5. ● Area and volume are additive. ● Students determine the formulas for area from their knowledge of the area of a rectangle and how it can be composed and

decomposed. ● Students use their prior knowledge of Module 2 to calculate the volume of right rectangular prisms with unit cubes with fractional

lengths. ● Students use their prior knowledge of Module 3 to determine area and surface area of polygons through distance calculations. ● Students use their prior knowledge of Module 4 to use equations to determine missing angles, as well as to use formulas for area,

volume, and surface area.

● Students apply formulas to real-life contexts and distinguish between the need to find surface area or volume within contextual situations.

● Scaffolding Focused: Amplify Language ● Scaffolding Focused: Move from Concrete to Representation to Abstract ● Scaffolding Focused: Give Specific Guidelines for Speaking, Reading, Writing, or Listening

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?

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.

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

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction 43 min Introduces Grade 6 Module 5• Grade 6 Module 5 PPT• Facilitator Guide

Review Grade 6 Module 5

Topic A: Area of Triangles,

40 min Explores the area of triangles, quadrilaterals, and other polygons

• Grade 6 Module 5 PPT• Facilitator Guide

Review Topic A

Quadrilaterals, and Polygons

through composition and decomposition.

Topic B: Polygons on the Coordinate Plane

36 min

Explores using absolute value to determine the distance between integers on the coordinate plane in order to find side lengths of polygons.


Review Topic B

Topic C: Volume of Right Rectangular Prisms

30 minExplores volume of right rectangular prisms


Review Topic C

Topic D: Nets and Surface Area

115 min Explores nets and surface area• Grade 6 Module 5 PPT• Facilitator Guide

Review Topic D

Session Roadmap

Section: Introduction Time: 43 minutes

In this section, you will be introduced to Grade 6 Module 5. Materials used include:• Grade 6 Module 5 PPT• Grade 6 Module 5 Facilitator Guide

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

0 min 1. NOTE THAT THIS SESSION IS DESIGNED TO BE 260 MINUTES IN LENGTH.Welcome! In this module focus session, we will examine Grade 6 – Module 5.

1 min 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.

As an overall theme of this NTI, we’ve been asked to pay special attention to the ways in which we can provide scaffolds to support specific student needs. Before we begin our examination of the mathematics in this module, let’s take a few minutes to review some of the principles we can use to support learning.

1 min 3. The mathematics modules were created based on the premise that scaffolding must be folded into the curriculum in such a way that it is part of its very DNA. The instruction in these modules is intentionally designed to provide multiple entry points for students at all levels.

Teachers are encouraged to pay particular attention to the manner in which knowledge is sequenced in the curriculum and to capitalize on that sequence when working with special student populations. Most lessons move from simple to complex allowing teachers to locate specific steps where students are struggling or need a challenge.

That said, there are specific resources to highlight and enhance strategies that can provide critical access for all students.

In developing the scaffolds already contained in the curriculum, Universal Design for Learning (UDL) has provided a structure for thinking about how to meet the needs of diverse learners. Broadly speaking, that structure asks teachers to consider multiple means of representation; multiple means of action and expression; and multiple means of engagement. These dimensions promote engagement of students and provide multiple approaches to the same content.

Individual lessons contain marginal notes to teachers (in text boxes) highlighting specific UDL information about scaffolds that might be

employed with particular intentionality when working with students. These tips are strategically placed in the lesson where the teacher might use the strategy to the best advantage.

Let’s now examine additional strategies that can be considered.In this module study, we will focus on three key ideas for developing scaffolds that can be adapted for your classroom to meet the needs of your students.

Explicit focus on the language of mathematics, using the development from concrete to representation to abstract in the building of concepts, and communicating clear expectations in instructions are areas that can provide multiple entry points for students and can be used to promote student learning.

1 min 4. Much of what we share in the mathematics classroom with students is embedded in language that is specific. Students learn casual language before academic language. This means they may sound comfortable and fluent, but may need additional support in their writing and speaking in an academic environment.

Presenters should stress that academic language is an essential component of closing the achievement gap and providing access to grade level content and beyond.

Students may have a preconceived or informal idea of the meaning of a mathematical term. Be specific in the definition or meaning that will be used.

Be cautions of words with multiple meanings that might be confusing• a garden plot and the request to plot points on a coordinate plane

Words with multiple meaning must be anticipated and then addressed, and teachers must also be prepared to pause and provide explanations when students identify words the teacher has not anticipated. Whenever possible, words with multiple means should be avoided on assessments, particularly when the meanings may be close enough to be confusing.

Make sure that Language is internally consistent (if practice problems ask students to solve, the assessments should use the same term). If language is not internally consistent, then different terms are highlighted and taught.• add, plus, sum, combine, all mean the same thing• prism, a rectangular prism, box, package all reference the same figure

in G6M5_L11

1 min 5. The more concrete and visual these ideas can be in foundational stages, the better!• Use contexts that are familiar to students in your classroom.• Use graphic organizers or other means for students to visually

organize thinking.

Note: Teachers should be thoughtful and purposeful about which graphic organizers they select. Are teachers introducing a new concept with a need to organize notes or are they connecting ideas comparing and contrasting? The goal is always to help students make those connections and not use a graphic organizer just for the novelty of it.

