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GRADE 6 Mission 1 Area and Surface Area In this mission students learn to find areas of polygons by decomposing rearranging and composing shapes They learn to understand and use the terms base and height and find areas of parallelograms and triangles Students approximate areas of non polygonal regions by polygonal regions They represent polyhedra with nets and find their surface areas

2023 Zearn Portions of this work Zearn Math are a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license Zearn is a registered trademark Printed in the U S A ISBN 979 8 88868 981 3

Table of Contents MISSION OVERVIEW viii ASSESSMENTS xvii TOPIC A REASONING TO FIND AREA LESSON 1 Tiling the Plane 1 LESSON 2 Finding Area by Decomposing and Rearranging 12 LESSON 3 Reasoning to Find Area 25 TOPIC B PARALLELOGRAMS LESSON 4 Parallelograms 41 LESSON 5 Bases and Heights of Parallelograms 56 LESSON 6 Area of Parallelograms 71 TOPIC C TRIANGLES LESSON 7 From Parallelograms to Triangles 82 LESSON 8 Area of Triangles 98 LESSON 9 Formula for the Area of a Triangle 107 LESSON 10 Bases and Heights of Triangles 120 TOPIC D POLYGONS LESSON 11 Polygons 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 132 iii

TOPIC E SURFACE AREA LESSON 12 What Is Surface Area 148 LESSON 13 Polyhedra 160 LESSON 14 Nets and Surface Area 185 LESSON 15 More Nets More Surface Area 198 LESSON 16 Distinguishing Between Surface Area and Volume 213 TOPIC F SQUARES AND CUBES iv LESSON 17 Squares and Cubes 225 LESSON 18 Surface Area of a Cube 236 LESSON 19 Designing a Tent 248 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH MISSION OVERVIEW G6M1 CURRICULUM MAP 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K M3 M2 M1 2D 3D Shapes Numbers to 10 Numbers to 5 Digital Activities 50 M1 G1 M1 G3 M2 Add Subtract Friendly Numbers Explore Length M1 Add Subtract Round G5 Place Value with Decimal Fractions G7 G8 Key Base Ten Operations M1 M2 Area and Surface Area Introducing Ratios M2 M1 Scale Drawings Introducing Proportional Relationships M1 Rigid Transformations and Congruence Whole Numbers and Operations M3 M2 M3 Rates and Percentages M4 M3 Add Subtract Fractions M4 Dividing Fractions Proportional Measuring Relationships Circles and Percentages M2 Dilations Similarity and Introducing Slope M3 Linear Relationships Expanding Whole Numbers and Operations to Fractions and Decimals M5 Rational Number Arithmetic M4 Linear Equations and Linear Systems M6 Equal Groups Fractions as Numbers M4 M5 M5 M7 Functions and Volume Algebraic Thinking and Reasoning Leading to Functions M6 Associations in Data Geometry M6 M9 M8 Rational Numbers Angles Triangles and Prisms Multiply Measure The Coordinate Plane M7 Expressions and Equations M7 Decimal Fractions Volume Area Shapes M6 M6 Shapes Measurement Display Data M6 Multiply and Divide Fractions Decimals Expressions Equations and Inequalities M7 M6 M5 Arithmetic in Base Ten M8 Shapes Time Fractions Length Money Data Equivalent Fractions M5 M6 Add Subtract to 100 M7 M5 M4 Construct Lines Angles Shapes Multiply Divide Big Numbers M5 M4 Find the Area Numbers to 20 Digital Activities 35 Work with Shapes Add Subtract Big Numbers M3 M3 M1 M4 M5 Multiply Divide Tricky Numbers Measure It Numbers to 15 Digital Activities 35 Add Subtract Big Numbers Add Subtract Solve M2 M2 M1 M3 Measure Length M4 Counting Place Value Multiply Divide Friendly Numbers G4 G6 M3 Measure Solve G2 M2 Meet Place Value M6 Analyzing Comparing Composing Shapes Numbers 10 20 Count to 100 by Ones and Tens Number Pairs Addition Subtraction to 10 Numbers to 10 Digital Activities 50 Add Subtract Small Numbers M5 M4 Comparison of Length Weight Capacity Numbers to 10 Putting It ALL Together 1 Data Sets and Distributions M8 Probability and Sampling M7 Exponents and Scientific Notation M9 Putting It ALL Together M8 Pythagorean Theorem and Irrational Numbers M9 Putting It ALL Together WEEK Measurement Statistics and Probability 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license v

ZEARN MATH MISSION OVERVIEW G6M1 Overview of Topics and Lesson Objectives Each mission is broken down into topics A topic is a group of lessons that teach the same concept There is a balance of Independent Digital Lessons and Concept Explorations in each topic of a mission to ensure every student learns with a mix of modalities feedback and support while engaging in grade level content Throughout each mission students work on grade level content with embedded remediation to address unfinished learnings Objective vi Topic A Reasoning to Find Area Lesson 1 Explore the meaning of area Lesson 2 Decompose and compose shapes to find area Lesson 3 Find the areas of polygons using reasoning strategies such as decomposing decomposing and rearranging and subtracting Topic B Parallelograms Lesson 4 Use the characteristics of a parallelogram and reasoning strategies to find the area of parallelograms Lesson 5 Understand the terms base and height and their relationship to the area of a parallelogram Lesson 6 Use the formula for area to find the area of any parallelogram Topic C Triangles Lesson 7 Explain the relationship between a pair of identical triangles and a parallelogram Lesson 8 Use parallelograms to find the area of triangles Lesson 9 Explain how to use a formula to find the area of a triangle Lesson 10 Identify a base and corresponding height of a triangle Topic D Polygons Lesson 11 Understand that you can find the area of a polygon by decomposing and rearranging it into rectangles and triangles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH MISSION OVERVIEW G6M1 Objective Mid Mission Assessment Topics A D Topic E Surface Area Lesson 12 Understand what the surface area of a three dimensional figure means Lesson 13 Investigate different polyhedra and their nets Lesson 14 Use nets to find surface area of polyhedra Lesson 15 Draw an accurate net of a polyhedron and use it to the calculate the surface area Lesson 16 Find and compare surface area and volume of different shapes Topic F Squares and Cubes Lesson 17 Use exponents to express the area of a square and volume of a cube Lesson 18 Express the surface area and volume of a cube in different ways given a side length Topic G Let s Put It to Work Lesson 19 Apply strategies and formulas for finding area of polygons to find surface area End of Mission Assessment Topics E G Note on Pacing for Differentiation If you are using the Zearn Math recommended weekly schedule that consists of four Core Days when students learn grade level content and one Flex Day that can be tailored to meet students needs we recommend omitting the optional lessons in this mission during the Core Days Students who demonstrate a need for further support can explore these concepts with you and peers as part of a flex day as needed This schedule ensures students have sufficient time each week to work through grade level content and includes built in weekly time you can use to differentiate instruction to meet student needs Optional lessons for G6M1 Lesson 16 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license vii

ZEARN MATH MISSION OVERVIEW G6M1 Mission Overview Work with area in grade 6 draws on earlier work with geometry and geometric measurement Students began to learn about two and three dimensional shapes in kindergarten and continued this work in grades 1 and 2 composing decomposing and identifying shapes Students work with geometric measurement began with length and continued with area Students learned to structure two dimensional space that is to see a rectangle with whole number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares In grade 3 students distinguished between perimeter and area They connected rectangle area with multiplication understanding why for whole number side lengths multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle They used area diagrams to represent instances of the distributive property In grade 4 students applied area and perimeter formulas for rectangles to solve real world and mathematical problems and learned to use protractors In grade 5 students extended the formula for the area of rectangles to rectangles with fractional side lengths Partition rectangles and circles into halves and quarters Compose figures in the plane Partition rectangles and circles into thirds Partition rectangles into squares and count them Find whole number areas Multiply to find areas of rectangles Distinguish between perimeter and area Apply area and perimeter formulas for rectangles in real world contexts Tile to find areas of rectangles with fractional side lengths Multiply to find these areas Understand the area of a triangle is half of the product of one of its sidelengths and its corresponding height Find areas of polygons In grade 6 students extend their reasoning about area to include shapes that are not composed of rectangles Doing this draws on abilities developed in earlier grades to compose and decompose shapes for example to see a rectangle as composed of two congruent right triangles Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them students build on these abilities and their knowledge of areas of rectangles to find the areas of polygons by decomposing and rearranging them to make figures whose areas they can determine They learn strategies for finding areas of parallelograms and triangles and use regularity in repeated reasoning to develop formulas for these areas using geometric properties to justify the correctness of these formulas They use these formulas to solve problems They understand that any polygon can be decomposed into triangles and use this knowledge to find areas of polygons Students find the surface areas of polyhedra with triangular and rectangular surfaces They study assemble and draw nets for polyhedra and use nets to determine surface areas Throughout they discuss their mathematical ideas and respond to the ideas of others Because grade 6 students will be writing algebraic expressions and equations involving the letter x and x is easily confused with these materials use the dot notation e g 2 3 for multiplication instead of the cross notation e g 2 3 The dot notation will be new for many students and they will need explicit guidance in using it Many of the lessons in this mission ask students to work on geometric figures that are not set in a real world context This design choice respects the significant intellectual work of reasoning about area Tasks set in real world contexts that involve areas of polygons are often contrived and hinder rather than help understanding Moreover mathematical contexts are legitimate contexts that are worthy of study Students do have an opportunity at the end of the mission to tackle a real world application viii 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH MISSION OVERVIEW G6M1 In grade 6 students are likely to need physical tools in order to check that one figure is an identical copy of another where identical copy is defined as follows One figure is an identical copy of another if one can be placed on top of the other so that they match up exactly In grade 8 students will understand identical copy of as congruent to and understand congruence in terms of rigid motions that is motions such as reflection rotation and translation In grade 6 students do not have any way to check for congruence except by inspection but it is not practical to cut out and stack every pair of figures one sees Tracing paper is an excellent tool for verifying that figures match up exactly and students should have access to this and other tools at all times in this mission Thus each lesson plan suggests that each student should have access to a geometry toolkit which contains tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Providing students with these toolkits gives opportunities for students to develop abilities to select appropriate tools and use them strategically to solve problems Note that even students in a digitally enhanced classroom should have access to such tools apps and simulations should be considered additions to their toolkits not replacements for physical tools In this grade all figures are drawn and labeled so that figures that look congruent actually are congruent in later grades when students have the tools to reason about geometric figures more precisely they will need to learn that visual inspection is not sufficient for determining congruence Also note that all arguments laid out in this mission can and should be made more precise in later grades as students geometric understanding deepens Progression of Disciplinary Language In this mission teachers can anticipate students using language for mathematical purposes such as comparing explaining and describing Throughout the mission students will benefit from routines designed to grow robust disciplinary language both for their own sense making and for building shared understanding with peers Teachers can formatively assess how students are using language in these ways particularly when students are using language to Compare geometric patterns and shapes Lesson 1 strategies for finding areas of shapes Lesson 3 and polygons Lesson 11 the characteristics of prisms and pyramids Lesson 13 the measures and units of 1 2 and 3 dimensional attributes Lesson 16 representations of area and volume Lesson 17 Explain how to find areas by composing Lesson 3 strategies used to find areas of parallelograms Lesson 4 and triangles Lesson 8 how to determine the area of a triangle using its base and height Lesson 9 strategies to find surface areas of polyhedra Lesson 14 Describe observations about decomposition of parallelograms Lesson 7 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license ix

ZEARN MATH MISSION OVERVIEW G6M1 information needed to find the surface area of rectangular prisms Lesson 12 the features of polyhedra and their nets Lesson 13 the features of polyhedra Lesson 15 relationships among features of a tent and the amount of fabric needed for the tent Lesson 19 In addition students are expected to justify claims about the base height or area of shapes generalize about the features of parallelograms and polygons interpret relevant information for finding the surface area of rectangular prisms and represent the measures and units of 2 and 3 dimensional figures Over the course of the mission teachers can support students mathematical understandings by amplifying not simplifying language used for all of these purposes as students demonstrate and develop ideas The table shows lessons where new terminology is first introduced including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing Terms from the glossary appear bolded Teachers should continue to support students use of a new term in the lessons that follow where it was first introduced New Terminology x Lesson Receptive 1 area region plane gap 2 compose decompose rearrange two dimensional 3 shaded strategy 4 parallelogram opposite sides or angles 5 base height corresponding expression represent 6 horizontal vertical Productive quadrilateral 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH MISSION OVERVIEW G6M1 New Terminology Lesson Receptive Productive 7 identical parallelogram base height compose decompose rearrange 8 9 opposite vertex 10 vertex edge 11 polygon horizontal vertical 12 face surface area area region 13 polyhedron net prism pyramid three dimensional polygon vertex edge face 15 volume prism pyramid 16 appropriate quantity two dimensional three dimensional 17 squared cubed exponent edge length 18 value of an expression squared cubed net 19 estimate description surface area volume 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license xi

ZEARN MATH MISSION OVERVIEW G6M1 Digital Lessons Students also learn the concepts from this mission in their Independent Digital Lessons There are 17 Digital Lessons for Mission 1 It s important to connect teacher instruction and digital instruction at the mission level So when you start teaching Mission 1 set students to the first digital lesson of Mission 1 The intentional balance of learning with teachers and peers and learning independently in digital lessons ensures every student has multiple opportunities to represent engage with and express their math reasoning In the digital lessons students explore the concepts through interactive problem solving with embedded support that launches at the moment of misconception As students complete digital lessons they will automatically progress to the next lesson Go online to Zearn org to explore more of the digital lessons for this mission Terminology Area Area is the number of square units that covers a two dimensional region without any gaps or overlaps For example the area of region A is 8 square units The area of the shaded region of B is 12 square unit Base of a parallelogram or triangle We can choose any side of a parallelogram or triangle to be the shape s base Sometimes we use the word base to refer to the length of this side Base of a prism or pyramid The word base can also refer to a face of a polyhedron A prism has two identical bases that are parallel A pyramid has one base A prism or pyramid is named for the shape of its base Compose Compose means put together We use the word compose to describe putting more than one figure together to make a new shape xii 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH MISSION OVERVIEW G6M1 Cubed We use the word cubed to mean to the third power This is because a cube with side length s has a volume of s s s or s3 Decompose Decompose means take apart We use the word decompose to describe taking a figure apart to make more than one new shape Edge Each straight side of a polygon is called an edge For example the edges of this polygon are segments AB BC CD DE and EA Exponent In expressions like 53 and 82 the 3 and the 2 are called exponents They tell you how many factors to multiply For example 53 5 5 5 and 82 8 8 Face Each flat side of a polyhedron is called a face For example a cube has 6 faces and they are all squares Height of a parallelogram or triangle The height is the shortest distance from the base of the shape to the opposite side for a parallelogram or opposite vertex for a triangle We can show the height in more than one place but it will always be perpendicular to the chosen base Net A net is a two dimensional figure that can be folded to make a polyhedron Here is a net for a cube 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license xiii

ZEARN MATH MISSION OVERVIEW G6M1 Opposite Vertex For each side of a triangle there is one vertex that is not on that side This is the opposite vertex For example point A is the opposite vertex to side BC Parallelogram A parallelogram is a four sided polygon with two pairs of parallel sides Here are two examples of parallelograms 5 4 24 45 135 135 45 4 24 9 34 27 2 152 8 4 5 152 8 9 34 4 27 2 Polygon A polygon is a closed two dimensional shape with straight sides that do not cross each other Figure ABCDE is an example of a polygon Polyhedron A polyhedron is a closed three dimensional shape with flat sides When we have more than one polyhedron we call them polyhedra Here are some drawings of polyhedra xiv 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH MISSION OVERVIEW G6M1 Prism A prism is a type of polyhedron that has two bases that are identical copies of each other The bases are connected by rectangles or parallelograms Here are some drawings of prisms Pyramid A pyramid is a type of polyhedron that has one base All the other faces are triangles and they all meet at a single vertex Here are some drawings of pyramids Quadrilateral A quadrilateral is a type of polygon that has 4 sides A rectangle is an example of a quadrilateral A pentagon is not a quadrilateral because it has 5 sides Region A region is the space inside of a shape Some examples of two dimensional regions are inside a circle or inside a polygon Some examples of three dimensional regions are the inside of a cube or the inside of a sphere Squared We use the word squared to mean to the second power This is because a square with side length s has an area of s s or s2 Surface Area The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron without any gaps or overlaps For example if the faces of a cube each have an area of 9 cm2 then the surface area of the cube is 6 9 or 54 cm2 Vertex A vertex is a point where two or more edges meet When we have more than one vertex we call them vertices The vertices in this polygon are labeled A B C D and E 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license xv