• Consider using non-verbal displays of mathematical relationships in your scaffolding.

• Use multiple representations and multiple approaches in explaining problems and allowing students to express solutions.

• Use pictures/ visuals/ illustrations are used to make content clearer.

1 min 6. Each day needs structured opportunities for students to speak and write in English.• Students can chorally repeat key vocabulary or phrases• Have them “turn to a neighbor and explain”

Clearly set expectations by the explicit instructions in student-friendly language.

Use visuals in your instructions.

Be direct about language.

• Pause to discuss a vocabulary term and discuss how it may be used in the lesson. Have students repeat the word chorally so that they can all hear and practice.

Provide sentence frames for anyone who may benefit.• “The volume of my prism is ___units cubed. I found this by ______.• “My idea is similar to _____’s because ____.”

Generic/ universal sentence frames may remain posted in the classroom throughout the year. These might include:• “I agree with ____ because ___” or “I think the answer is _____

because...”

2 min 7. Let’s review some key points of scaffolding instruction.

As we study the module for this session, be thinking about specific scaffolds that might be most helpful for your classroom. We will pause at various points in the session to intentionally examine and discuss suggestions for scaffolds.

8. Note to presenter:Insert this slide at appropriate points in the module study for an in-depth look at scaffolds. You may highlight a scaffold that already exists and discuss it or locate a point where a student might encounter difficulty and explore options.

Delete the slide from this current sequence after you’ve inserted it in appropriate places throughout your session.

Note to presenter: When you have inserted the slide, list several suggestions for scaffolds that would address the situation.

9. Note to presenter:If applicable, insert this slide at an appropriate point in the module study for an in-depth examination of a problem or task for multiple entry points through the principles of the Universal Design for Learning (UDL).

Delete this slide from this current sequence after you’ve used it elsewhere as needed.

REPRESENTATION: The “what” of learning.How does the task present information and content in different ways?How students gather facts and categorize what they see, hear, and read.How are they identifying letters, words, or an author's style?

In this task, teachers can ...Pre-teach vocabulary and symbols, especially in ways that build a connection to the learners’ experience and prior knowledge by providing text based examples and illustrations of fields. Integrate numbers and symbols into word problems.

ACTION/EXPRESSION: The “how” of learning.How does the task differentiate the ways that students can express what they know?How do they plan and perform tasks?How do students organize and express their ideas?

In this task, teachers can...Anchor instruction by pre-teaching critical prerequisite concepts through demonstration or models (i.e. use of two dimensional representations of space and geometric models).

ENGAGEMENT: The “why” of learning.How does the task stimulate interest and motivation for learning?How do students get engaged?How are they challenged, excited, or interested?

In this task, teachers can...Optimize relevance, value and authenticity by designing activities so that

learning outcomes are authentic, communicate to real audiences, and reflect a purpose that is clear to the participants.

If available, reviewing student work would provide participants with the opportunity to deeply understand the benefits of students sharing their thinking in working the problem. Assessments in the module have rubrics that clearly outline expectations and could be used in the discussion.

1 min 10. 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.

1 min 11. The third module in Grade 6 is called Area, Surface Area and Volume Problems (click for red ring). The module is allotted 25 instructional days.

6 min 12. Locate “K-6, Geometry” progressions document in your extra materials.

Take a few minutes to read and highlight interesting areas and key take aways that you would like to discuss with the whole group.

(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.

3 min 13. 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 four topics which span 19 lessons with a bonus model lesson after lesson 19. 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.

7 min 14. Turn to the Module Overview document.

Take four minutes to read through and highlight key areas that you would like to discuss with the whole group.

Three minutes to share with whole group.

3 min 15. 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. What questions might you have about the vocabulary?

1 min 16. The Module is broken into four topics which span 19 lesson with an additional modeling lesson after lesson 19. 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: Topic A: Area of Triangles, Quadrilaterals, and Polygons

Time: 40 minutes

In this section, you will discover the area of triangles, quadrilaterals, and other polygons through composition and decomposition.

Materials used include:• Grade 6 Module 5 PPT• Grade 6 Module 5 Facilitator Guide• Grade 6 Module 5 Topic Opener A


0 min 17. Let’s take a closer look at the development of key understandings in Topic A.

4 min 18. 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?

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.

3 min 19. Read title and outcomes. Discuss the opening exercise where students name shapes (right triangle, parallelogram, acute triangle, rectangle, trapezoid) and pay special attention to the shape “parallelogram” where it is applicable to two quadrilaterals in the set; and although a rectangle is a parallelogram, it is commonly called a rectangle because it is a more specific name. They then progress into naming all of the quadrilaterals they know in order to introduce the lesson on finding the area of a parallelogram, using their knowledge of rectangles to help.