ZEARN MATH MISSION OVERVIEW G6M1 Required Materials Copies of template Demonstration nets with and without flaps Geometry toolkits Tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Glue or gluesticks Nets of polyhedra Pre assembled or commercially produced polyhedra Pre assembled or commercially produced tangrams Pre printed slips cut from copies of the template Rulers Scissors Snap cubes Sticky notes Tape xvi 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 1 Name Date GRADE 6 MISSION 1 Mid Mission Assessment 1 Find the area of the shaded region Show your reasoning 2 Find the area of each parallelogram Show your work a 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 2 G6M1 MID MISSION ASSESSMENT b 3 Use the image below to answer the questions 10 cm a Find the area of the parallelogram Explain or show your reason 8 5 cm 8 cm b Was there a length measurement you did not use If so explain why it wasn t used 4 Find the area of the triangle Explain or show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 3 G6M1 MID MISSION ASSESSMENT 5 Identify a base and corresponding height of the triangle and use them to find the area 7 in n 8i 4 12 in 5 7 in 1 6 Calculate the area of the shaded region Show your work 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 4 G6M1 MID MISSION ASSESSMENT 7 LaWanda makes pennant flags like the one shown below 24 in 12 in a How much fabric will LaWanda need to make 5 total flags Show your work b LaWanda alters the design of her pennant flag as shown below How much fabric will she need to make a single flag of her new design Show your work 24 in 12 in 10 in 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 1 Name Date GRADE 6 MISSION 1 End of Mission Assessment 1 Determine the surface area Show your work a b 2 Find the surface area of this polyhedron Show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 2 G6M1 END OF MISSION ASSESSMENT 3 Find the surface area of this polyhedron Show your reasoning 5 cm 5 cm 4 cm 3 cm 3 cm 4 cm 4 cm 5 cm 4 cm 5 cm 3 cm 3 cm 4 Below there is an image of a polyhedra and its net Use the images below to answer the questions a Label the edge lengths in the net b Calculate the surface area Show your work 13 in 12 in 10 in 5 in 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 END OF MISSION ASSESSMENT PAGE 3 5 A square has a side length of 6 ft Use an exponent to express its area and evaluate Draw a picture to support your work 6 A cube has an edge length of 9 in Use an exponent to express its volume and evaluate Draw a picture to support your work 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

PAGE 4 G6M1 END OF MISSION ASSESSMENT 7 Lutz is making a toy chest for his grandson The chest is 4 feet wide 2 12 feet deep and 2 12 feet tall Lutz plans to paint the outside of the chest blue using a pint of blue paint A pint of paint should cover 50 sq feet a Label the edge lengths of the toy chest b Does Lutz have enough paint Show your work or explain your answer 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 1 Tiling the Plane G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Compare orally areas of the shapes that make up a geometric pattern LEARNING GOALS Comprehend that the word area orally and in writing refers to how much of the plane a shape covers LEARNING GOALS STUDENT FACING Let s look at tiling patterns and think about area LEARNING TARGETS STUDENT FACING I can explain the meaning of area REQUIRED MATERIALS REQUIRED PREPARATION Geometry toolkits Assemble geometry toolkits It would be best if students have access to these toolkits at all times throughout the mission Toolkits include tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 1

G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Students start the first lesson of the school year by recalling what they know about area note that students studied the areas of rectangles with whole number side lengths in grade 3 and with fractional side lengths in grade 5 The mathematics they explore is not complicated so it offers a low threshold for entry The lesson does however uncover two important ideas If two figures can be placed one on top of the other so that they match up exactly then they have the same area The area of a region does not change when the region is decomposed and rearranged At the end of this lesson students are asked to write their best definition of area It is important to let them formulate their definition in their own words For English learners it is especially important that they be encouraged to use their own words and also to use words of their peers In the next lesson students will revisit the definition of area as the number of square units that cover a region without gaps or overlaps As the first set of problems in a problem based curriculum students will also begin their year long work on making sense of problems and persevering in solving them This opening lesson leaves space for teachers to begin setting classroom routines and their expectations for mathematical discourse In all of the lessons in this mission students should have access to their geometry toolkits which should contain tracing paper graph paper colored pencils scissors and an index card Students may not need all or even any of these tools to solve a particular problem However to make strategic choices about when to use which tools students need to have opportunities to make those choices Notes on terminology In these materials when we talk about a figure such as a rectangle triangle or circle we usually mean the boundary of the figure e g the sides of the rectangle not including the region inside However we also use shorthand language such as the area of a rectangle to mean the area of the region inside the rectangle The term shape could refer to a figure with or without its interior Although the terms figure region and shape are used without being defined precisely for students help students understand that sometimes our focus is on the boundary which in this mission will always be composed of black or gray line segments and sometimes it is on the region inside which in this mission will be referred to as the shaded region 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 1 Warm Up WHICH ONE DOESN T BELONG TILINGS ZEARN MATH TEACHER LESSON MATERIALS Instructional Routine Which One Doesn t Belong This warm up prompts students to compare four geometric patterns explain their reasoning and hold mathematical conversations It allows you to hear how students use terminology in describing geometric characteristics YOUR NOTES Observing patterns gives every student an entry point Each figure has at least one reason it does not belong The patterns also urge students to think about shapes that cover the plane without gaps and overlaps which supports future conversations about the meaning of area Before students begin consider establishing a small discreet hand signal that students can display to indicate they have an answer they can support with reasoning This signal could be a thumbs up a certain number of fingers that tells the number of responses they have or another subtle signal This is a quick way to see if students have had enough time to think about the problem It also keeps students from being distracted or rushed by hands being raised around the class Anticipate students to describe the patterns in terms of Colors blue yellow white or no color Size of shapes or other measurements Geometric shapes polygons squares pentagons hexagons etc Relationships of shapes whether each side of the polygons meets the side of another polygon what polygon is attached to each side whether there is a gap between polygons etc LAUNCH Arrange students in groups of 2 4 Display the four patterns for all to see Give students quiet think time and ask them to indicate when they have noticed one pattern that does not belong and can explain why Encourage them to think of more than one possibility After students have had time to think give them time to share their response with their group and then together find at least one reason if possible that each pattern doesn t belong 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 3

G6M1 LESSON 1 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 Your teacher will show you 4 patterns Which pattern doesn t belong YOUR NOTES STUDENT RESPONSE Answers vary Sample responses A It doesn t have any yellow Groups of four pentagons make hexagonal shapes that interlock without gaps B It doesn t have any blue Groups of six pentagons make flower like shapes that interlock without gaps C It doesn t have any pentagons It has octagons and squares The polygons that make up the patterns are very different in size D It has gaps between the shapes Not all of the colored polygons meet another colored polygon on all sides It has white or non filled shapes that are more complex than other colored shapes It is the only one where all the polygons have the same side length 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 1 DISCUSSION GUIDANCE ZEARN MATH TEACHER LESSON MATERIALS After students shared their observations in groups invite each group to share one reason why a particular figure might not belong Record and display the responses for all to see After each response poll the rest of the class to see if others made the same observation YOUR NOTES Since there is no single correct answer to the question of which pattern does not belong attend to students explanations and make sure the reasons given are correct Prompt students to explain the meaning of any terminology they use names of polygons or angles parts of polygons area etc and to substantiate their claims For example a student may claim that Pattern D does not belong because its polygons all have the same side length Ask how they know that is the case and whether that is true for the white or non filled polygon Explain to students that covering a two dimensional region with copies of the same shape or shapes such that there are no gaps or overlaps is called tiling the plane Patterns A B and C are examples of tiling Tell students that we explore more tilings in upcoming activities Concept Exploration Activity 1 MORE GRAY BLACK OR WHITE Instructional Routines Anticipate Monitor Select Sequence Connect Think Pair Share MLR2 Collect and Display This activity asks students to compare the amounts of the plane covered by two tiling patterns with the aim of supporting two big ideas of the mission If two figures can be placed one on top of the other so that they match up exactly then they have the same area A region can be decomposed and rearranged without changing its area Students are likely to notice that in each pattern The same three polygons triangles rhombuses and trapezoids are used as tiles The shapes are arranged without gaps and overlaps but their arrangements are different A certain set of smaller tiles form a larger hexagon Each hexagon has 3 trapezoids 4 rhombuses and 7 triangles The entire tiling pattern is composed of these hexagons 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 5

G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Expect some students to begin their comparison by counting each shape either within a hexagon or the entire pattern To effectively compare how much of the plane is covered by each shape however they need to be aware of the relationships between the shapes For example two white triangles can be placed on top of a gray rhombus so that they match up exactly which tells us that two white triangles cover the same amount of the plane as one gray rhombus Monitor for such an awareness It is not necessary for students to use the word area in their explanations At this point phrases such as they match up or two triangles make one rhombus suffice If students are not sure how to approach the questions encourage them to think about whether any tools in their geometry toolkits could help For example they could use tracing paper to trace entire patterns or certain shapes to make comparisons or use a straightedge to extend lines within the pattern Some students may be inclined to cut out and compare the shapes Pattern tiles if available can be offered as well During the partner discussion monitor for groups who discuss the following ideas so that they can share later in this sequence Relationships between two shapes E g 2 triangles make a rhombus and 3 triangles make a trapezoid Relative overall quantities E g there are 56 white triangles 32 gray rhombuses which have the same area as 64 triangles and 24 black trapezoids which have the same area as 72 triangles so there is more black Relative quantities in a hexagon E g in each hexagon there are 7 white triangles 4 rhombuses which have the same area as 8 triangles and 3 trapezoids which have the same area as 9 triangles LAUNCH Arrange students in groups of 2 Ask one partner to analyze Pattern A and the other to analyze Pattern B Tell students that their job is to compare the amount of the plane covered by each shape in the pattern Before students begin introduce students to the geometry toolkits and explain that they can use the toolkits for help if needed Give students plenty of quiet think time Then ask students to share their responses with their partners and follow with a group discussion 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 1 ACTIVITY 1 TASK 1 2 ZEARN MATH TEACHER LESSON MATERIALS Your teacher will assign you to look at Pattern A or Pattern B YOUR NOTES In your pattern which shape covers more of the plane gray rhombuses black trapezoids or white triangles Explain how you know Pattern A Pattern B STUDENT RESPONSE In both Patterns A and B more of the plane is covered by black trapezoids than white triangles or gray rhombuses Possible explanations Patterns A and B are each made of 56 white triangles 32 gray rhombuses 24 black trapezoids One black trapezoid covers the same amount of the plane as 3 white triangles so 24 black trapezoids cover the same amount of the plane as 72 white triangles which are more than the 56 white triangles Each gray rhombus covers the same amount of the plane as 2 white triangles so the 32 rhombuses cover the same amount of the plane as 64 white triangles which are also more than the 56 white triangles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 7

G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS Each pattern is composed up of 8 hexagons In each hexagon there are 3 black trapezoids 4 gray rhombuses and 7 white triangles Two black trapezoids can be arranged into a small hexagon Three rhombuses can also be arranged into the same small hexagon This means 2 trapezoids cover the same amount of the plane as 3 rhombuses Each large hexagon has 3 black trapezoids and 4 gray rhombuses Since 2 trapezoids are equal to 3 rhombuses we can just compare 1 trapezoid and 1 rhombus We can see that 1 black trapezoid covers more of the plane than 1 rhombus Each large hexagon has 3 black trapezoids and 7 white triangles One trapezoid covers the same amount of the plane as 3 triangles so 3 trapezoids cover the same amount of the plane as 9 triangles which are more than 7 white triangles YOUR NOTES DISCUSSION GUIDANCE Select previously identified students or groups to share their answers and explanations Sequence the explanations in the order listed in the activity narrative Then make it explicit that when we ask Which type of shape covers more of the plane we are asking them to compare the areas covered by the different types of shapes To recast the comparisons of the shapes in terms of area ask questions such as How does the area of the trapezoid compare to the area of the triangle The area of the trapezoid is three times the area of the triangle How does the area of the rhombus compare to the area of the triangle The area of the rhombus is twice the area of the triangle Is it possible to compare the area of the rhombuses in Pattern A and the area of the triangles in Pattern B How Yes we can count the number of rhombuses in A and the number of triangles in B Because 2 triangles have the same area as 1 rhombus we divide the number of triangles by 2 to compare them ANTICIPATED MISCONCEPTIONS Students may say more of the area is covered by the color they see the most in each image saying for example It just looks like there is more black Ask these students if there is a way to prove their observations Students may only count the number of white triangles black trapezoids and gray rhombuses but not account for the area covered by each shape If they suggest that the shape with the largest number of pieces covers the most amount of the plane ask them to test their hypothesis For example ask Do 2 triangles cover more of the plane than 1 trapezoid 8 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Students may not recall the terms trapezoid rhombus and triangle Consider reviewing the terms although they do not need to know the formal definitions to work on the task G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS SUPPORT FOR ENGLISH LANGUAGE LEARNERS YOUR NOTES Conversing Speaking Listening Math Language Routine 2 Collect and Display This is the first time Math Language Routine 2 is suggested as a support in this course In this routine the teacher circulates and listens to students talk while writing down the words phrases or drawings students produce The language collected is displayed visually for the whole class to use throughout the lesson and mission Generally the display contains different examples of students using features of the disciplinary language functions such as interpreting justifying or comparing The purpose of this routine is to capture a variety of students words and phrases in a display that students can refer to build on or make connections with during future discussions and to increase students awareness of language used in mathematics conversations Design Principle s Support sense making How It Happens 1 After assigning students to work on Pattern A or B circulate around the room and collect examples of language students are using to compare areas of polygons Focus on capturing a variety of language describing the relationship between the size of two shapes comparing overall quantities of shapes to equivalent areas of other shapes and comparing relevant quantities in a hexagon Aim to capture a range of student language that includes formal precise complete ideas and informal ambiguous and partial ideas Plan to publicly update and revise this display throughout the lesson and mission If pairs are stuck consider using these questions to elicit conversation How many triangles rhombuses and trapezoids are in each pattern Three triangles is equivalent to how many trapezoids and Which shapes make up a hexagon 2 Create a display that includes visual representations of the words and phrases collected Group language about Pattern A on one side of the display and language about Pattern B on the other side 3 Close this conversation by posting the display in the front of the classroom for students to reference for the remainder of the lesson and then have students move on to discussing other aspects of the activity Continue to publicly update and revise the display throughout the lesson and mission SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives For students who benefit from hands on activities provide pattern tiles or pre cut an extra copy of each pattern for students to compare the shapes 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 9

G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS Digital Lesson Marvin says that there are more white triangles than black squares so the white triangles must cover more area than the black squares YOUR NOTES Do you agree or disagree with Marvin Explain your reasoning I agree disagree with Marvin because STUDENT RESPONSE I disagree with Marvin I disagree with Marvin because the black squares take up more area than the white triangles 2 white triangles take up the same area as 1 black square so the 8 white triangles in the figure take up the same area as 4 black squares Since there are 6 black squares the white triangles take up less area than the black squares Wrap Up In this lesson we have started to reason about what it means for two shapes to have the same area We started doing mathematics and thinking about tools that can help us LESSON SYNTHESIS Ask students What are some of the tools in the geometry toolkit and what are they used for Draw two shapes that you know do not have the same area How can you tell Tell students that we will continue to think about area to do and talk about mathematics and to learn to use tools strategically 10 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS TERMINOLOGY Area Area is the number of square units that covers a two dimensional region without any gaps or overlaps YOUR NOTES For example the area of region A is 8 square units The area of the shaded region of B is 12 square unit A B Region A region is the space inside of a shape Some examples of two dimensional regions are inside a circle or inside a polygon Some examples of three dimensional regions are the inside of a cube or the inside of a sphere EXIT TICKET The purpose of this exit ticket is to check how students are thinking about area after engaging in the activities While the task prompts students to reflect on the work in this lesson ideas about area from students prior work in grades 3 5 may also emerge Knowing the range of student thinking will help to inform the next day s lesson TASK Think about your work today and write your best definition of area STUDENT RESPONSE Answers vary Sample responses The amount of space inside a two dimensional shape The measurement of the inside of a polygon The number of square units inside a shape 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 11