Students cut out the parallelogram and predict how they could calculate the area of the shape. They are prompted with recalling their prior knowledge of the area of a rectangle, and how they can transform the parallelogram in front of them into a rectangle. They are led to understand that if they cut a triangle off one end of the parallelogram and move it to the other side, then they can create a rectangle. Students draw a dotted perpendicular line to show the triangle they will cut. They cut on the line and move the triangle

[ADVANCE to demonstrate the visual] to the opposite side and glue it in place, thus resulting in a rectangle with the SAME area as the original parallelogram. [ADVANCE] Nothing has been added or taken away. Just moved. Then students choose appropriate tools in order to measure the lengths of the base and height of the rectangle [ADVANCE] and note that the base and the height are the same in both the rectangle and parallelogram, [ADVANCE] thus understanding that the area of a parallelogram is the same as the area of a rectangle.

3 min 20. What are some common misconceptions students form about finding the area of a parallelogram?[ADVANCE] parallelogram. How could we construct a rectangle from this parallelogram? Why can’t we use the same method we used previously? It’s slanted in a way that we can’t determine the height because the dotted line does not go through the entire parallelogram. Students are provided opportunity to struggle drawing the height and then are instructed to cut the shape out. To solve this problem, students cut the parallelogram into four equal portions using the appropriate measurement tool to determine where to make the cuts, after demonstration from the teacher. [ADVANCE] Now we have four parallelograms. How can we use them to calculate the area of the parallelograms? Decompose them to compose rectangles. From each of these rectangles [ADVANCE] students cut triangles from each end and move to the opposite side. [ADVANCE] How can we show that the original parallelogram forms a rectangle? If we push all the rectangles together, they will form one rectangle. Therefore, it does not matter how tilted a parallelogram is. The formula to calculate the area will always be the same as the area formula of a rectangle. From there, [ADVANCE] students identify the bases and the heights of the rectangle and parallelograms are the same, thus allowing students to use the same formula for finding the area.

1 min 21. Read the title and outcomes. [ADVANCE] Read the lesson notes. [ADVANCE] to show the visuals. [ADVANCE] to show the progression and read.

3 min 22. Read through title and outcomes. Explain that students will be participating in a mathematical modeling exercise for the majority of the lesson. Students will need the triangle template found at the end of the lesson and a ruler to complete this exercise. To save class time, cut out the triangles ahead of time. Discuss the difference between height and altitude: The height of a triangle does not always have to be a side of the triangle. The height of a triangle is also called the altitude, which is a line segment from a vertex of the triangle and is perpendicular to the opposite side. Students are directed to fold the triangle where the altitude would be located, and then draw the altitude (or the height) of the triangle [ADVANCE]. Relying on information from the last lesson, students notice that by drawing the altitude they have created two right triangles and can calculate the area of the entire triangle. Students measure each base and height and round their measurements to the nearest half-inch, calculate the area of each right triangle [ADVANCE], and then add the areas to determine the area of the entire triangle [ADVANCE]. In order to expedite finding area, students are encouraged to [ADVANCE] note that a triangle has already been determined to be exactly half of its corresponding rectangle. They then refer to the area formula for the triangle, determine the length of the entire base and the height, and utilize the formula to find the area of the whole triangle. [ADVANCE] They note that the area they found using the formula is the same as decomposing the triangle into two right triangles, and further determine that the area formula applies not only to right triangles, but acute triangles as well.

2 min 23. Students further this study using a graphic organizer and continue to test that [ADVANCE] decomposing acute triangles into two right triangles and then finding the sum of the areas results in the same as when they apply the area formula directly to the entire triangle [ADVANCE]. From there, students use rectangle facts and triangle facts and the idea that area is additive to find the area of complex figures [ADVANCE TWICE] through decomposition.

2 min 24. Read the title and the outcomes. Students further their study of the area of triangles to obtuse triangles. The lesson begins with students examining these triangles from the opening exercise and find what is different about them. They determine that the height or altitude is in a different location for each triangle. The first triangle [ADVANCE] has an altitude inside the triangle. The second triangle [ADVANCE] has a side length that is an altitude, and the third triangle [ADVANCE] has an altitude that is outside of the triangle. [ADVANCE] Students are then prompted to determine what formula they can use to find the area of all three triangles with the purpose of students choosing the area of triangles formula for all three triangles. [ADVANCE]

2 min 25. From there, students work in small groups to show that the area formula is the same for all three types of triangles shown in the opening exercise through decomposing shapes using triangle templates, scissors, rulers and glue. Read through the problem and continue advancing the slide to show the decomposition.

1 min 26. Continue advancing through the slide to show decomposition of the triangle into exactly half of the rectangle.

1 min 27. Continue advancing through the slide to show decomposition of the triangle into exactly half of the rectangle.

10 min 28. Allow participants to create the proofs with the templates. Allow 2 minutes of discussion (included within the 10 minutes) to rectify any concerns about the lesson.

1 min 29. Read through the title and outcomes. Advance the shapes to show the difference between the sides of the shape and the area in which those side lengths encompass.

2 min 30. [ADVANCE] to show the complex shape that students will decompose into familiar shapes. [ADVANCE] to show one option students can choose and speak to [ADVANCE] the reasoning students will use to determine the additive area of the whole shape. Continue advancing through the duration of the slide and discuss the reasoning behind using each decomposition and the composition into a rectangle and use missing area to determine area.