G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 1 LESSON 2 Finding Area by Decomposing and Rearranging Calculate the area of a region by decomposing it and rearranging the pieces and explain orally and in writing the solution method LEARNING GOALS Recognize and explain orally that if two figures can be placed one on top one other so that they match up exactly they must have the same area Show that area is additive by composing polygons with a given area LEARNING GOALS STUDENT FACING Let s create shapes and find their areas I know what it means for two figures to have the same area LEARNING TARGETS STUDENT FACING I can explain how to find the area of a figure that is composed of other shapes I know how to find the area of a figure by decomposing it and rearranging the parts Template for Concept Exploration Activity 1 REQUIRED MATERIALS Pre assembled or commercially produced tangrams Geometry toolkits including tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles REQUIRED PREPARATION Prepare 1 set of tangrams that contains 1 square 4 small 1 medium and 2 large right triangles for every 2 students Print and cut out the template printing on card stock is recommended or use commercially available tangrams Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram Make sure students have access to their geometry toolkits which should include tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 12 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

This lesson begins by revisiting the definitions for area that students learned in earlier grades The goal here is to refine their definitions and come up with one that can be used by the class for the rest of the mission They also learn to reason flexibly about two dimensional figures to find their areas and to communicate their reasoning clearly The area of two dimensional figures can be determined in multiple ways We can compose that figure using smaller pieces with known areas We can decompose a figure into shapes whose areas we can determine and add the areas of those shapes We can also decompose it and rearrange the pieces into a different but familiar shape so that its area can be found The two key principles in this lesson are G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Figures that match up exactly have equal areas If two figures can be placed one on top of the other such that they match up exactly then they have the same area A figure can be decomposed and its pieces rearranged without changing its area The sum of the areas of the pieces is equal to the area of the original figure Likewise if a figure is composed of non overlapping pieces its area is equal to the sum of the areas of the pieces In other words area is additive Students have used these principles since grade 3 but mainly to decompose squares rectangles and their composites e g an L shape and rearrange them to form other such figures A note about two figures that match up exactly In grade 8 students will learn to refer to such figures as congruent and to describe congruence in terms of rigid motions reflections rotations and translations In these materials the word congruent is not used in grade 6 A possibility is to use an informal term such as identical so that students can talk about one figure being an identical copy of another What identical means however might also require clarification e g that it is independent of color and orientation Warm Up WHAT IS AREA Instructional Routine Think Pair Share This warm up activates and refines students prior knowledge of area It prompts students to articulate a definition of area that can be used for the rest of the mission This definition of area is not new but rather reiterates what students learned in grades 3 5 Before this lesson students explored tiling and tile patterns Here they analyze four ways a region is being tiled or otherwise fitted with squares They decide which arrangements of squares can be used to find the area of the region and why and use their analysis to write a 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 13

G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS definition of area In identifying the most important aspects that should be included in the definition students attend to precision Students initial definitions may be incomplete During partner discussions note students who mention these components so they can share later YOUR NOTES Plane or two dimensional region Square units Covering a region completely without gaps or overlaps Limit the whole class discussion to leave enough time for the work that follows LAUNCH Arrange students in groups of 2 Give students quiet think time for the first question and ask them to be ready to explain their decision Then give partners time to share their responses and to complete the second question together WARM UP TASK 1 14 You may recall that the term area tells us something about the number of squares inside a two dimensional shape Which drawings show squares that could be used to find the area of the shape A B C D 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

1 Select all drawings whose squares could be used to find the area of the shape Be prepared to explain your reasoning G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS 2 Write a definition of area that includes all the information that you think is important STUDENT RESPONSE YOUR NOTES 1 A and D B could be considered if the larger squares and the smaller ones are distinguished when determining area 2 Answers vary but the working definition should contain all of these components The area of a two dimensional region in square units is the number of unit squares that cover the region without gaps or overlaps DISCUSSION GUIDANCE For each drawing in the first question ask students to indicate whether they think the squares could or couldn t be used to find the area From their work in earlier grades students are likely to see that the number of squares in A and D can each tell us about the area Given the recent work on tiling students may decide that C is unhelpful Discuss students decisions and ask What is it about A and D that can help us find the area The squares are all the same size They are unit squares What is it about C that might make it unhelpful for finding area The squares overlap and do not cover the entire region so counting the squares won t give us the area If you think B cannot be used to find area why not We can t just count the number of squares and say that the number is the area because the squares are not all the same size If you think you can use B to find area how Four small squares make a large square If we count the number of large squares and the number of small squares separately we can convert one to the other and find the area in terms of either one of them If time permits discuss How are A and D different A uses larger unit squares and D uses smaller ones Each size represents a different unit Will they give us different areas They will give us areas in different units such as square inches and square centimeters Select a few groups to share their definitions of area or what they think should be included in the class definition of area The discussion should lead to a definition that conveys key aspects 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 15

G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS of area The area of a two dimensional region in square units is the number of unit squares that cover the region without gaps or overlaps Display the class definition and revisit as needed throughout this mission Tell students this will be a working definition that can be revised as they continue their work in the mission YOUR NOTES ANTICIPATED MISCONCEPTIONS Students may focus on how they have typically found the area of a rectangle by multiplying its side lengths instead of thinking about what the area of any region means Ask them to consider what the product of the side lengths of a rectangle actually tells us For example if they say that the area of a 5 by 3 rectangle is 15 ask what the 15 means Some students may think that none of the options including A and D could be used to find area because they involve partial squares or because the partial squares do not appear to be familiar fractional parts Use of benchmark fractions may help students see that the area of a region could be a non whole number For example ask students if the area of a rectangle could be say 8 12 or 2 14 square units SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Concept Exploration Activity 1 COMPOSING SHAPES MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 1 Instructional Routines Think Pair Share MLR2 Collect and Display In grade 3 students recognized that area is additive They learned to find the area of a rectilinear figure by decomposing it into non overlapping rectangles and adding their areas Here students extend that understanding to non rectangular shapes They compose tangram pieces consisting of triangles and a square into shapes with certain areas The square serves as a unit square Because students have only one square they need to use these principles in their reasoning If two figures can be placed one on top of the other so that they match up exactly then they have the same area 16 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

If a figure is decomposed and rearranged to compose another figure then its area is the same as the area of the original figure G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS Each question in the task aims to elicit discussions about these two principles Though they may seem obvious these principles still need to be stated explicitly at the end of the lesson as more advanced understanding of the area of complex figures depends on them YOUR NOTES The terms compose decompose and rearrange will be formalized in an upcoming lesson but throughout this lesson look for opportunities to demonstrate their use as students describe their work with the tangram pieces When students use make or build break and move around recast their everyday terms using the more formal ones As students work notice how they compose the pieces to create shapes with certain areas Look for students whose reasoning illustrates the ideas outlined in the Discussion Guidance Demonstrate the use of the word compose by repeating students everyday language use and then recast using the formal terms here LAUNCH Give each group of 2 students the following set of tangram pieces from the template or from commercially available sets Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram Square 1 Small triangles 4 Medium triangle 1 Large triangles 2 It is important not to give them more than these pieces Give students quiet think time for the first three questions Ask them to pause afterwards and compare their solutions to their partner s If they created the same shape for each question ask them to create a different shape that has the same given area before moving on Then ask them to work together to answer the remaining questions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 17

G6M1 LESSON 2 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 Use the shapes in front of you to make new shapes in your Zearn Student Notes YOUR NOTES 1 Notice that you can put together two small triangles to make a square What is the area of the square composed of two small triangles Be prepared to explain your reasoning 2 Use your shapes to create a new shape with an area of 1 square unit that is not a square Trace your shape on paper 3 Use your shapes to create a new shape with an area of 2 square units Trace your shape 4 Use your shapes to create a different shape with an area of 2 square units Trace your shape 5 Use your shapes to create a new shape with an area of 4 square units Trace your shape STUDENT RESPONSE 1 The area of the square made from two small triangles is 1 square unit because it is identical to the given square with area 1 square unit Identical means you can put one on top of the other and they match up exactly 2 Any composite of two small triangles small small small small 3 Any composite of four small triangles or two small triangles and one medium triangle Sample responses small l al sm l al sm l al sm small al sm medium l 4 Any composite of four small triangles or two small triangles and one medium triangle 18 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 2 5 Any composite with an area of 4 square units Some possibilities ZEARN MATH TEACHER LESSON MATERIALS medium large small small YOUR NOTES large large DISCUSSION GUIDANCE Invite previously identified students whose work illustrates the ideas shown here to share Name these moves explicitly as they come up compose decompose and rearrange First question Two small triangles can be composed into a square that matches up exactly with the given square piece This means that the two squares the composite and the unit square have the same area Tell students We say that if a region can be placed on top of another region so that they match up exactly then they have the same area Second question Two small triangles can be rearranged to compose a different figure but the area of that composite is still 1 square unit These three shapes each composed of two triangles have the same area If we rotate the first figure it can be placed on top of the second so that they match up exactly The third one has a different shape than the other two but because it is made up of the same two triangles it has the same area small small small small small small Emphasize If a figure is decomposed and rearranged as a new figure the area of the new figure is the same as the area of the original figure Third and fourth questions The composite figures could be formed in several ways with only small triangles with two triangles and a medium triangle or with two small triangles and a square Last question A large triangle is needed here To find its area we need to either compose 4 smaller triangles into a large triangle or to see that the large triangle could be decomposed into 4 smaller triangles which can then be composed into 2 unit squares 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 19

G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS ANTICIPATED MISCONCEPTIONS Students may consider the area to be the number of pieces in the compositions instead of the number of square units Remind them of the meaning of area or prompt them to review the definition of area discussed in the warm up activity YOUR NOTES Because the 2 large triangles in the tangram set can be arranged to form a square students may consider that square to be the square unit rather than the smaller square composed of 2 small triangles Ask students to review the task statement and verify the size of the unit square SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Conversing MLR 2 Collect and Display While students are working circulate and listen to students talk about the ways they describe composing decomposing and rearranging the shapes Write down common phrases you hear students say about each e g building breaking apart moving Write the students words onto a visual display including any pictures or drawing This will help students use mathematical language during their paired and discussions Design Principle s Maximize meta awareness SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Executive Functioning Eliminate Barriers Chunk this task into more manageable parts e g presenting one question at a time which will aid students who benefit from support with organizational skills in problem solving Organize Your Work Have students create a word web or graphic organizer 20 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 2 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS This is the figure from the last problem Explain how you found the area Key 1 square 1 sq unit 2 small triangles 1 sq unit 1 medium triangle 2 small triangles 1 sq unit The area of this figure is YOUR NOTES square units I know this because STUDENT RESPONSE The area of this figure is 7 square units I know this because 1 square is 1 square unit There are 6 small triangles which is 3 square units And there are 3 medium triangles which is 3 square units Since 1 plus 3 plus 3 is 7 the area is 7 square units Wrap Up LESSON SYNTHESIS There are two principles that can help us reason about area 1 If two figures can be placed one on top of the other so that they match up exactly then they have the same area 2 The area of a figure can be found by adding the areas of its parts If we compose put together a new figure from smaller pieces without overlapping them then the sum of the areas of the pieces is the area of the new figure Likewise if we decompose cut or break apart a given figure into pieces then the area of the given figure is the sum of the areas of the pieces Even if we rearrange the pieces the overall area does not change 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 21

G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Here is an example Suppose we know the area of a small triangle and wish to find the area of a large triangle Demonstrate the following using the tangram pieces if possible We can use 4 small triangles to compose a large triangle Here are two ways to do so If we place a large triangle on top of a composition of 4 small triangles and they match up exactly we know that the area of the large triangle is equal to the combined area of 4 small triangles We can decompose the large triangle into 4 small triangles Again we can reason that the area of one large triangle is equal to the combined area of 4 small triangles Suppose we don t know the area of a small triangle but we do know the area of a square that is composed of 2 small triangles We can decompose the large triangle into 4 small triangles and then rearrange them into 2 squares We can reason that the area of the large triangle is equal to the combined area of 2 squares This is because when the 4 rearranged small triangles are placed on top of two squares they match up exactly We will look more deeply into these strategies in the next lesson TERMINOLOGY Area Area is the number of square units that covers a two dimensional region without any gaps or overlaps For example the area of region A is 8 square units The area of the shaded region of B is 12 square unit A B Compose Compose means put together We use the word compose to describe putting more than one figure together to make a new shape 22 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Decompose Decompose means take apart We use the word decompose to describe taking a figure apart to make more than one new shape G6M1 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES EXIT TICKET LAUNCH Give students access to the tangram shapes and geometry toolkits Tell students that this figure is composed of two small right triangles two medium right triangles and a square just like the ones they used earlier Note that students might not at first see the square in the middle as a square or they might think of it a diamond with unequal angles Make sure that everyone understands that squareness does not depend on how we turn the paper A square is a rectangle with all four angles being right angles that has 4 equal sides TASK The square in the middle has an area of 1 square unit What is the area of the entire rectangle in square units Explain your reasoning STUDENT RESPONSE The area is 4 square units Possible strategies Put together the two small triangles to make a square Its area is 1 square unit Decompose each medium triangle into two small triangles that can be arranged as a square Each of these squares has an area of 1 square unit Together with the square in the middle the sum of the areas of these pieces is 4 square units A small triangle has an area of unit 1 1 1 12 12 4 1 2 square unit and a medium triangle has an area of 1 square 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 23

Template for Lesson 2 Concept Exploration Activity 1 page 1 of 1 Blackline Master for Classroom Activity 6 1 2 2 Composing Shapes 24 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 3 Reasoning to Find Area G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Compare and contrast orally different strategies for calculating the area of a polygon LEARNING GOALS Find the area of a polygon by decomposing rearranging subtracting or enclosing shapes and explain orally and in writing the solution method Include appropriate units in spoken and written language when stating the area of a polygon LEARNING GOALS STUDENT FACING Let s decompose and rearrange shapes to find their areas LEARNING TARGETS STUDENT FACING I can use different reasoning strategies to find the area of shapes Copies of Template for Warm Up REQUIRED MATERIALS REQUIRED PREPARATION Geometry toolkits including tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Make sure students have access to items in their geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles For the warm up activity prepare several copies of the pair of figures on the template in case students propose cutting them out to compare the areas 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 25

G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS This lesson is the third of three lessons that use the following principles for reasoning about figures to find area If two figures can be placed one on top of the other so that they match up exactly then they have the same area YOUR NOTES If a figure is composed from pieces that don t overlap the sum of the areas of the pieces is the area of the figure If a given figure is decomposed into pieces then the area of the given figure is the sum of the areas of the pieces Following these principles students can use several strategies to find the area of a figure They can Decompose it into shapes whose areas they can calculate Decompose and rearrange it into shapes whose areas they can calculate Consider it as a shape with one or more missing pieces calculate the area of the shape then subtract the areas of the missing pieces Enclose it with a figure whose area they can calculate consider the result as a region with missing pieces and find its area using the previous strategy Use of these strategies involves looking for and making use of structure explaining them involves constructing logical arguments For now rectangles are the only shapes whose areas students know how to calculate but the strategies will become more powerful as students repertoires grow This lesson includes one figure for which the enclosing strategy is appropriate however that strategy is not the main focus of the lesson and is not included in the list of strategies at the end Note that these materials use the dot notation for example 2 3 to represent multiplication instead of the cross notation for example 2 3 This is because students will be writing many algebraic expressions and equations in this course sometimes involving the letter x used as a variable This notation will be new for many students and they will need explicit guidance in using it Warm Up COMPARING REGIONS MATERIALS TEMPLATE FOR WARM UP This activity prompts students to use reasoning strategies from earlier lessons to compare the areas of two figures It is also an opportunity to use or introduce tracing paper as a way to illustrate decomposing and rearranging a figure 26 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

As students work look for students who are able to explain or show how they know that the areas are equal Some students may simply look at the figures and say with no justification that they have the same area Urge them to think of a way to show that their conclusion is true G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH YOUR NOTES Give students access to their geometry toolkits and allow for quiet think time Ask them to be ready to support their answer and remind them to use the tools at their disposal Have copies of the template ready for students who propose cutting the figures out for comparison or as a way to differentiate the activity WARM UP TASK 1 Is the area of Figure A greater than less than or equal to the area of the shaded region in Figure B Be prepared to explain your reasoning A B STUDENT RESPONSE The areas are equal Possible strategies Measuring Measure the side lengths of the small unfilled rectangle and the small rectangle that is on the side of Figure B Both rectangles have the same side lengths so their areas are equal This means the rectangle on the side fills the hole in the middle Measure the side lengths of the large shaded square in Figure A and then in Figure B Both have the same side lengths so their areas are equal Using scissors Cut off the little square on the side of Figure B and use it to fill the hole in the middle of Figure B Then you get a square that matches up exactly with Figure A Using tracing paper Trace the boundary of the little square on the side of Figure B move the tracing paper over the unshaded hole and you can see that the little shaded square is exactly 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 27