2 min 31. In this example, a parallelogram is bisected along a diagonal. The resulting triangles are congruent, with the same base and height of the parallelogram. Students should see that the area for a parallelogram is equal to the base times the height, regardless of how much the bases are skewed. Students are asked how they could find the area using only triangles. [ADVANCE] through the slide to show decomposing into two triangles, finding the areas and then adding the areas together to determine the area of the parallelogram. Continue advancing to show that it does not matter which direction the bisector is drawn, the final area is the same.

1 min 32. The study continues with the use of decomposition to determine the area of a trapezoid. Students decompose the trapezoid into two triangles, find the area of each triangle and find the sum of both areas to determine the area of the parallelogram.

1 min 33. Similar to what they learned when finding the area of the parallelogram, it does not matter where the bisector is placed, that it still separates the trapezoid into two triangles and students can determine the additive area. [ADVANCE] Students further their study to use rectangle and triangle facts to decompose the rectangle by taking away the areas of the triangles to find the difference, which is the area of the trapezoid.

1 min 34. Read title and outcomes. Students take their work and apply what they have learned in lessons 1 – 5 and apply it to real-life contextual situations through composition and decomposition. Walk through the next two slides and discuss the contextual problem.

Section: Topic B: Polygons on the Coordinate Plane Time: 36 minutes

In this section, you will focus on using absolute value to determine the distance between integers on the coordinate plane in order to find side lengths of polygons.

Materials used include:• Grade 6 Module 5 PPT• Grade 6 Module 5 Facilitator Guide

• Grade 6 Module 5 Topic Opener B


0 min 35. Let’s take a closer look at the development of key understandings in Topic B.

4 min 36. Turn to Topic Opener B in your materials. Read the Topic Opener to yourself.

What material is included in Lessons 7-10?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.

5 min 37. Read title and outcome(s). [ADVANCE] Read teacher notes and explain further if need be that students are bringing with them knowledge of absolute value starting in Module 3 with horizontal and vertical number lines and then build their skills to apply absolute value to the coordinate plane. Students will use this knowledge to calculate distances between integers in order to find side lengths of polygons. [ADVANCE]

Students begin the lesson with an example where they identify and organize the coordinate points of all the line segments in the figures to the right. They are asked what they notice about each pair of points and students determine that in each pair, either the x-coordinates are the same or the y-

coordinates are the same. They then move to calculating the lengths of the segments using only the coordinates of their endpoints and discover that they only use the coordinates that are different. So, looking at the example, they see that line segment AB has coordinates (-2,8) and (9,8). They see that the y-coordinates are the same, so they must determine the distance between the x-coordinates. They can use their prior knowledge of absolute value and determine that -2 is two units from zero, and that 9 is 9 units from zero. Together, the total distance is combined and determined to be 11. [ADVANCE] Further, students discover that they subtract absolute values of the endpoints if they have the same sign, or if both endpoints are either positive or both negative. If the signs of the endpoints are different, then students discover that they add the absolute values. Through this discovery process, students complete the last column in their organizer and use absolute value to prove the distances they previously determined are accurate. [ADVANCE]

Further discussion has students determining how the distances from one point to another in this example would change if each square unit on the plane were 2 units or 3 units, and students find that the distances would double and triple, respectively.

1 min 38. Read title and outcome. Students are moving from using coordinates to determine segment lengths in Lesson 7 to using coordinates to draw polygons in order to use distances to determine the area of the shapes they create. Briefly discuss the vocabulary for the lesson focusing on relating the coordinate plane to the verb “coordinate” and speaking to the plural form of vertex as vertices. From there, students use prior knowledge to plot the points using the x and y coordinates in each ordered pair and then connecting the points starting at A, connecting to B, then connecting B to C. From there, students determine the distance between A and C to find the base of the triangle as 5 units and determine the distance between A and B to find the height of the triangle as 5 units. Using their knowledge of the area formula for triangles, students are able to determine the area of this triangle they have drawn on the coordinate plane.

3 min 39. Students move to a more difficult drawing of a triangle on the coordinate plane where the base and the height are not as easily detected using the distances between the three ordered pairs. They continue to plot the points and connect the points with line segments, however the representation on the plane does not allow them to use calculations on the plane to find the dimensions of the triangle. So, the students will use their prior knowledge of composition and decomposition to create a square around the triangle and decompose that square, subtracting the areas of the outside right triangles it formed. Once these areas are subtracted from the square, the remaining area will represent the area of the original triangle they started with. [ADVANCE] Students will begin by using the distance on the coordinate plane to determine a side length of the square in order to determine the area. Here, the distance is 6 units and students find the area to be 36 square units. Then, they begin subtracting each of the right triangles surrounding the original triangle. [ADVANCE] The first triangle has a base of 1 unit and height of 6 units. Students use the area formula for triangles and determine that this portion of the square is 3 square units. [ADVANCE] They move to the second right triangle, determine its base and height and calculate the area to be 9 square units. Remember, here, it is important that students can determine the base as 6 and the height as 3, OR the base as 3 and the height as 6. Either way, because of the commutative property and the fact that bases are determined by the altitude’s perpendicular position either measurement is correct. [ADVANCE] Finally, students determine the area of the third triangle they need to subtract and find that to be 7.5 square units. So, decomposing that square would involve taking the area of the square and repeatedly subtracting the areas of the triangles that need to be removed in order to determine the area of the original triangle. [ADVANCE] The area of the triangle will be the difference of 36 square units and the areas of each of these triangles. After subtracting each, students determine the area of the triangle as 16.5 square units.