G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES the same size as the hole If you moved that little shaded square to fill the unshaded hole you would get a big shaded square If you trace the boundary of that big shaded square and put the tracing paper over Figure A you can show that the boundary of that square matches up exactly with Figure A DISCUSSION GUIDANCE Start the discussion by asking the students to indicate which of the three possible responses area of Figure A is greater area of Figure B is greater or the areas are equal they choose Select previously identified students to share their explanations If no student mentioned using tracing paper demonstrate the following Decomposing and rearranging Figure B Place a piece of tracing paper over Figure B Draw the boundary of the small square making a dotted auxiliary line to show its separation from the large square Move the tracing paper so that the boundary of the small square matches up exactly with the boundary of the square shaped hole in Figure B Draw the boundary of the large square Explain that the small square matches up exactly with the hole so we know the small shaded square and the hole have equal area Matching the two figures Move the tracing paper over Figure A so that the boundary of the rearranged Figure B matches up exactly with that of Figure A Say When two figures that are overlaid one on top of another match up exactly their areas are equal Highlight the strategies and principles that are central to this mission Tell students We just decomposed and rearranged Figure B so that it matches up exactly with Figure A When two figures that are overlaid one on top of another match up exactly we can say that their areas are equal ANTICIPATED MISCONCEPTIONS Students may interpret the area of Figure B as the entire region inside the outer boundary including the unfilled square Clarify that we want to compare the areas of only the shaded parts of Figure B and Figure A SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Processing Time Provide the image to students who benefit from extra processing time to review prior to implementation of this activity 28 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 3 Concept Exploration Activity 1 ON THE GRID ZEARN MATH TEACHER LESSON MATERIALS Instructional Routines Anticipate Monitor Select Sequence Connect Think Pair Share MLR3 Critique Correct and Clarify MLR2 Collect and Display YOUR NOTES This activity gives students opportunities to find areas of regions using a variety of strategies When working with a grid students may start by counting squares as they had done in earlier grades However the figures have been chosen to elicit the strategies listed in the Lesson Narrative Figure A can be easily decomposed into rectangles Figure B can be decomposed into rectangles Or more efficiently it can be seen as a square with a missing piece and the area of the inner unshaded square can be subtracted from the area of the larger square Figure C can also be seen as having a missing piece but subtracting the area of the unshaded shape does not work because the side lengths of the inner square are unknown Instead the shaded triangles can be decomposed and rearranged into rectangles Figure D can be decomposed and rearranged into rectangles It can also be viewed as the inner square of Figure C As students work identify students who use these strategies and can illustrate or explain them clearly Ask them to share later Look for at least two strategies being used for each figure one strategy as shown in the Student Response and at least one other LAUNCH Tell students that they will find the areas of various figures on a grid To encourage students to use a more grade appropriate strategy for finding areas show them a strategy from earlier grades As a class find the area of Figure A by counting the squares one by one aloud Confirm that there are 24 square units and then ask students to think about other ways to find the area of Figure A and other figures besides counting each square Arrange students in groups of 2 Ask one partner to start with Figures A and C and the other with B and D Give students quiet think time and provide access to their geometry toolkits Then give them a few minutes to share their responses with their partner Emphasize that as one partner explains the other should listen carefully and see if they agree or disagree with the answer and explanation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 29

G6M1 LESSON 3 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 YOUR NOTES Each grid square is 1 square unit Find the area in square units of each shaded region without counting every square Be prepared to explain your reasoning A B C D STUDENT RESPONSE A Strategies vary A 24 square units Sample strategy Decompose the figure into rectangles One way is shown here 2 6 4 3 24 B 27 square units Sample strategies Decompose the figure into four rectangles Subtract the area of the inner square from the larger square 6 6 3 3 27 C C 16 square units Sample strategies Decompose into right triangles and rearrange into rectangles 2 4 4 2 16 30 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Find the area of Figure D first and then subtract it from the 6 by 6 square D 20 square units Sample strategies B Decompose the shaded square into four right triangles and a 2 by 2 square Rearrange the right triangles into two rectangles that are each 2 units by 4 units with a combined area of 16 square units Adding the area of the small square 4 square units gives a total of 20 square units G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Notice that the shaded square is the inner square of Figure C enclose it in a square as in Figure C and subtract the areas of the four right triangles i e the area of Figure C from the area of the enclosing square 6 6 16 20 DISCUSSION GUIDANCE Discussion should center around how different strategies decomposing rearranging subtracting and enclosing are used to find area For each figure select two students with different strategies to share their work if possible Sequence students presentations so that for each figure a subtracting strategy comes last as that is typically the most challenging Before sharing begins explain to students that they should notice similarities and differences in the strategies shared and be ready to explain them As students share their strategies consider recording the moves on each figure for all to see After each person shares name the strategy and poll the group to see if anyone else reasoned the same way If one of these strategies does not appear in students work illustrate it for the group Decomposing A and B Decomposing and rearranging C and D Subtracting B Enclosing then subtracting D If time permits give partners a minute to talk about the similarities and differences they saw in the strategies used to find the areas of the four figures Consider displaying sentence starters such as The strategies used to find the areas of figures and are alike because The strategies used to find the areas of figures because and are different 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 31

G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Some students may count both complete and partial grid squares instead of looking for ways to decompose and rearrange larger shapes Ask them if they can find a way to find the area by decomposing and rearranging larger pieces The discussion at the end during which everyone sees a variety of strategies is especially important for these students SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Visual Aids Create an anchor chart i e composing decomposing rearranging publicly displaying important definitions rules formulas or concepts for future reference Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allow students who struggle with fine motor skills to dictate how to decompose the figures as needed SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 2 Collect and Display While pairs are working circulate and collect student talk about the approaches to determine the area for each shaded region Someplace where all students can see write down common or important phrases you hear students say about each representation e g I decomposed the area or I rearranged the triangles This will help students use mathematical language during their paired and group discussions Design Principle s Support sense making Cultivate conversation Speaking Math Language Routine 3 Clarify Critique Correct This is the first time Math Language Routine 3 is suggested as a support in this course In this routine students are given an incorrect or incomplete piece of mathematical work This may be in the form of a written statement drawing problem solving steps or another mathematical representation Pairs of students analyze reflect on and improve the written work by correcting errors and clarifying meaning Typical prompts are Is anything unclear or Are there any reasoning errors The purpose of this routine is to engage students in analyzing mathematical thinking that is not their own and to solidify their knowledge through communicating about conceptual errors and ambiguities in language 32 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Design Principle s Maximize meta awareness How It Happens 1 In the discussion for Figure C of this activity provide the following draft explanation I cut a square in the middle then I saw that it was a bunch of triangles so then I figured those out to get my answer Prompt students to identify the ambiguity of this response Ask students What do you think this person is trying to say What is unclear Did the author use any of the strategies we ve been using to find area 2 Give students individual time to respond to the questions in writing and then a few minutes to discuss with a partner As pairs discuss provide these sentence frames for scaffolding I think the author is trying to use the strategy _ because I think what the author meant by figured it out was and The part that is most unclear to me is because Encourage the listener to press for detail by asking follow up questions to clarify the intended meaning of the statement Allow each partner to take a turn as the speaker and listener 3 Invite students to improve the draft response using the target vocabulary and structures The targeted vocabulary includes the names of the strategies decomposing rearranging subtracting enclosing or any combination of them and other terms from this lesson such as alike different area and polygon The targeted structures of the response should include an explanation of each step order time transition words first next then etc and or reasons for decisions made during steps Here is one example of an improved response First I cut out the square in the middle of the shaded regions Next I noticed that all the shaded regions were a bunch of right triangles Then I rearranged the triangles into rectangles Lastly I figured out the area of each rectangle and added them together to get my answer 4 Ask each pair of students to contribute their improved response to a poster the whiteboard or digital projection Call on 2 3 pairs of students to present their response to the whole group and invite the group to make comparisons among the responses shared and their own responses Listen for responses that include the strategy of decomposing and then rearranging the triangles Emphasize that this is one way of figuring out the area of the shaded region 5 Share one more improved response discuss to reach a general understanding and then move on to Figure D G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Concept Exploration Activity 2 OFF THE GRID Instructional Routine MLR7 Compare and Connect In this activity students apply the strategies they learned to find the areas of figures that are not on a grid 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 33

G6M1 LESSON 3 Figure A can easily be decomposed and rearranged ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Figure B can be decomposed and rearranged into rectangles the same as for Figure C of the previous task Students cannot use the strategy of subtracting the area of the inner square from that of the outer square because the side lengths of the inner square are unknown For Figure C students must subtract the area of the inner square from that of the outer square because there is not enough information to decompose and rearrange the shaded regions into rectangles As students discuss their approaches in groups support them in naming the strategies and by asking clarifying questions Notice any groups that may be stuck in a disagreement on the area of a particular figure Identify students who observed that the same area reasoning strategies can be applied both on and off the grid Students may not remember from earlier grades that if the measurements of side lengths of a rectangle are given in a particular unit then the area is given in square units Look for students who have trouble giving the appropriate area units square centimeters for these figures LAUNCH Tell students that they will now find areas of figures that are not on a grid Give students access to their geometry toolkits Allow for quiet think time to find all three areas Then arrange students into groups of 4 and ask each group to discuss their answers and strategies using these guiding questions 1 What units did you use for each area 2 Compare your answers and strategies for finding the area of each figure 34 How are your strategies the same How are they different Which strategies are similar to the ones you used in the previous activity 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 3 ACTIVITY 2 TASK 1 3 ZEARN MATH TEACHER LESSON MATERIALS Find the area of the shaded region s of each figure Explain or show your reasoning YOUR NOTES A B 3 cm 4 cm 5 cm 5 cm 2 cm 4 cm C 2 cm 2 cm 5 cm 3 cm 5 cm 2 cm 4 cm 2 cm 4 cm 2 cm STUDENT RESPONSE Reasoning varies Figure A The area is 15 square centimeters Sample reasoning Decompose the two triangles and rearrange them to form a rectangle with side lengths of 5 centimeters and 3 centimeters Figure B The area is 16 square centimeters Sample reasoning Decompose the triangles and rearrange them to form two rectangles with side lengths of 4 centimeters and 2 centimeters Figure C The area is 21 square centimeters Sample reasoning Subtract the area of the inner square from the area of the outer square 25 4 21 DISCUSSION GUIDANCE Reconvene the group briefly to discuss the question Which strategies are similar to the ones you used in the previous activity Select 1 2 previously identified students to share those who noticed that they decomposed rearranged enclosed and subtracted in both activities Emphasize that the same strategies for finding area can be used whether we use the measurements indicated by a grid or whether the measurements are given directly without a grid ANTICIPATED MISCONCEPTIONS In Figure B students may estimate the side lengths of the inner square so that its area could be subtracted from that of the outer square They may struggle to see how the triangles could be rearranged Suggest that they use tracing paper to help them in their thinking 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 35

G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS In Figures B and C students may confuse finding area with finding perimeter Remind them that area refers to the number of square units it takes to cover a region without gaps or overlaps Students might not be familiar with the symbols that indicate right angles and might think these symbols indicate square units Remind them that those symbols indicate 90 degree angles YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Executive Functioning Eliminate Barriers Chunk this task into more manageable parts e g presenting one question at a time which will aid students who benefit from support with organizational skills in problem solving SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Math Language Routine 7 Compare and Connect This is the first time Math Language Routine 7 is suggested as a support in this course In this routine students are given a problem that can be approached using multiple strategies or representations and are asked to prepare a visual display of their method Students then engage in investigating the strategies by means of a teacher led gallery walk partner exchange group presentation etc compare approaches and identify correspondences between different representations A typical discussion prompt is What is the same and what is different regarding their own strategy and that of the others The purpose of this routine is to allow students to make sense of mathematical strategies by identifying comparing contrasting and connecting other approaches to their own and to develop students awareness of the language used through constructive conversations Design Principle s Maximize meta awareness How It Happens 1 Identify which Figure A B or C generated the most variety among students strategies Invite students to create a visual display showing how they made sense of this figure Students should include these features on their display the drawing of the figure appropriate units a representation of how the area was calculated the name of the strategy used to find the area decomposing rearranging subtracting enclosing or any combination of them 36 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

2 Arrange students in groups of 4 and invite them to investigate each other s work Allow time for each display and signal when it is time to switch Next give each student the opportunity to add detail to their own display for a short time 3 Circulate around the room and invite 2 3 students to present their display to the whole group Be sure to select a variety of strategies 4 After the pre selected students have finished presenting their displays lead a discussion comparing contrasting and connecting the different approaches Consider using these prompts to amplify student language while comparing and contrasting the different approaches Why did different approaches for Figure _ lead to the same outcome What worked well in this approach for Figure _ What did not work well What would make this strategy for Figure _ more complete or easy to understand Consider using these prompts to amplify student language while connecting the different approaches Can you find any connections between the representations Where are units used in each strategy Is it possible to use the strategy of decomposing for this figure What mathematical features do you see present in all of the representations 5 Close the discussion by inviting 3 students to revoice the strategies used in the presentations and then transition back to the Lesson Synthesis and Exit Ticket G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Digital Lesson Without solving explain the steps you would take to find the area of the shaded region in the figure 6 cm 3 cm 6 cm 2 cm First I would Then I would Finally I would 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 37

G6M1 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE Responses vary Here is one possibility First I would find the area of the large rectangle Then I would find the area of the small rectangle Finally I would subtract the area of the small rectangle from the area of the large rectangle to find the area of the shaded region YOUR NOTES Wrap Up This lesson was all about identifying strategies for finding area and applying them to various figures LESSON SYNTHESIS We reasoned about the area of a figure on and off a grid by decomposing it into familiar shapes decomposing it and rearranging the pieces into familiar shapes or considering it as a shape with missing pieces then subtracting the areas of the missing pieces from the area of the shape Ask students to go back through the activities and find problems in which these strategies were used one strategy at a time Tell students we will have lots of opportunity to use these strategies in upcoming lessons EXIT TICKET This task does not explicitly ask students to state area units because one purpose of the task is to assess if students understand what units are appropriate given the information presented LAUNCH Give students access to their geometry toolkits 38 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 3 TASK ZEARN MATH TEACHER LESSON MATERIALS 8 in A maritime flag is shown What is the area of the shaded part of the flag Explain or show your reasoning 6 in YOUR NOTES 6 in 4 in 4 in STUDENT RESPONSE 72 square inches Reasoning varies Sample reasoning If we draw a line down the middle of the shaded area we would have a 4 inch by 12 inch rectangle on the left and two right triangles The 4 by 12 rectangle has an area of 48 square inches The two triangles on the right can be composed into a 4 inch by 6 inch rectangle so their combined area is 24 square inches 48 24 72 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 39

Template for Lesson 3 Warm Up page 1 of 1 40 A B A B 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 4 Parallelograms G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Compare and contrast orally different strategies for determining the area of a parallelogram LEARNING GOALS Describe orally and in writing observations about the opposites sides and opposite angles of parallelograms Explain orally and in writing how to find the area of a parallelogram by rearranging or enclosing it in a rectangle LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS Let s investigate the features and area of parallelograms I can use reasoning strategies and what I know about the area of a rectangle to find the area of a parallelogram I know how to describe the features of a parallelogram using mathematical vocabulary Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Extra copies of Activity 1 and Activity 2 tasks in case students wish to cut the parallelograms out 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 41