2 min 40. Students move from triangles and squares on a coordinate plane to finding the area of polygons on the coordinate plane through the use of decomposition. In this example students can decompose this polygon into two familiar shapes: triangles. [ADVANCE] They use triangle facts in order to determine the composite area of this polygon. From here, students determine the area of the first triangle whose base is 6 units and height is 4 units to be [ADVANCE] 12 square units. They move onto to the second triangle they created and determine that its base is 2 units and its height is 3 units. Using the area formula, students determine the area of this triangle to be [ADVANCE] 3 square units. From here, students find a total area of the polygon by adding the areas of both triangles. 12 square units and 3 square units equals a total area of 15 square units.

3 min 41. Students move forward to determine the area of more complex polygons through their knowledge of triangle and rectangle facts, as well as decomposition. [ADVANCE] Students are then posed with this problem: [ADVANCE] Read the problem. What are some things we know? We know that the length of the line segment between points A and B is 5 units because we are looking at the difference between the y coordinates. We also know that the area of the rectangle is 30 square units. How can we determine where the other two vertices should be drawn? I know that if the length is 5 and the area is 30, then the width has to be the quotient of the two, so the width must be 6 units. Where do I draw the six units? [ADVANCE] There are two places students can draw. They can draw 6 units to the right of the original line segment [ADVANCE] and determine the area that way, or they can [ADVANCE] draw the length of the distance to the left of the original segment.

1 min 42. Read through title, outcomes and teacher notes.

2 min 43. Given this polygon, students must use distances to determine the lengths of each line segment creating this polygon. [ADVANCE] They begin by finding the length of each line segment as (read the lengths). Using their knowledge of finding perimeter, students simply [ADVANCE] add each of the side lengths. But, how can they determine the area of this irregular figure? They must use their knowledge of decomposition to break this shape into familiar shapes in which they can find the area. In this example, (ADVANCE TWICE] students will use a line segment as a horizontal cut from (1, 1) to point C. From there, [ADVANCE] students determine the area of the first rectangle and determine the area of the second rectangle [ADVANCE], combine the areas [ADVANCE] and determine the area of this polygon as 76 square units. This, however, is not the only way the area could have been determined. Students could have drawn a line segment between point B and (5, -5) and calculated the distances of those familiar rectangles.

4 min 44. Read the problem aloud. Note that method one, students are using decomposition in order to find the area of the polygon, [ADVANCE] but in the second method, students are composing that corresponding rectangle around the figure to use subtraction to find the area of the original polygon [ADVANCE]. What’s new here is the students are now being asked to represent their methods using an expression. Up until now, we have been modeling representing our work using an expression, but now students will represent the expression on their own. So for the first method [ADVANCE], students can represent their work with this expression. Note that the areas of triangle 1 and 4 are the same, so that expression can be written as 2 times the quantity ½ times 4 times 3. Also note that the areas of triangles 2 and 3 are the same, so that expression can be written as 2 times the quantity ½ times 8 times 3. Combined the expression that represents the entire work is 2 times the quantity ½ times 4 times 3 plus 2 times the quantity ½ times 8 times 3. Similarly, students will use an expression to represent their work as [ADVANCE] first finding the area of the rectangle as 8 times 6, and then subtracting from it the areas of all 4 of those same triangles as 4 times the quantity ½ times 2 times 3. Note here, that this is where students are bringing in their prior knowledge form Module 4 and using it as a tool to represent their methods.

1 min 45. Click to advance each and discuss the student reasoning behind choosing those methods.

1 min 46. Read title, outcome and teacher notes. Note that the only thing in addition to finding the area in these real world examples is now students are finding the perimeter as well and representing each with an expression.

9 min 47. Following Topics A and B is the Mid-Module Assessment. There are 4 questions; but you will complete just one right now – question 4. 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? Which Mathematical Practices are embodied in the task? 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: Topic C: Volume of Right Rectangular Prisms Time: 30 minutes

In this section, you will focus on exploring the volume of right rectangular prisms.

Materials used include:• Grade 6 Module 5 PPT• Grade 6 Module 5 Facilitator Guide• Grade 6 Module 5 Topic Opener C


1 min 48. Let’s take a closer look at the development of key understandings in Topic C.

4 min 49. Turn to Topic Opener C in your materials. Read the Topic Opener to yourself.

What material is included in Lessons 11-14?

How do Topics A and B prepare students for Topic C?

1 min 50. Read title, outcome and teacher notes.

1 min 51. The opening exercise is a good revisit to what knowledge they are bringing with them from elementary grades. They discuss first how many unit cubes will fit in the base and then use the layers to determine how many will fit in each of the prisms. From there, students determine which holds more prisms and how many in order for them to prepare for finding missing dimensions of volume later in the lesson.