G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Students were introduced to parallel lines in grade 4 While students may not have worked explicitly with parallelograms in grades 3 5 students should have learned about and explored quadrilaterals of all kinds In this lesson students analyze the defining attributes of parallelograms observe other properties that follow from that definition and use reasoning strategies from previous lessons to find the areas of parallelograms By decomposing and rearranging parallelograms into rectangles and by enclosing a parallelogram in a rectangle and then subtracting the area of the extra regions students begin to see that parallelograms have related rectangles that can be used to find the area Throughout the lesson students encounter various parallelograms that because of their shape encourage the use of certain strategies For example some can be easily decomposed and rearranged into a rectangle Others such as ones that are narrow and stretched out may encourage students to enclose them in rectangles and subtract the areas of the extra pieces two right triangles After working with a series of parallelograms students attempt to generalize informally the process of finding the area of any parallelogram Note that these materials use the dot notation for example 2 3 to represent multiplication instead of the cross notation for example 2 3 This is because students will be writing many algebraic expressions and equations in this course sometimes involving the letter x used as a variable This notation will be new for many students and they will need explicit guidance in using it Warm Up FEATURES OF A PARALLELOGRAM Instructional Routine Notice and Wonder Prior to grade 6 students learned that lines are parallel if they never intersect In this activity students look at the defining attributes of parallelograms a quadrilateral whose opposite sides are parallel They observe other properties that follow from that definition that opposite sides of a parallelogram have the same length and opposite angles have the same measure Students initial investigation of parallelograms should involve lots of examples and nonexamples giving them opportunities to look for and express regularity in repeated reasoning and seek and make use of structure This activity assumes that students have had some exposure to parallelograms but is also accessible to students who have not 42 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 4 LAUNCH ZEARN MATH TEACHER LESSON MATERIALS Display the image of figures A F for all to see Give students time to observe it and to prepare to share at least one thing they notice and one thing they wonder Invite students to share their responses with the class YOUR NOTES Students may notice that all except one figure E are quadrilaterals figures A and B have two pairs of equal sides figures B and C are rectangles none of the sides in D are parallel two of the sides in F are parallel They may wonder why a hexagon is in the set if the sides of figure A are all equal if figure C is a parallelogram One or more students are likely to mention parallelogram in their observations or questions Tell students that they will look closely at parallelograms in this lesson Read aloud the opening sentences in the task statement Clarify that A B and C are examples of parallelograms and that D E and F are non examples i e they are not parallelograms Arrange students into groups of 2 and give them geometry toolkits Give students quiet think time to complete the task Afterwards give them time to discuss their answers and observations with their partner 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 43

G6M1 LESSON 4 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 Figures A B and C are parallelograms Figures D E and F are not parallelograms YOUR NOTES A D B E C F Study the examples and non examples What do you notice about 1 the number of sides that a parallelogram has 2 opposite sides of a parallelogram 3 opposite angles of a parallelogram STUDENT RESPONSE 1 Parallelograms have four sides 2 Opposite sides of parallelograms are parallel and have equal length 3 Opposite angles of parallelograms have equal size DISCUSSION GUIDANCE Ask a few students to share their responses to the questions After each response ask students to indicate whether they agree If a student disagrees discuss the disagreement Record the agreed upon responses for all to see and highlight that A parallelogram is a polygon with four sides and both pairs of opposite sides are parallel 44 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Opposite sides have equal length Opposite angles have equal measure G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS Tell students that for now we will just take properties about parallelograms as facts and that later on in their schooling they will learn some ways to prove that they are always true YOUR NOTES If time permits revisit figures D E and F Ask students to explain why these are non examples and see if students connect their explanations to the properties of parallelogram Why is figure D not a parallelogram Why is figure E not a parallelogram What about F ANTICIPATED MISCONCEPTIONS Students may have trouble seeing C as a square because of its orientation They may also think that squares and rectangles are not parallelograms Explain that the definition we are using for parallelograms is a quadrilateral where both pairs of opposite sides are parallel By that definition rectangles and squares are special kinds of parallelograms If students wonder how they would know if two sides are parallel explain that a consequence of never intersecting is that the length of a perpendicular line segment between them always has the same length Students can use an index card to check this in figures A and C SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Graphic Organizers Provide a t chart for students to record what they notice and wonder prior to being expected to share these ideas with others Executive Functioning Visual Aids Add parallelogram to the classroom vocabulary anchor chart 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 45

G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 1 AREA OF A PARALLELOGRAM Instructional Routine Anticipate Monitor Select Sequence Connect MLR2 Collect and Display YOUR NOTES In this activity students explore different methods of decomposing a parallelogram and rearranging the pieces to find its area Presenting the parallelograms on a grid makes it easier for students to see that the area does not change as they decompose and rearrange the pieces This investigation lays a foundation for upcoming work with area of triangles and other polygons Here are the two key approaches for finding the area of parallelograms Decompose the parallelogram rearrange the parts into a rectangle and multiply the side lengths of the rectangle to find the area Enclose the parallelogram in a rectangle and subtract the combined area of the extra regions As students work and discuss monitor for these approaches Identify students whose reasoning can highlight the usefulness of a related rectangle for finding the area of a parallelogram Some students may begin by counting squares because it is a strategy used in earlier grades This strategy is not reinforced here Instead encourage students to listen for and try more sophisticated grade appropriate methods shared during the class discussion LAUNCH Arrange students in groups of 2 4 Ask students to find the area of the parallelogram using recently learned strategies and tools Prepare extra copies of the task in case students wish to cut the parallelogram out Give students quiet think time and access to their toolkits Ask them to share their strategies with their group afterwards To encourage students to be mindful of their strategies and to plant the seed for the discussion display and read aloud the following reflection questions before students begin working Why did you decompose the parallelogram the way you did Why did you rearrange the pieces the way you did 46 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 4 ACTIVITY 1 TASK 1 2 ZEARN MATH TEACHER LESSON MATERIALS Find the area of each parallelogram Show your reasoning YOUR NOTES a b STUDENT RESPONSE a 36 square units Possible strategies Count the squares This strategy is not encouraged or to be shared but it may arise 31 1 2 3 4 5 6 7 8 9 10 34 32 11 12 13 14 15 16 17 18 19 20 35 33 21 22 23 24 25 26 27 28 29 30 36 Enclose the parallelogram within a rectangle and subtract the extra pieces To subtract the area of the two right triangles students may count the squares or put them together to form a rectangle 6 9 6 3 36 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 47

G6M1 LESSON 4 Decompose the parallelogram as shown and move the right triangle to form a rectangle 6 6 36 Decompose the parallelogram as shown and move the right triangle to form a rectangle 6 6 36 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES b 18 square units Reasonings vary but should involve decomposition and rearrangement or other area reasoning strategies DISCUSSION GUIDANCE Invite selected students to share how they found the area of the first parallelogram Begin with students who decomposed the parallelogram in different ways Follow with students who enclosed the parallelogram and rearranged the extra right triangles As students share display and list the strategies for all to see Restate them in terms of decomposing rearranging and enclosing as needed The list will serve as a reference for upcoming work If one of the key strategies is not mentioned illustrate it and add it to the list Use the reflection questions in the Launch to help highlight the usefulness of rectangles in finding the area of parallelograms SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Listening MLR 2 Collect and Display While pairs are working circulate and listen to student talk about the similarities and differences between the two approaches to finding the area Write down common or important phrases you hear students say about each approach using a visual display with the labels Decompose Rearrange Enclose This will help students use mathematical language during their paired and whole group discussions 48 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Design Principle s Support sense making Maximize meta awareness G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 2 YOUR NOTES LOTS OF PARALLELOGRAMS Instructional Routine Anticipate Monitor Select Sequence Connect MLR3 Clarify Critique Correct In this activity students continue to reason about areas of parallelograms both on and off the grid While there is not one correct way to find the area of a parallelogram each parallelogram here is designed to elicit a particular strategy Parallelograms A and C encourage decomposing and rearranging the parallelograms into a rectangle Parallelogram B is not as easy to decompose and rearrange though some students are likely to first try that approach and may prompt students to enclose the parallelograms and subtract the areas of the extra pieces The grid and its absence allow students to reason concretely and abstractly respectively about the measurements that they need to find the area As students work monitor for uses of the two main strategies decompose and rearrange and enclose and subtract The three measurements in Parallelogram C may give students pause Notice students who are able to reason about why the side that is 4 5 inches long is not needed for finding the area of the parallelogram With repeated reasoning students may begin to see the segments and measurements that tend to be useful in finding area There are no references to bases or heights in this lesson but highlighting the segments and measurements that helped students reason about area here can help to support students future work In the next lesson awareness of the measurements that affect the area of a parallelogram will be formalized LAUNCH Keep students in groups of 2 4 and ask them to find the areas of several more parallelograms Give them quiet think time followed by time to share their strategies with their groups Ask students to attempt at least the first two questions individually before discussing with their group Prepare extra copies of the task in case students wish to cut the parallelograms out and provide access to their geometry toolkits To prepare for the discussion consider asking students to think as they work through the problems about what measurements of the parallelogram seem to be helpful for finding its area 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 49

G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS ACTIVITY 2 TASK 1 3 Find the area of the following parallelograms Show your reasoning YOUR NOTES a b c 4 cm 4 5 cm 6 cm STUDENT RESPONSE a 15 square units Possible strategy Decompose and rearrange the pieces to form a rectangle and multiply the side lengths of the rectangle to find the area 5 3 15 50 3 units 5 units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

b 12 square units Possible strategy Enclose the parallelogram and subtract the area of the extra pieces The area of the extra pieces is found by rearranging the triangles to form a rectangle 6 8 6 6 12 6 units ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 6 units 8 units G6M1 LESSON 4 6 units c 24 square units Possible strategy Decompose the parallelogram and rearrange into a rectangle Multiply the side lengths of the rectangle 6 4 24 6 units 4 units DISCUSSION GUIDANCE For each parallelogram select 1 2 students with differing strategies to share their work with the group starting with the less efficient strategy If an important strategy is not mentioned bring it up and illustrate it Briefly poll students to see how others approached the problem After hearing from students on each problem consider asking questions such as the following Focus the discussion on parallelograms B and C Is Parallelogram A different than others you have seen so far How so Students may answer yes or no but some may see it as different because it has a pair of vertical sides Can it still be decomposed and rearranged into a shape whose area we know how to calculate Yes Which strategy decomposing and rearranging or enclosing and subtracting seems more practical for finding a parallelogram such as B Why Enclosing and subtracting because it can be done in fewer steps Decomposing the figure into small pieces could get confusing and lead to errors If you decomposed C into a right triangle and another shape how do you know that the cut out piece actually fits on the other side given that there s no grid to verify The two opposite sides are of a parallelogram are parallel so the longest slanted side of the right triangle that is rearranged would match up perfectly with the segment on the other side Make available tracing paper or a copy of the drawing for cutting and rearranging physically so students can verify Three measurements are shown for Parallelogram C Which ones did you use Which ones did you not use Why and why not The 4 units and 6 units are side lengths of a rectangle 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 51

G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS that has the same area of the parallelogram If we decompose the parallelogram with a vertical cut and move the piece on the left to the right to make a rectangle the 4 5 unit length is no longer relevant Why did your strategy make the most sense to you for this parallelogram YOUR NOTES To help students make connections and generalize their observations ask questions such as When you decomposed and rearranged the parallelogram into another shape did the area change No Why use a rectangle We know how to find the area of a rectangle we can multiply the two side lengths For those of you who enclosed the parallelogram with a rectangle how did the two right triangles help you They can be combined into a rectangle whose area we can find and subtract from the area of a large rectangle Which measurements or lengths were useful for finding the area of the parallelogram One side length of the parallelogram and the length of a perpendicular segment between that side and the opposite side Which lengths did you not use The other side length ANTICIPATED MISCONCEPTIONS Some students may think that it is not possible to decompose and rearrange Parallelogram A because it has a pair of vertical sides instead of a pair of horizontal sides Rotating their paper 90 degrees and back might help them see that they could still use the same reasoning strategy Or they may find it helpful to first reason about area with the parallelogram rotated 90 degrees and then rotating it back to its original orientation SUPPORT FOR ENGLISH LANGUAGE LEARNERS Writing Speaking MLR 3 Clarify Critique Correct Before students share their work present the following incorrect statement to the class To calculate the area of parallelogram A I counted 3 units on the diagonal side and 5 units on the vertical side Then I multiplied 3 x 5 to get 15 square units Give students a few minutes to correct this statement in writing before sharing their improvements with a partner Identify students who can explain why the length of a diagonal segment cannot be easily used to find the area of a parallelogram Select them to share their reasoning with the class and highlight how they describe measurements that can be helpful for finding the area This will support students use of mathematical language when the class discusses the two main strategies decompose and rearrange and enclose and subtract 52 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Design Principle s Optimize output for explanation Maximize meta awareness G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS Digital Lesson YOUR NOTES This is a parallelogram Without calculating write how you would explain to a friend the steps to take to find the area Consider using either the decompose and rearrange or the enclose and subtract strategies To find the area first I would Then I would Finally I would STUDENT RESPONSE Answers vary Sample response To find the area first I would decompose the parallelogram into a rectangle and 2 triangles Then I would move one of the triangles over so the whole shape formed a rectangle Finally I would find the area of the rectangle by multiplying the length by the width That s the same as the area of the parallelogram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 53

G6M1 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Wrap Up LESSON SYNTHESIS Revisit the definition of a parallelogram with students A parallelogram has four sides The opposite sides of a parallelogram are parallel Remind them that as a result of the sides being parallel it is also true that The opposite sides of a parallelogram have equal length The opposite angles of a parallelogram have equal measure b x y a a y x b Tell students that while we are just taking these properties as facts for now in later grades they will be able to prove these for themselves Briefly revisit the last task displaying for all to see the multiple area strategies students used Point out that in some cases students chose to decompose and rearrange parts and in others they chose to enclose the parallelogram with a rectangle and subtract the area of the extra pieces from the area of the rectangle Ask about a couple of the parallelograms What was it about that parallelogram that prompted that particular choice TERMINOLOGY Parallelogram A parallelogram is a four sided polygon with two pairs of parallel sides Here are two examples of parallelograms 5 4 24 45 54 45 135 4 24 135 27 2 4 9 34 152 8 152 8 9 34 4 27 2 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 4 EXIT TICKET ZEARN MATH TEACHER LESSON MATERIALS This activity sets the stage for the next lesson which formalizes how to find the area of any parallelogram Notice the strategies students are currently using to help make connections to the algebraic expression b h that they will see in the next lesson YOUR NOTES TASK How would you find the area of this parallelogram Describe your strategy STUDENT RESPONSE Answers vary Sample responses Decompose a triangle from one side of the parallelogram and move it to the other side to make a rectangle Multiply the side lengths of the rectangle Draw a rectangle around the parallelogram multiply the side lengths of the rectangle to find the area of the rectangle and subtract the combined area of the triangles that do not belong to the parallelogram Count how many squares are across the bottom of the parallelogram and how many squares tall it is and multiply them 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 55

G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 1 LESSON 5 Bases and Heights of Parallelograms Comprehend the terms base and height to refer to one side of a parallelogram and the perpendicular distance between that side and the opposite side LEARNING GOALS Generalize orally a process for finding the area of a parallelogram using the length of a base and the corresponding height Identify a base and the corresponding height for a parallelogram and understand that there are two different base height pairs for any parallelogram LEARNING GOALS STUDENT FACING Let s investigate the area of parallelograms some more I can identify base and height pairs of a parallelogram LEARNING TARGETS STUDENT FACING I can write and explain the formula for the area of a parallelogram I know what the terms base and height refer to in a parallelogram REQUIRED MATERIALS 56 Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Students begin this lesson by comparing two strategies for finding the area of a parallelogram This comparison sets the stage both for formally defining the terms base and height and for writing a general formula for the area of a parallelogram Being able to correctly identify a baseheight pair for a parallelogram requires looking for and making use of structure The terms base and height are potentially confusing because they are sometimes used to refer to particular line segments and sometimes to the length of a line segment or the distance between parallel lines Furthermore there are always two base height pairs for any parallelogram so asking for the base and the height is not technically a well posed question Instead asking for a base and its corresponding height is more appropriate As students clarify their intended meaning when using these terms they are attending to precision of language In these materials the words base and height mean the numbers unless it is clear from the context that it means a segment and that there is no potential confusion G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES By the end of the lesson students both look for a pattern they can generalize to the formula for the area of a rectangle and make arguments that explain why this works for all parallelograms Warm Up A PARALLELOGRAM AND ITS RECTANGLES Instructional Routine Think Pair Share In this warm up students compare and contrast two ways of decomposing and rearranging a parallelogram on a grid such that its area can be found It serves a few purposes to reinforce the work done in the previous lesson to allow students to practice communicating their observations and to shed light on the features of a parallelogram that are useful for finding area a base and a corresponding height The flow of key ideas to be uncovered during discussion and gradually throughout the lesson is as follows There are multiple ways to decompose a parallelogram with one cut and rearrange it into a rectangle whose area we can determine The cut can be made in different places but to compose a rectangle the cut has to be at a right angle to two opposite sides of the parallelogram The length of one side of this newly composed rectangle is the same as the length of one side of the parallelogram We use the term base to refer to this side The length of the other side of the rectangle is the length of the cut we made to the parallelogram We call this segment a height that corresponds to the chosen base 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 57

G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS We use these two lengths to determine the area of the rectangle and thus also the area of the parallelogram As students work and discuss identify those who make comparisons in terms of the first two points so they could share later Be sure to leave enough time to discuss the first four points as a class YOUR NOTES LAUNCH Arrange students in groups of 2 Give students quiet think time and access to geometry toolkits Ask them to share their responses with a partner afterwards WARM UP TASK 1 Use the diagrams below to answer the questions Elena found the area of a parallelogram like this Tyler found the area of the same parallelogram like this How are the two strategies for finding the area of a parallelogram the same How are they different 58 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE Answers vary Sample responses Similar They both cut off a piece from the left of the parallelogram and moved it over to the right to make a rectangle The rectangles they made are identical YOUR NOTES Different They cut the parallelogram at different places Elena cut a right triangle from the left side and Tyler cut off a trapezoid The rectangles they made are not in the same place DISCUSSION GUIDANCE Ask a few students to share what was the same and what was different about the methods they observed Highlight the following points on how Elena and Tyler s approaches are the same though do not expect students to use the language Instead rely on pointing and gesturing to make clear what is meant If any of these are not mentioned by the students share them The rectangles are identical they have the same side lengths Label the side lengths of the rectangles The cuts were made in different places but the length of the cuts was the same Label the lengths along the vertical cuts The horizontal sides of the parallelogram have the same length as the horizontal sides of the rectangle Point out how both segments have the same length The length of each cut is also the distance between the two horizontal sides of the parallelogram It is also the vertical side length of the rectangle Point out how that distance stays the same across the horizontal length of the parallelogram 7 7 6 6 7 6 6 6 7 7 7 7 6 7 7 6 7 7 6 6 6 7 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 59

G6M1 LESSON 5 Begin to connect the observations to the terms base and height For example explain ZEARN MATH TEACHER LESSON MATERIALS The two measurements that we see here have special names The length of one side of the parallelogram which is also the length of one side of the rectangle is called a base The length of the vertical cut segment which is also the length of the vertical side of the rectangle is called a height that corresponds to that base YOUR NOTES Here the side of the parallelogram that is 7 units long is also called a base In other words the word base is used for both the segment and the measurement Tell students that we will explore bases and heights of a parallelogram in this lesson SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Eliminate Barriers Provide an enlarged version of the visual Conceptual Processing Processing Time Begin with a physical demonstration of how Elena decomposed the parallelogram which will provide access for students who benefit from concrete contexts Concept Exploration Activity 1 THE RIGHT HEIGHT Instructional Routines Notice and Wonder Think Pair Share MLR8 Discussion Supports Previously students saw numerical examples of a base and a height of a parallelogram This activity further develops the idea of base and height through examples and non examples and error analysis Some important ideas to be uncovered here In a parallelogram the term base refers to the length of one side and height to the length of a perpendicular segment between that side and the opposite side Any side of a parallelogram can be a base There are always two base height pairs for a given parallelogram 60 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 5 LAUNCH Display the image of examples and non examples of bases and heights for all to see Read aloud the description for examples and non examples Give students time to observe it and to prepare to share at least one thing they notice and one thing they wonder about Invite students to share their responses with the class and record these for all to see It isn t necessary to address their questions at this time ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Students may notice Both sets of diagrams show the same 2 pairs of parallelograms and the same sides labeled base All the examples show a right angle mark a dashed segment and a side labeled base Only one of the non examples show a right angle mark but all of them show a dashed segment In both examples and non examples there is one parallelogram with a dashed segment and a right angle shown outside of it If the dashed segments are used to cut the first three parallelograms in the examples the cut out pieces could be rearranged to form a rectangle The same cannot be done for the dashed segments in the non examples They may wonder why some dashed segments are inside the parallelogram and some are outside what the rule might be for a dashed segment to be considered a height what the bases and heights have to do with area Arrange students in groups of 2 Give students time to complete the first question with their partner Ask them to pause for a group discussion after the first question Select a student or a group to make a case for whether each statement is true or false If one or more students disagree ask them to explain their reasoning and discuss to reach a consensus Before moving on to the next question be sure students record the verified true statements so that they can be used as a reference later Give students quiet time to answer the second question and then time to share their responses with a partner Ask them to focus partner conversations on the following questions displayed for all to see How do you know the parallelogram is labeled correctly or incorrectly Is there another way a base and height could be labeled on this parallelogram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 61

G6M1 LESSON 5 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 YOUR NOTES Each parallelogram has a side that is labeled base Study the examples and non examples of bases and heights of parallelograms Then answer the questions Examples The dashed segment in each drawing represents the corresponding height for the given base Non examples The dashed segment in each drawing does not represent the corresponding height for the given base Base Base Base Base Base Base Base Base Select all statements that are true about bases and heights in a parallelogram a Only a horizontal side of a parallelogram can be a base b Any side of a parallelogram can be a base c A height can be drawn at any angle to the side chosen as the base d A base and its corresponding height must be perpendicular to each other e A height can only be drawn inside a parallelogram f A height can be drawn outside of the parallelogram as long as it is drawn at a 90 degree angle to the base g A base cannot be extended to meet a height STUDENT RESPONSE 1 Statements B D and F are true 62 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 5 ACTIVITY 1 TASK 2 3 ZEARN MATH TEACHER LESSON MATERIALS Five students labeled a base b and a corresponding height h for each of these parallelograms Are all drawings correctly labeled Explain how you know b A YOUR NOTES C B h h h b D b E b h h b STUDENT RESPONSE 2 A C and D are correct B and E are not correct because in each the segment labeled with an h is not perpendicular to the side labeled with a b DISCUSSION GUIDANCE Poll the group with a quick agree or disagree signal on whether each figure in the last question is labeled correctly with b and h After each polling ask a student to explain how they know it is correct or incorrect If a parallelogram is incorrectly labeled ask where a correct height could be If it is correctly labeled ask students if there is another base and height that could be labeled on this parallelogram Be sure students understand which parallelograms are labeled correctly before moving forward in this lesson An important point to emphasize We can choose any side of a parallelogram as a base To find the height that corresponds to that base draw a segment that joins the base and its opposite side that segment has to be perpendicular to both 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 63

G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Students may not yet internalize that any side of parallelogram can be a base they may think that a base must be the bottom horizontal side or that the height needs to be perpendicular to the base Point out where the right angle symbols are located and how they relate to the height Students may think a segment showing the height cannot be drawn outside of the parallelogram as in Parallelogram C Students may relate how they think about the side lengths of a rectangle and inaccurately apply it to Parallelogram E SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Graphic Organizers Provide a t chart for students to record what they notice and wonder prior to being expected to share these ideas with others Conceptual Processing Processing Time Check in with individual students as needed to assess for comprehension during each step of the activity SUPPORT FOR ENGLISH LANGUAGE LEARNERS Listening Speaking Conversing MLR 8 Discussion Supports Use this support to practice verbal use of the mathematical language base and height As students are comparing parallelograms and examples non examples of correct heights encourage them to re voice their partners reasoning Reinforce the meaning of perpendicular by using visuals e g manipulatives drawings gestures of right angles Design Principle s Support sense making 64 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 5 Concept Exploration Activity 2 FINDING THE FORMULA FOR AREA OF PARALLELOGRAMS ZEARN MATH TEACHER LESSON MATERIALS Instructional Routines Think Pair Share MLR7 Compare and Connect In previous lessons students reasoned about the area of parallelograms by decomposing rearranging and enclosing them and by using what they know about the area of rectangles They also identified base height pairs in parallelograms Here they use what they learned to find the area of new parallelograms generalize the process and write an expression for finding the area of any parallelogram YOUR NOTES As students discuss their work monitor conversations for any disagreements between partners Support them by asking clarifying questions How did you choose a base How can you be sure that is the height How did you find the area Why did you choose that strategy for this parallelogram Is there another way to find the area and to check our answer LAUNCH Keep students in groups of 2 Give students access to their geometry toolkits and quiet think time to complete the first four rows of the table Ask them to be prepared to share their reasoning If time is limited consider splitting up the work have one partner work independently on parallelograms A and C and the other partner on B and D Encourage students to use their work from earlier activities on bases and heights as a reference Ask students to pause after completing the first four rows and to share their responses with their partner Then they should discuss how to write the expression for the area of any parallelogram Students should notice that the area of every parallelogram is the product of a base and its corresponding height ACTIVITY 2 TASK 1 43 Find the base height and area of these parallelograms For each parallelogram Identify a base and a corresponding height and record their lengths in the table that follows 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 65

G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS Find the area and record it in the right most column In the last row write an expression using b and h for the area of any parallelogram A B C D YOUR NOTES Parallelogram Base units Height units b h Area sq units A B C D Any parallelogram STUDENT RESPONSE While there are two possible base height pairs these are the easiest ones for students to use given the orientation of each parallelogram on the grid Parallelogram Base units Height units Area sq units A 6 or 4 4 or 6 24 B 5 3 15 C 2 3 6 D 4 2 8 Any parallelogram b h b h DISCUSSION GUIDANCE Display the parallelograms and the table for all to see Select a few students to share the correct answers for each parallelogram As students share highlight the base height pairs on each parallelogram and record the responses in the table Although only one base height pair is 66 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

named for each parallelogram reiterate that there is another pair Show the second pair on the diagram or ask students to point it out G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS After all answers for the first four rows are shared discuss the following questions displayed for all to see How did you determine the expression for the area for any parallelogram The areas of parallelograms A D are each the product of base and height YOUR NOTES Suppose you decompose a parallelogram with a cut and rearrange it into a rectangle Does this expression for finding area still work Why or why not Yes One side of the rectangle will be the same as the base of the parallelogram The height of the parallelogram is also the height of the rectangle both are perpendicular to the base Do you think this expression will always work Be sure everyone has the correct expression for finding the area of a parallelogram by the end of the discussion The second discussion question is meant to elicit connections to the parallelogram s related rectangle as they decomposed and rearranged to find the area The third question about whether the expression will always work is not meant to be proven here so speculation on students part is expected at this point It is intended to prompt students to think of other differently shaped parallelograms beyond the four shown here ANTICIPATED MISCONCEPTIONS Finding a height segment outside of the parallelogram may still be a rather unfamiliar idea to students Have examples from the The Right Height section visible so they can serve as a reference in finding heights Students may say that the base of Parallelogram D cannot be determined because as displayed it does not have a horizontal side Remind students that in an earlier activity we learned that any side of a parallelogram could be a base Ask students to see if there is a side whose length can be determined SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing Conversing MLR 7 Compare and Connect When students share their responses for the first four rows of the table ask students to identify what is the same and what is different about how they determined the base and height for each parallelogram Listen for and amplify student discussions that attempt to explain why their different approaches led to the same area Then ask students if any side of a parallelogram can be used as the base This will help students understand how the area of every parallelogram is product of the base and its corresponding height 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 67

G6M1 LESSON 5 Design Principle s Support sense making ZEARN MATH TEACHER LESSON MATERIALS SUPPORT FOR STUDENTS WITH DISABILITIES YOUR NOTES Conceptual Processing Visual Aids Use color coding and annotations to highlight important concepts or connections between representations in a problem Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Digital Lesson Draw a height that corresponds to the base labeled on the parallelogram b Explain how you identified the height I know the height needs to be perpendicular parallel to the base so first I STUDENT RESPONSE Height should be drawn perpendicular to side labeled b and the side opposite it Explanations vary Sample response I know the height needs to be perpendicular to the base So first I used a corner to draw a line perpendicular to the side labeled b I made that height go the entire distance between the base and the opposite side 68 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 5 Wrap Up ZEARN MATH TEACHER LESSON MATERIALS LESSON SYNTHESIS YOUR NOTES In this lesson we identified a base and a corresponding height in a parallelogram and then wrote an algebraic expression for finding the area of any parallelogram How do you decide the base of a parallelogram Any side can be a base Sometimes one side is preferable over another because its length is known or easy to know Once we have chosen a base how can we identify a height that corresponds to it Identify a perpendicular segment that connects that base and the opposite side find the length of that segment In how many ways can we identify a base and a height for a given parallelogram There are two possible bases For each base many possible segments can represent the corresponding height What is the relationship between the base and height of a parallelogram and its area The area is the product of base and height GLOSSARY Base of a parallelogram or triangle We can choose any side of a parallelogram or triangle to be the shape s base Sometimes we use the word base to refer to the length of this side Base Base Base Height of a parallelogram or triangle The height is the shortest distance from the base of the shape to the opposite side for a parallelogram or opposite vertex for a triangle We can show the height in more than one place but it will always be perpendicular to the chosen base Height 2 Height 1 Base 2 Base 1 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 69

G6M1 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS EXIT TICKET Parallelograms S and T are each labeled with a base and a corresponding height S YOUR NOTES T h b h b 1 What are the values of b and h for each parallelogram Parallelogram S b h Parallelogram T b h 2 Use the values of b and h to find the area of each parallelogram Area of Parallelogram S Area of Parallelogram T STUDENT RESPONSE 1 Parallelogram S b 7 h 6 Parallelogram T b 3 h 6 Area of Parallelogram S 42 square units 7 6 42 Area of Parallelogram T 18 square units 3 6 18 2 70 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 6 Area of Parallelograms G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES LEARNING GOALS Apply the formula for area of a parallelogram to find the area the length of the base or the height and explain orally and in writing the solution method Choose which measurements to use for calculating the area of a parallelogram when more than one base or height measurement is given and explain orally and in writing the choice LEARNING GOALS STUDENT FACING Let s practice finding the area of parallelograms LEARNING TARGETS STUDENT FACING I can use the area formula to find the area of any parallelogram REQUIRED MATERIALS Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 71

G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES This lesson allows students to practice using the formula for the area of parallelograms and to choose the measurements to use as a base and a corresponding height Through repeated reasoning they see that some measurements are more helpful than others For example if a parallelogram on a grid has a vertical side or horizontal side both the base and height can be more easily determined if the vertical or horizontal side is used as a base Along the way students see that parallelograms with the same base and the same height have the same area because the products of those two numbers are equal even if the parallelograms look very different This gives us a way to use given dimensions to find others Warm Up MISSING DOTS Instructional Routine MLR8 Discussion Supports In this warm up students determine the number of dots in an image without counting and explain how they arrive at that answer The activity also gives students a chance to use decomposition and structure to quantify something in a setting that is slightly different than what they have seen in this mission To arrive at the total number of dots students need to visualize and articulate how the dots can be decomposed and use what they know about arrays multiplication and area to arrive at the number of interest To encourage students to refer to the image in their explanation ask students how they saw the dots instead of how they found the number of dots As in an earlier warm up consider establishing a small discreet hand signal that students can display to indicate that they have an answer they can support with reasoning This signal could be a thumbs up a certain number of fingers that tells the number of responses they have or another subtle signal This is a quick way to see if the students have had enough time to think about the problem It also keeps students from being distracted or rushed by hands being raised around the class LAUNCH Give students quiet think time and ask them to give a signal showing how many solutions they have Encourage students who have found one way of seeing the dots to think about another way while they wait 72 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 6 WARM UP TASK 1 ZEARN MATH TEACHER LESSON MATERIALS How many dots are in the image How do you see them YOUR NOTES STUDENT RESPONSE 30 dots Strategies vary Sample strategies Decomposing the image into parts then multiplying and adding 2 6 3 4 6 30 3 6 3 2 6 30 Multiply to find the dots in the entire array and subtract the missing array of dots 6 6 2 3 30 DISCUSSION GUIDANCE Ask students to share how many dots they saw and how they saw them Record and display student explanations for all to see To involve more students in the conversation consider asking some of the following questions Who can restate the way saw the dots in different words 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 73