1 min 52. Students move from finding volume of prisms using cubes with whole unit sized lengths to finding volume with cubes with fractional lengths. Here is where scaffolding will be available for students who cannot visualize.

5 min 53. Read through slide. Address any questions or concerns.

1 min 54. Continually advance through slide and discuss each talking point.

1 min 55. Continually advance through slide and discuss each talking point. Make a note that the product represents the volume of the original prism with whole unit measurements.

4 min 56. Students will use what they know about placing cubes with fractional lengths inside the prism and determine that it takes 6 cubes with ¼ lengths to fill the length, 4 to fill the width and 15 to fill the height. When we multiply 6 times 4 times 15 we determine that it will take 360 cubes to fill that prism. Since we know we can find the volume of that prism by multiplying the number of cubes by the volume of one of those cubes [ADVANCE] 360 times 1/64 is 5 and 5/8 cubic inches. But, how else can we find the volume using the formula? Can we simply multiply the length times the width times the height? Let’s try to see if it matches the volume we just found: [ADVANCE] And yes, in fact it matches. This activity is extremely important for students to understand HOW the volume formula relates to packing with cubes and WHY the volume of prisms with fractional side lengths can be determined simply by using the figure. The students continue to find the volumes of prisms with fractional side lengths using both methods for the duration of this lesson in order for students to continue to make the connection between the two.

2 min 57. Read the title and outcomes and continuously advance through the slide to discuss its contents alluding to the fact that if students simply take the area of the two dimensional base and multiply it by the height, the volume of the prism can still be determined accurately.

2 min 58. Read through example and discuss how now matter which way the prism is positioned, any of the faces can be used as the base, and then the base can be used to find the area, and then multiply that by the height. This is also [ADVANCE] confirmed because the commutative property allows us to rearrange the factors and regardless of their position, the product will still be the same.

1 min 59. Read through the title and outcome. Continually advance through slide discussing that when the base remains a constant area, and the height is doubled, the entire volume doubles.

1 min 60. Students create a table to determine further that if the base area remains constant, if the height is multiplied, the volume is multiplied by that same amount. Continually advance to talk through the slide examples. From here, students are able to write expressions to represent finding the volume of a prism when the base is doubled, and this final expression will remain the same, only the coefficient will change based on how the height of the prism is changing. If the height is tripled, the coefficient will be three. If the height is quadrupled, the coefficient will be 4, and so on.

2 min 61. Students will rely on the any grouping, any order rule they discovered in Module 4 and note that… read through the rests of the slide

2 min 62. Read title and outcome. Make connections to students finding area that is additive and now students will use real world examples to discover that volume is also additive. Then begin with a prism and determine the volume. [ADVANCE] Next they are posed with a compound prism where they need to determine how to find the volume. They can use their prior knowledge and find the volume of the missing piece and subtract it from the original volume, or they can deconstruct the compound prism into two prisms, find their volumes and add them together.

1 min 63. One last real life example is using this diagram to determine how much water the tank can hold. They use the dimensions of 3/5 times ¼ times ¾ to determine the volume. Then they can use the dimensions to find the volume of the water that is present by multiplying 3/8 times ¾ times ¼, and then subtract those volumes to determine the volume of the empty portion of the tank. This is going to lead the students into our next session where student will be completing a modeling cycle exercise in order to determine volume of water in aquariums.

Section: Topic D: Nets and Surface Area Time: 115 minutes

In this section, you will focus on exploring nets and surface area. Materials used include:• Grade 6 Module 5 PPT• Grade 6 Module 5 Facilitator Guide• Grade 6 Module 5 Topic Opener D


0 min 64. Let’s take a closer look at the development of key understandings in Topic D.

3 min 65. Turn to Topic Opener D in your materials. Read the Topic Opener to yourself.

What material is included in Lessons 15-19?

How do Topics A, B and C prepare students for Topic D?

2 min 66. Advance through each slide and read the lesson notes for the teacher.

3 min 67. Advance through each slide and read the lesson notes for the teacher.

2 min 68. Discuss the net of the cereal box in the picture. Once you have the net of the cereal box deconstructed, display the net with the unprinted side out, perhaps using magnets on a whiteboard. You will direct students to discuss what they find about the cardboard (cereal box) and they will hopefully determine that the shape is irregular, it has three sets of identical rectangles all the vertices are right angles, and maybe that it can be folded into a box or a three-dimensional shape.

3 min 69. Then, display (ADVANCE) the nets, here and discuss with students the similarities or differences of these nets to the net(s) of the cereal box. The similarities are there are 6 sections in each; they can be folded to create a 3-D shape, etc. The differences are that one is made of rectangles (the cereal box) and the others are made of squares (displayed nets) and the sizes are different. From here, you will turn the cereal box over to demonstrate how it was cut. Then you will reassemble it to resemble the intact box. From there, you will direct the students’ attention to the six square arrangements and ask what they think the six square shapes will fold into. A cube. And if it were to fold into a cube, how many faces would it have? Six. Then, demonstrate that, in fact, each fold into cubes and each cube has six faces.

3 min 70. What about this six-square arrangement? Do you think it will fold into a cube? Why or why not? Allow for answers. Teachers will demonstrate, or could also provide students templates of this arrangement in order for them to discover that it, in fact, does not fold into a net. From there teachers will formally define the term net as two-dimensional figures that can be folded to create a three-dimensional solid.