G6M1 LESSON 6 Did anyone see the dots the same way but would explain it differently ZEARN MATH TEACHER LESSON MATERIALS Does anyone want to add an observation to the way saw the dots Do you agree or disagree Why YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Visual Spatial Processing Visual Aids Suggest students use their own copy of the task in their workbook tablet or provide them with a copy of the task to draw on or highlight SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Provide sentence frames to support students with explaining their strategies For example I noticed that ______ or First I ________ because ________ When students share their answers with a partner prompt them to rehearse what they will say when they share with the full group Rehearsing provides opportunities to clarify their thinking Design Principle s Optimize output for explanation Concept Exploration Activity 1 MORE AREAS OF PARALLELOGRAMS Instructional Routine MLR8 Discussion Supports This activity allows students to practice finding and reasoning about the area of various parallelograms on and off a grid Students need to make sense of the measurements and relationships in the given figures identify an appropriate pair of base height measurements to use and recognize that two parallelograms with the same base height measurements or with different base height measurements but the same product have the same area As they work individually notice how students determine base height pairs to use As they work in groups listen to their discussions and identify those who can clearly explain how they found the area of each of the parallelograms 74 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Arrange students in groups of 4 Give each student access to their geometry toolkits and quiet time to find the areas of the parallelograms in the first question Then assign each student one parallelogram A B C or D Ask each student to explain to the group one at a time how they found the area of the assigned parallelogram After each student shares check for agreement or disagreement from the rest of the group Discuss any disagreement and come to a consensus on the correct answer before moving to the next parallelogram YOUR NOTES Afterwards give students more quiet work time to complete the rest of the activity ACTIVITY 1 TASK 1 2 Find the area of each parallelogram Show your reasoning b a 6 cm 8 cm 10 cm 15 cm c 10 cm 9 cm d 7 cm 1 cm 1 cm 8 cm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 75

G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 a 10 6 60 square centimeters YOUR NOTES b 15 8 120 square centimeters c 9 7 63 square centimeters d 7 5 35 square centimeters ACTIVITY 1 TASK 2 3 In Parallelogram B what is the corresponding height that is 10 cm long Explain or show your reasoning b 8 cm 15 cm 10 cm STUDENT RESPONSE 2 12 centimeters Sample reasoning We found the area of the parallelogram to be 120 square centimeters If the side that is 10 centimeters is the base then 10 h must equal 120 so the height must be 120 10 or 12 centimeters 76 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 6 ACTIVITY 1 TASK 3 43 Two different parallelograms P and Q both have an area of 20 square units Neither of the parallelograms are rectangles On the grid draw two parallelograms that could be P and Q ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES STUDENT RESPONSE 3 Answers vary Sample responses One parallelogram has a base of 10 units and a height of 2 units another one has a base that is 4 units and a height that is 5 units One parallelogram has a base of 5 units and a height of 4 units another one has a base that is 4 units and a height that is 5 units Two parallelograms with equal base and equal height but with different orientations or with the pair of parallel bases positioned differently DISCUSSION GUIDANCE Use the discussion to draw out three important points 1 We need base and height information to help us calculate the area of a parallelogram so we generally look for the length of one side and the length of a perpendicular segment that connects the base to the opposite side Other measurements may not be as useful 2 A parallelogram generally has two pairs of base and height Both pairs produce the same area it s the same parallelogram so the product of one pair of numbers should equal the product of the other pair 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 77

G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS 3 Two parallelograms with different pairs of base and height can have the same area as long as their products are equal So a 3 by 6 rectangle and a parallelogram with base 1 and height 18 will have the same area because 3 6 1 18 To highlight the first point ask how students decided which measurements to use when calculating area YOUR NOTES When multiple measurements are shown how did you know which of the measurements would help you find area Which pieces of information in parallelograms B and C were not needed Why not To highlight the second point select 1 2 previously identified students to share how they went about finding the missing height in the second question Emphasize that the product 8 15 and that of 10 and the unknown h must be equal because both give us the area of the same parallelogram To highlight the last point invite a few students to share their pair of parallelograms with equal area and an explanation of how they know the areas are equal If not made explicit in students explanations stress that the base height pairs must have the same product ANTICIPATED MISCONCEPTIONS Some students may continue to use visual reasoning strategies decomposition rearranging enclosing and subtracting to find the area of parallelograms This is fine at this stage but to help them gradually transition toward abstract reasoning encourage them to try solving one problem both ways using visual reasoning and their generalization about bases and heights from an earlier lesson They can start with one method and use the other to check their work SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Executive Functioning Eliminate Barriers Chunk this task into more manageable parts e g presenting one question at a time which will aid students who benefit from support with organizational skills in problem solving 78 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Use this routine to support discussion when students explain how they know they created two parallelograms with equal area After students have shared their explanations call on other students to restate and or revoice their peers descriptions using mathematical terms e g area product base height etc If students are not able to restate they should ask for clarification This will provide more students with an opportunity to produce the language needed to describe the final point G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Design Principle s Support sense making Maximize meta awareness Digital Lesson Which side could be the base for this parallelogram What would be the corresponding height Label a base and height and then use the measurements to find the area STUDENT RESPONSE Answers vary A base and corresponding height should be labeled on parallelogram such as a base labeled 11 units and height labeled 9 units Area 99 square units Wrap Up LESSON SYNTHESIS We used the formula for area to practice finding the area of various parallelograms 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 79

G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES When a parallelogram is on a grid how do we know which side to choose for a base Can we use any side It is helpful to use a horizontal or a vertical side as a base it would be easier to tell the length of that side and of its corresponding height Off a grid how do we know which measurements can help us find the area of a parallelogram We need the length of one side of the parallelogram and of a perpendicular segment that connects that side to the opposite side Do parallelograms that have the same area always look the same No Can you show an example Do parallelograms that have the same base and height always look the same No Can you show an example How can we draw two different parallelograms with the same area We can find any two pairs of base height lengths that have the same product We can also use the same pair of numbers by drawing parallelograms differently EXIT TICKET Give students access to geometry toolkits TASK 1 Find the area of the parallelogram Explain or show your reasoning 6 cm 2 Was there a length measurement you did not use to find the area If so explain why it was not used 9 cm 7 5 cm STUDENT RESPONSE 1 54 sq cm A base is 9 cm and its corresponding height is 6 cm 9 6 54 2 The 7 5 cm length was not used Explanations vary Sample explanations 80 If the side that is 7 5 cm was used to find area we would need the length of a perpendicular segment between that side and the opposite side as its corresponding height We don t have that information 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

The parallelogram can be decomposed and rearranged into a rectangle by cutting it along the horizontal line and moving the right triangle to the bottom side Doing this means the side that is 7 5 cm is no longer relevant The rectangle is 6 cm by 9 cm we can use those side lengths to find area G6M1 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 81

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 1 LESSON 7 From Parallelograms to Triangles LEARNING GOALS Understand and explain that any two identical triangles can be composed into a parallelogram Describe how any parallelogram can be decomposed into two identical triangles by drawing a diagonal LEARNING GOALS STUDENT FACING Let s compare parallelograms and triangles LEARNING TARGETS STUDENT FACING I can explain the special relationship between a pair of identical triangles and a parallelogram Rulers REQUIRED MATERIALS Pre printed slips cut from copies of the Template for Concept Exploration Activity 2 Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles REQUIRED PREPARATION 82 Print pairs of triangles from the Template for Concept Exploration Activity 2 If students are cutting out the triangles use the first page only If the triangles are to be pre cut by the teacher print the second and third pages Prepare enough sets so that each group of 3 4 students has a complete set 2 copies each of triangles P U 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

This lesson prepares students to apply what they know about the area of parallelograms to reason about the area of triangles Highlighting the relationship between triangles and parallelograms is a key goal of this lesson The activities make use of both the idea of decomposition of a quadrilateral into triangles and composition of two triangles into a quadrilateral The two way study is deliberate designed to help students view and reason about the area of a triangle differently Students see that a parallelogram can always be decomposed into two identical triangles and that any two identical triangles can always be composed into a parallelogram G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Because a lot happens in this lesson and timing might be tight it is important to both prepare all the materials and consider grouping arrangements in advance Warm Up SAME PARALLELOGRAMS DIFFERENT BASES This warm up reinforces students understanding of bases and heights in a parallelogram In previous lessons students calculated areas of parallelograms using bases and heights They have also determined possible bases and heights of a parallelogram given a whole number area They saw for instance that finding possible bases and heights of a parallelogram with an area of 20 square units means finding two numbers with a product of 20 Students extend that work here by working with decimal side lengths and area As students work notice students who understand that the two identical parallelograms have equal area and who use that understanding to find the unknown base Ask them to share later LAUNCH Give students quiet work time and access to their geometry toolkits Students should be adequately familiar with bases and heights to begin the warm up If needed however briefly review the relationship between a pair of base and height in a parallelogram using questions such as Can we use any side of a parallelogram as a base Yes Is the height always the length of one of the sides of the parallelogram No Once we have identified a base how do we identify a height It can be any segment that is perpendicular to the base and goes from the base to the opposite side Can a height segment be drawn outside of a parallelogram Yes 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 83

G6M1 LESSON 7 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 YOUR NOTES Here are two copies of a parallelogram Each copy has one side labeled as the base b and a segment drawn for its corresponding height and labeled h h b b h 1 The base of the parallelogram on the left is 2 4 meters its corresponding height is 1 meter Find its area in square meters 2 The height of the parallelogram on the right is 2 meters How long is the base of that parallelogram Explain your reasoning STUDENT RESPONSE 1 The area of the first parallelogram is 2 4 square meters 2 4 1 2 4 2 The area of the second parallelogram is also 2 4 square meters Since the base and height must multiply to the same area of 2 4 the base must be 1 2 meters because 1 2 2 2 4 DISCUSSION GUIDANCE Select 1 2 previously identified students to share their responses If not already explained by students emphasize that we know the parallelograms have the same area because they are identical which means that when one is placed on top of the other they would match up exactly Before moving on ask students How can we verify that the height we found is correct or that the two pairs of bases and heights produce the same area We can multiply the values of each pair and see if they both produce 2 4 ANTICIPATED MISCONCEPTIONS Some students may not know how to begin answering the questions because measurements are not shown on the diagrams Ask students to label the parallelograms based on the information in the task statement 84 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Students may say that there is not enough information to answer the second question because only one piece of information is known the height Ask them what additional information might be needed Prompt them to revisit the task statement and see what it says about the two parallelograms Ask what they know about the areas of two figures that are identical Students may struggle to find the unknown base in the second question because the area of the parallelogram is a decimal and they are unsure how to divide a decimal Ask them to explain how they would reason about it if the area was a whole number If they understand that they need to divide the area by 2 since the height is 2 cm see if they could reason in terms of multiplication i e 2 times what number is 2 4 or if they could reason about the division using 4 fractions i e 2 4 can be seen as 2 10 or 24 10 what is 24 tenths divided by 2 G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Processing Time Check in with individual students as needed to assess for comprehension during each step of the activity Receptive Language Processing Time Read all statements aloud Students who both listen to and read the information will benefit from extra processing time Concept Exploration Activity 1 A TALE OF TWO TRIANGLES PART 1 In earlier lessons students saw that a square can be decomposed into two identical isosceles right triangles They concluded that the area of each of those triangles is half of the area of the square They used this observation to determine the area of composite regions This activity helps students see that parallelograms other than squares can also be decomposed into two identical triangles by drawing a diagonal They check this by tracing a triangle on tracing paper and then rotating it to match the other copy The process prepares students to see any triangle as occupying half of a parallelogram and consequently as having one half of its area To generalize about quadrilaterals that can be decomposed into identical triangles students need to analyze the features of the given shapes and look for structure There are a number of geometric observations in this mission that must be taken for granted at this point in students mathematical study This is one of those instances Students have only seen examples of a parallelogram being decomposable into two copies of the same triangle or have only verified this conjecture through physical experimentation but for the time being it 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 85

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS can be considered a fact Starting in grade 8 they will begin to prove some of the observations they have previously taken to be true LAUNCH YOUR NOTES Arrange students in groups of 3 4 Give students access to geometry toolkits and allow for quiet think time for the first two questions Then ask them to share their drawings with their group and discuss how they drew their lines If group members disagree on whether a quadrilateral can be decomposed into two identical triangles they should note the disagreement but it is not necessary to come to an agreement They will soon have a chance to verify their responses Next ask students to use tracing paper to check that the pairs of triangles that they believe to be identical are indeed so i e they would match up exactly if placed on top of one another Tell students to divide the checking work among the members of their group to optimize time Though students have worked with tracing paper earlier in the mission some may not recall how to use it to check the congruence of two shapes some explicit guidance might be needed Encourage students to work carefully and precisely A straightedge can be used in tracing but is not essential and may get in the way Once students finish checking the triangles in their list and verify that they are identical or correct their initial response ask them to answer the last question ACTIVITY 1 TASK 1 2 Two polygons are identical if they match up exactly when placed one on top of the other 1 Draw one line to decompose each of the following polygons into two identical triangles if possible Use a straightedge to draw your line A D 86 B E C F G 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

2 Which quadrilaterals can be decomposed into two identical triangles G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS Pause here for a discussion 3 Study the quadrilaterals that can in fact be decomposed into two identical triangles What do you notice about them Write a couple of observations about what these quadrilaterals have in common YOUR NOTES STUDENT RESPONSE 1 Answers vary Sample response A D B E C F G 2 Answers vary Quadrilaterals A B D F and G can be decomposed into two identical triangles 3 Answers vary Sample responses They have two pairs of parallel sides and each pair has equal length They are all parallelograms The triangles are formed by drawing a diagonal connecting opposite vertices Some triangles are right triangles some are acute and some are obtuse For some quadrilaterals there is more than one way to decompose it into two identical triangles DISCUSSION GUIDANCE The discussion should serve two goals to highlight how quadrilaterals can be decomposed into triangles and to help students make generalizations about the types of quadrilaterals that can be decomposed into two identical triangles Consider these questions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 87

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS How did you decompose the quadrilaterals into two triangles Connect opposite vertices i e draw a diagonal Did the strategy of drawing a diagonal work for decomposing all quadrilaterals into two triangles Yes Are all of the resulting triangles identical No YOUR NOTES What is it about C and E that they cannot be decomposed into two identical triangles They don t have equal sides or equal angles Their opposite sides are not parallel What do A B and D have that C and E do not A B and D have two pairs of parallel sides that are of equal lengths They are parallelograms Ask students to complete this sentence starter For a quadrilateral to be decomposable into two identical triangles it must be or have If time permits discuss how students verified the congruence of the two triangles How did you check if the triangles are identical Did you simply stack the traced triangle or did you do something more specific They may notice that it is necessary to rotate one triangle or to reflect one triangle it twice before the triangles could be matched up Did anyone use another way to check for congruence Students may also think in terms of the parts or composition of each triangle E g Both triangles have all the same side lengths they both have a right angle ANTICIPATED MISCONCEPTIONS It may not occur to students to rotate triangles to check congruence If so tell students that we still consider two triangles identical even when one needs to be rotated to match the other SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allowing students who struggle with fine motor skills to dictate how to decompose the quadrilaterals and use the tracing paper as needed 88 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 7 Concept Exploration Activity 2 A TALE OF TWO TRIANGLES PART 2 ZEARN MATH TEACHER LESSON MATERIALS MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 2 Previously students decomposed quadrilaterals into two identical triangles The work warmed them to the idea of a triangle as a half of a familiar quadrilateral This activity prompts them to think the other way to compose quadrilaterals using two identical triangles It helps students see that two identical triangles of any kind can always be joined to produce a parallelogram Both explorations prepare students to make connections between the area of a triangle and that of a parallelogram in the next lesson YOUR NOTES A key understanding to uncover here is that two identical copies of a triangle can be joined along any corresponding side to produce a parallelogram and that more than one parallelogram can be formed As students work look for different compositions of the same pair of triangles Select students using different approaches to share later When manipulating the cutouts students are likely to notice that right triangles can be composed into rectangles and sometimes squares and that non right triangles produce parallelograms that are not rectangles Students may not immediately recall that squares and rectangles are also parallelograms Consider preparing a reference for students to consult Here is an example Quadrilaterals Parallelograms Rhombuses Squares Rectangles Trapezoids As before students make generalizations here that they don t yet have the tools to justify them This is appropriate at this stage Later in their mathematical study they will learn to verify what they now take as facts 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 89