25 min 71. Discuss that there are 20 templates at the end of the lesson that students can cut out and wrap around their cubes to determine if the arrangements are nets. To save time, have these cut out ahead of time, placed in baggies with 8 unifix cubes. Allow 25 minutes for participants to cut out and determine which arrangements are actually nets.

2 min 72. Read the directive. Students will shade in their student materials the arrangements that are nets.

2 min 73. Read directive. These nets are found at the end of the lesson. (ADVANCE) Discuss the scaffolding. (ADVANCE) Read definitions.

2 min 74. Students will explore the nets as a means to understand how the prism is created. Why are the faces of this square pyramid triangles? (ADVANCE) They note that the base is a square, but the other shapes around that meet at the apex, or vertex are triangles. (ADVANCE) Read directive. (ADVANCE) Discuss scaffolding.

2 min 75. Students continue their study with right rectangular prisms and determine why the faces of the prism are parallelograms. (ADVANCE) Read the directive. (ADVANCE) How are these parallelograms related to the shape and size of the base? (ADVANCE) Read the directive and show that because the base has four sides, there will be four lateral faces. There are two bases in this example, the top and the bottom, where the four lateral faces are here: direct to the ne

4 min 76. If the bases are hexagons, does this mean the prism must have six faces? Let’s take a look: (ADVANCE) hexagonal prism. Notice that there are six faces, but we cannot omit the bases as faces, as any one of these faces could be the base. So, with the six faces and the two bases (ADVANCE) there is a total of eight faces. (ADVANCE) So, what is the relationship between the number of sides on a polygonal base and the number of total faces on the prism? Let’s take a look back at the last slide. (Rewind back to Slide 12) Note that the rectangular prism had 4 lateral faces and two bases, for a total of six faces, altogether. Note that the base is a parallelogram with 4 sides. (ADVANCE back to Slide 13) Note here, that there are six lateral faces and two bases, for a total of 8 faces in all. See that the base is a hexagon and has 6 sides. (ADVANCE) So, the total number of faces in a prism will be two more than the number of sides in the polygonal base. (ADVANCE) So, what additional information do you know about a prism if its base is a regular polygon? (ADVANCE) All sides of the prism will be identical.

NOTE: This is quite an in-depth look at one lesson for many reasons. It sets the foundation for the duration of the module. Students must first understand how these nets represent three-dimensional figures in order to understand the construction of three-dimensional figures, as well as in the next lesson where students will construct nets. Their knowledge from this lesson will be instrumental in understanding how to calculate surface area later in the module.

2 min 77. Read the title and outcome. Click to advance the directive and picture and read the directive. Explain that the measurement of each of these dimensions will set up questioning on what students can do with those dimensions, as well as provide background knowledge on how to deconstruct solids later in the lesson. From there, students participate in an opening discussion where they are identifying the angle measures, as well as the faces, and determine that the shape formed is a right rectangular prism.

3 min 78. Students take their understanding from the opening exercise and apply it to this solid where they first discuss with a partner how they could create a net for this solid. These conversations will lead into a whole-class discussion on how the dimensions of a rectangular solid can determine the dimensions of the polygons that make up its net. (ADVANCE) question and read. (ADVANCE) the answer and discuss, using the pointer to show the measurements. (ADVANCE) the questions and answers and discuss the progression of questioning in order to determine the measurements of each face.

2 min 79. There are three different rectangles, but two copies of each will be needed to make the solid. The (ADVANCE) top face is identical to the (ADVANCE) bottom face, the (ADVANCE) left side face is identical to the (ADVANCE) right side face, and the (ADVANCE) front is identical to the (ADVANCE) back.

3 min 80. Students then use the templates provided at the end of the lesson and work in pairs to arrange the rectangles into the shape (ADVANCE) here and use tape to attach them. Students can also draw this on graph paper, where each square on the paper represents one unit. Some scaffolding for this activity includes (ADVANCE), read the scaffolding. NOTE: If this is truly a net of the solid, it will fold into a right rectangular prism. Students should fold the net into the solid to prove that it was indeed a net. Be prepared for questions about other arrangements of these rectangles that are also nets of the right rectangular prism. There are many various arrangements.

2 min 81. Students move from the hands on example to a pictorial representation of the net. They are asked to use the dimensions given to create a drawing of the net of the prism. From there:Pose the problem and elicit participants to create an expression that will represent how to find the total area for the net on whiteboards.

1 min 82. From these exercises, students use previous knowledge from Lesson 15 to deconstruct pyramids and other prisms into nets. They are also provided templates at the end of this lesson in order for them to deconstruct.

3 min 83. Read title and outcome. Discuss directive about scaffolds for English Learners. Students will move from their learning in Lesson 16 to being able to determine surface area of prisms. Here, (ADVANCE) they directly move from the net, and the measurements of each rectangle to determine (ADVANCE) locating where each portion of the net is represented in the actual prism. Show the direct correlation between each section of the net and where they are represented in the diagrams of the prism. Note that from each of the faces on the prism, the area can be determined using the measurements from the net.

2 min 84. From here, students will determine that to find the area of the entire surface of the prism, they have to add (ADVANCE) the areas of the front, back, left side, right side, top and bottom. From there, students replace the faces with their measurements, add them together and determine a final surface area of 40 square cm.

2 min 85. Students continue their study of surface area by using nets to calculate the surface area of a figure (ADVANCE) and through discussion with the teacher. They discuss that when finding surface area, they are determining that the surface area of a three dimensional figure is the calculated by adding the area of each face. Once they are given or they determine the dimensions of each face (ADVANCE), students calculate the area of each face of the rectangular prism (ADVANCE). From there, a discussion details and the teacher models how to write the calculations using an expression.

2 min 86. The students will be responsible throughout the lesson to (ADVANCE) name the solid the net creates, (ADVANCE) then write an expression for the surface area and calculate the surface area, then (ADVANCE) explain how the expression represents the figure. Read the example.

2 min 87. Lesson 17 leads directly into Lesson 18 where (read the title and outcomes) Lesson 18 begins with a review of Lesson 17 (labeling faces, determining the area of each face, then creating and using an expression to determine surface area, in order for students to then…

3 min 88. …develop a formula for surface area. Using a graphing organizer, students begin with (ADVANCE) the dimensions of each face, then record the area of each face (ADVANCE) and then further detail the formulas for finding those areas (ADVANCE). What did we do to determine the area of the top/base? We multiplied the length times the width. What did we do to determine the area of the bottom/base? Why, we multiplied the length times the width. We did that formula twice, so we can represent that as (ADVANCE) 2(l x w). How did we find the area of the front? We multiplied length times height. We did that again for the area of the back, so we can represent that with (ADVANCE) 2(l x h). What about the area of the left side? How did we determine that? We multiplied the width times the height. And, we did that again to determine the area of the right side. We can represent that as (ADVANCE) 2(w x h). Now, how did we determine the surface area in our original expression? We added the areas, so, (ADVANCE TWICE) we will also add the areas in this formula/expression. (ADVANCE) And, now we have created a formula for finding the surface area of right rectangular prisms: SA= 2(l x w) + 2(l x h) + 2(w x h). (ADVANCE) The lesson extends to finding the area of cubes using the formula for surface area they just created. But then…

1 min 89. …they determine there is a specific formula to find the surface area of a cube. Read through slide.

1 min 90. Continue reading through slide.

1 min 91. Read title and outcome. Note that the opening exercise has students utilizing the formula for surface area they determined in Lesson 18.

3 min 92. This lesson also affords students the opportunity to decide which formulas they have discovered in this module are to be used in real-life contexts. In this example, the problem isn’t asking to determine the surface area, or how much cardboard it takes to create the juice box, it is asking how much juice will go into the juice box. The students need to determine that they are looking for volume (ADVANCE), instead of surface area. (ADVANCE) Students are then to explain why they chose the formulas for each example, further reiterating the specific need for each formula and that they are far from interchangeable. Should students have further complications choosing the appropriate formula, this graphic organizer (ADVANCE) has been provided for discussion. Read through the graphic organizer.

2 min 93. Lesson 19 continues to have students using the formulas for surface area and volume for specific purposes. Continue advancing through slide.

2 min 94. Read title and outcomes. Lesson 19a is a culmination of not only the module in real-life context, but a culmination of Modules 1 – 5. Please note that this lesson was created to introduce the modeling cycle in a simple form to prepare students for high school. There are activities in the lesson that are posted as “optional” because we could not guarantee that the materials involved in this lesson would be readily available. It is, of course, highly recommended that students have the opportunity to participate in this hands-on experimental activity. Students begin the lesson with a review from Lesson 18, but this aquarium will also be a reference point throughout the lesson.

1 min 95. Students begin the activity by trying to figure out how many cubic inches of water is in a regular ten gallon tank. They start with finding the unit rate of cubic inches to gallons using this table.

1 min 96. From there, students use a tape diagram to determine if there are 231 in3 for every gallon, that in ten gallons there are 2,310 cubic inches, relying on their knowledge from Module 1. (ADVANCE) Read through Optional Exercise 1. (ADVANCE) Read through Exercise 1.

1 min 97. CONTINUE READING THROUGH THE SLIDE.




10 min 101. Let’s take a look at the End-of-Module Assessment: questions 1 and 5. 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 D? We have come full-circle back to where we began today’s session: the progressions document. You are encouraged 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.

1 min 102. Let’s take a few minutes to reflect on the purpose of today’s session.

2 min 103. 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.

6 min 104. 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. Then, each table will share out.

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 5 PPT● Grade 6 Module 5 Facilitator Guide● Grade 6 Module 5 Module Overview● Grade 6 Module 5 Topic Overview A● Grade 6 Module 5 Topic Overview B● Grade 6 Module 5 Topic Overview C● Grade 6 Module 5 Topic Overview D

● Grade 6 Module 5 End-of-Module Assessment

Additional Suggested Resources

● How to Implement A Story of Ratios● A Story of Ratios Year Long Curriculum Overview● A Story of Ratios CCLS Checklist

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