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES LAUNCH Keep students in the same groups Give each group one set of triangles labeled P U two copies of each triangle from the template and access to scissors if the triangles are not pre cut The set includes different types of triangles isosceles right scalene right obtuse acute and equilateral Ask each group member to take 1 2 pairs of triangles Reiterate that students learned that certain types of quadrilaterals can be decomposed into two identical triangles Explain that they will now see if it is possible to compose quadrilaterals out of two identical triangles and if so to find out what types of quadrilaterals would result Give students quiet work time and then some time to discuss their responses and answer the second question with their group ACTIVITY 2 TASK 1 3 Your teacher will give your group several pairs of triangles Each group member should take 1 2 pairs 1 a Which pair s of triangles do you have b Can each pair be composed into a rectangle A parallelogram 2 Discuss with your group your responses to the first question Then complete each of the following statements with all some or none Sketch 1 2 examples to illustrate each completed statement a of these pairs of identical triangles can be composed into a rectangle b of these pairs of identical triangles can be composed into a parallelogram 90 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 a Answers vary b Answers vary For rectangles answer is yes for triangles R and U but no for the rest For parallelograms answer is yes for all triangles YOUR NOTES 2 a Some of these pairs of triangles can be composed into a rectangle Examples U R R U b All of these pairs of triangles can be composed into a parallelogram Examples P U Q R P Q R U DISCUSSION GUIDANCE The focus of this discussion would be to clarify whether or not two copies of each triangle can be composed into a rectangle or a parallelogram and to highlight the different ways two triangles could be composed into a parallelogram Ask a few students who composed different parallelograms from the same pair of triangles to share Invite the small group to notice how these students ended up with different parallelograms To help them see that a triangle can be joined along any side of its copy to produce a parallelogram ask questions such as Here is one way of composing triangles S into a parallelogram Did anyone else do it this way Did anyone obtain a parallelogram a different way How many different parallelograms can be created with any two copies of a triangle Why 3 ways because there are 3 sides along which the triangles could be joined What kinds of triangles can be used to compose a rectangle How Right triangles by joining two copies along the side opposite of the right angle 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 91

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS What kinds of triangles can be used to compose a parallelogram How Any triangle by joining two copies along any side with the same length ANTICIPATED MISCONCEPTIONS YOUR NOTES Students may draw incorrect conclusions if certain pieces of their triangles are turned over to face down or if it did not occur to them that the pieces could be moved Ask them to try manipulating the pieces in different ways Seeing that two copies of a triangle can always be composed into a parallelogram students might mistakenly conclude that any two copies of a triangle can only be composed into a parallelogram i e no other quadrilaterals can be formed from joining two identical triangles Showing a counterexample may be a simple way to help students see that this is not the case SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Processing Time Check in with individual students as needed to assess for comprehension during each step of the activity Digital Lesson Draw one line to decompose the parallelogram into 2 identical triangles STUDENT RESPONSE 92 or 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 7 Wrap Up ZEARN MATH TEACHER LESSON MATERIALS Display and revisit representative works from the two main activities Draw out key observations about the special connections between triangles and parallelograms YOUR NOTES LESSON SYNTHESIS First we tried to decompose or break apart quadrilaterals into two identical triangles What strategy allowed us to do that Drawing a segment connecting opposite vertices Which types of quadrilaterals could always be decomposed into two identical triangles Parallelograms Can quadrilaterals that are not parallelograms be decomposed into triangles Yes but the resulting triangles may not be identical Then we explored the relationship between triangles and quadrilaterals the other way around We tried to compose or create quadrilaterals from pairs of identical triangles What types of quadrilaterals were you able to compose with a pair of identical triangles Parallelograms some of them are rectangles Does it matter what type of triangles was used No Any two copies of a triangle could be composed into a parallelogram Was there a particular side along which the two triangles must be joined to form a parallelogram No Any of the three sides could be used We saw how two identical copies of a triangle can be combined to make a parallelogram This is true for any triangle The reverse is also true any parallelogram can be split into two identical triangles In grade 8 we will acquire some tools to prove these observations For now we will take the special relationships between triangles and parallelograms as a fact We will use them to find the area of any triangle in upcoming lessons EXIT TICKET Give students access to their geometry toolkits if needed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 93

G6M1 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS TASK 1 Here are some quadrilaterals A B C D E F YOUR NOTES a Circle all quadrilaterals that you think can be decomposed into two identical triangles using only one line b What characteristics do the quadrilaterals that you circled have in common 2 Here is a right triangle Show or briefly describe how two copies of it can be composed into a parallelogram STUDENT RESPONSE 1 a Quadrilaterals B C D and F should be circled b They all have two pairs of parallel sides They are all parallelograms 2 Answers vary Sample response Joining two copies of the triangle along a side that is the same length e g the shortest side of one and the shortest side of the other would make a parallelogram Three parallelograms are possible since there are three sides at which the triangles could be joined One of the parallelograms is a rectangle 94 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 7 Concept Exploration Activity 2 page 1 of 3 Use this template if triangles are to be cut by students 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 95

Template for Lesson 7 Concept Exploration Activity 2 page 2 of 3 Use this template triangles are pre cut by teacher 96 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 7 Concept Exploration Activity 2 page 3 of 3 Use this template triangles are pre cut by teacher 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 97

G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 1 LESSON 8 Area of Triangles YOUR NOTES LEARNING GOALS Reason about the area of any triangle using the base and height of an associated parallelogram Recognize that a triangle has half the area of a parallelogram LEARNING GOALS STUDENT FACING Let s use what we know about parallelograms to find the area of triangles LEARNING TARGETS STUDENT FACING I can use what I know about parallelograms to reason about the area of triangles REQUIRED MATERIALS REQUIRED PREPARATION 98 Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Make sure students have access to their geometry toolkits which should include tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

This lesson builds on students earlier work decomposing and rearranging regions to find area It leads students to see that in addition to using area reasoning methods from previous lessons they can use what they know to be true about parallelograms i e that the area of a parallelogram is b h to reason about the area of triangles Students begin to see that the area of a triangle is half of the area of the parallelogram of the same height or that it is the same as the area of a parallelogram that is half its height They build this intuition in several ways G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES by recalling that two copies of a triangle can be composed into a parallelogram by recognizing that a triangle can be recomposed into a parallelogram that is half the triangle s height or by reasoning indirectly using one or more rectangles with the same height as the triangle They apply this insight to find the area of triangles both on and off the grid Warm Up COMPOSING PARALLELOGRAMS Instructional Routines Notice and Wonder Think Pair Share This warm up has two aims to solidify what students learned about the relationship between triangles and parallelograms and to connect their new insights back to the concept of area Students are given a right triangle and the three parallelograms that can be composed from two copies of the triangle Though students are not asked to find the area of the triangle they may make some important observations along the way They are likely to see that The triangle covers half of the region of each parallelogram The base height measurements for each parallelogram involve the numbers 6 and 4 which are the lengths of two sides of the triangle All parallelograms have the same area of 24 square units These observations enable them to reason that the area of the triangle is half of the area of a parallelogram in this case any of the three parallelograms can be used to find the area of the triangle In upcoming work students will test and extend this awareness generalizing it to help them find the area of any triangle 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 99

G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Display the images of the triangle and the three parallelograms for all to see Give students time to observe them Ask them to be ready to share at least one thing they notice and one thing they wonder Give students time to share their observations and questions with a partner YOUR NOTES Give students time to complete the activity and provide access to their geometry toolkits Follow with a whole class discussion WARM UP TASK 1 Answer the following questions about these figures Here is Triangle M M Han made a copy of Triangle M and composed three different parallelograms using the original M and the copy as shown here M M M 1 For each parallelogram Han composed identify a base and a corresponding height and write the measurements on the drawing 2 Find the area of each parallelogram Han composed Show your reasoning STUDENT RESPONSE 1 First parallelogram b 6 and h 4 second parallelogram b 4 and h 6 third parallelogram b 6 and h 4 6 M 4 6 100 4 M 4 M 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

2 The area of all parallelograms is 24 square units The base and height measurements for the parallelograms are 4 units and 6 units or 6 units and 4 units 4 6 24 and 6 4 24 G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS DISCUSSION GUIDANCE Ask one student to identify the base height and area of each parallelogram as well as how they reasoned about the area If not already brought up by students in their explanations discuss the following questions YOUR NOTES Why do all parallelograms have the same area even though they all have different shapes They are composed of the same parts two copies of the same right triangles They have the same pair of numbers for their base and height They all call be decomposed and rearranged into a 6 by 4 rectangle What do you notice about the bases and heights of the parallelograms They are the same pair of numbers How are the base height measurements related to the right triangle They are the lengths of two sides of the right triangles Can we find the area of the triangle How Yes the area of the triangle is 12 square units because it is half of the area of every parallelogram which is 24 square units ANTICIPATED MISCONCEPTIONS When identifying bases and heights of the parallelograms some students may choose a nonhorizontal or non vertical side as a base and struggle to find its length and the length of its corresponding height Ask them to see if there s another side that could serve as a base and has a length that can be more easily determined SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Eliminate Barriers Provide an enlarged version of the visual 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 101

G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 1 MORE TRIANGLES Instructional Routines Anticipate Monitor Select Sequence Connect YOUR NOTES In this activity students apply what they have learned to find the area of various triangles They use reasoning strategies and tools that make sense to them Students are not expected to use a formal procedure or to make a general argument They will think about general arguments in an upcoming lesson Here are some anticipated paths students may take from more elaborate to more direct Also monitor for other approaches Draw two smaller rectangles that decompose the given triangle into two right triangles Find the area of each rectangle and take half of its area Add the areas of the two right triangles This is likely used for B and D For Triangle C some students may choose to draw two rectangles around and on the triangle as shown here find half of the area of each rectangle and subtract one area from the other Enclose the triangle with one rectangle find the area of the rectangle and take half of that area This is likely used for right triangle A Duplicate the triangle to form a parallelogram find the area of the parallelogram and take half of its area Likely used with any triangle Monitor the different strategies students use Consider asking each student that uses a unique strategy to create a visual display of their work and to share it with the class later LAUNCH Tell students that they will now apply their observations from the past few activities to find the area of several triangles Arrange students in groups of 2 3 Give students quiet work time and time to discuss their work with a partner Ask them to confer with their group only after each person has attempted to find the area of at least two triangles Provide access to their geometry toolkits especially tracing paper 102 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 8 ACTIVITY 1 TASK 1 2 ZEARN MATH TEACHER LESSON MATERIALS Find the areas of at least two of the triangles below Show your reasoning YOUR NOTES A B C D STUDENT RESPONSE Diagrams and explanations vary Sample responses A A 8 square units 8 2 16 so the area of the rectangle is 16 square units The area of the triangle is half of that of the rectangle so it is 8 square units 2 units 8 units B 10 5 square units 5 3 15 so the area of the left rectangle is 15 square units The area of the left triangle is then 7 5 square units 2 3 6 so the area of the right rectangle is 6 square units so area of the right triangle is 3 square units The sum of the areas of the small triangles which make up the large triangle is 7 5 3 10 5 so the large triangle has area 10 5 square units B 3 units 5 units C 10 square units If we make a copy of the triangle rotate it and join them along the longest side we would get a parallelogram The base length is 5 units and the height is 4 units so the area of the parallelogram is 20 square units The area of the triangle is half of that area so it is 10 square units D 12 square units Decompose the triangle into a trapezoid and a small triangle by drawing a vertical line 3 units from the left side Rotate the small triangle to line up with the bottom side of the trapezoid to create a parallelogram To get the area of that parallelogram 4 3 12 D 4 units 3 units 2 units C 4 units 5 units D 3 units 4 units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 103

G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS DISCUSSION GUIDANCE Though students may have conferred with one or more partners during the task take a few minutes to come together as a small group so that everyone has a chance to see a wider range of approaches YOUR NOTES Select previously identified students to explain their approach and display their reasoning for all to see Start with the most elaborate strategy most likely a strategy that involves enclosing a triangle and move toward the most direct most likely duplicating the triangle to compose a parallelogram After each student presents ask the small group Did anyone else reason the same way Did anyone else draw the same diagram but think about the problem differently Can this strategy be used on another triangle in this set Which one Is there a triangle for which this strategy would not be helpful Which one and why not ANTICIPATED MISCONCEPTIONS At this point students should not be counting squares to determine area If students are still using this approach steer them in the direction of recently learned strategies decomposing rearranging enclosing or duplicating Students may not recognize that the vertical side of Triangle D could be the base and try to measure the lengths the other sides If so remind them that any side of a triangle can be the base SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Eliminate Barriers Assist students to see the connections between new problems and prior work Students may benefit from a review of different representations to activate prior knowledge Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors 104 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 8 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS Without calculating explain the steps you would take to find the area of this triangle Consider using pictures and words to answer YOUR NOTES STUDENT RESPONSE Answers vary Sample response I can duplicate this triangle to make a parallelogram I could then find the area of the parallelogram and take half of that to find the area of the triangle Here is a picture to illustrate this process Wrap Up LESSON SYNTHESIS In this lesson we practiced using what we know about parallelograms to reason about areas of triangles We duplicated a triangle to make a parallelogram decomposed and rearranged a triangle into a parallelogram or enclosed a triangle with one or more rectangles What can we say about the area of a triangle and that of a parallelogram with the same height The area of the triangle is half of the area of the related parallelogram In the second activity we cut a triangle along a line that goes through the midpoints of two sides and rearranged the pieces into a parallelogram What did we notice about the area and the height of the resulting parallelogram It has the same area as the original triangle but half its height How might we start finding the area of any triangle in general Start by finding the area of a related parallelogram whose base is also a side of the triangle EXIT TICKET Students have explored several ways to reason about the area of a triangle This exit ticket prompts them to articulate at least one way to do so Not all methods will be equally intuitive 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 105

G6M1 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS or clear to them In writing a commentary about at least one approach students can show what makes sense to them at this point TASK YOUR NOTES Elena Lin and Noah all found the area of Triangle Q to be 14 square units but reasoned about it differently as shown in the diagrams Explain at least one student s way of thinking and why his or her answer is correct Q Q Elena Lin Q Noah STUDENT RESPONSE Explanations vary Sample responses Elena drew two rectangles that decomposed the triangle into two right triangles She found the area of each right triangle to be half of the area of its enclosing rectangle This means that the area of the original triangle is the sum of half of the area of the rectangle on the left and half of the rectangle on the right Half of 4 5 plus half of 4 2 is 10 4 so the area is 14 square units Lin saw it as half of a parallelogram with the base of 7 units and height of 4 units and thus an area of 28 square units Half of 28 is 14 Noah decomposed the triangle by cutting it at half of the triangle s height turning the top triangle around and joining it with the bottom trapezoid to make a parallelogram He then calculated the area of that parallelogram which has the same base length but half the height of the triangle 7 2 14 so the area is 14 square units SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Expressive Language Support Allow for multiple methods of the articulation of student understanding such as drawing oral written etc 106 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 9 Formula for the Area of a Triangle G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Compare contrast and critique orally different strategies for determining the area of a triangle LEARNING GOALS Generalize a process for finding the area of a triangle and justify orally and in writing why this can be abstracted as 12 b h Recognize that any side of a triangle can be considered its base choose a side to use as the base when calculating the area of a triangle and identify the corresponding height LEARNING GOALS STUDENT FACING Let s write and use a formula to find the area of a triangle I can use the area formula to find the area of any triangle LEARNING TARGETS STUDENT FACING I can write and explain the formula for the area of a triangle I know what the terms base and height refer to in a triangle REQUIRED MATERIALS Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 107

G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS In this lesson students begin to reason about area of triangles more methodically by generalizing their observations up to this point and expressing the area of a triangle in terms of its base and height YOUR NOTES Students first learn about bases and heights in a triangle by studying examples and counterexamples They then identify base height measurements of triangles use them to determine area and look for a pattern in their reasoning to help them write a general formula for finding area Students also have a chance to build an informal argument about why the formula works for any triangle Warm Up BASES AND HEIGHTS OF A TRIANGLE Instructional Routine Notice and Wonder In this activity students think about the meaning of base and height in a triangle by studying examples and non examples The goal is for them to see that in a triangle Any side can be a base A segment that represents a height must be drawn at a right angle to the base but can be drawn in more than one place The length of this perpendicular segment is the distance between the base and the vertex opposite it A triangle can have three possible bases and three corresponding heights Students may draw on their experience with bases and heights in a parallelogram and observe similarities Encourage this as it would help them conceptualize base height pairs in triangles As students discuss with their partners listen for how they justify their decisions or how they know which statements are true LAUNCH Display the examples and non examples of bases and heights for all to see Give students time to observe them Ask them to be ready to share at least one thing they notice and one thing they wonder Give the class time to share some of their observations and questions Tell students they will now use the examples and non examples to determine what is true about bases and heights in a triangle Arrange students in groups of 2 Give them quiet think time and then time to share their response with a partner 108 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 9 WARM UP TASK 1 ZEARN MATH TEACHER LESSON MATERIALS Study the examples and non examples of bases and heights in a triangle Then answer the questions YOUR NOTES These dashed segments represent heights of the triangle base base base These dashed segments do not represent heights of the triangle base base base Select all the statements that are true about bases and heights in a triangle a Any side of a triangle can be a base b There is only one possible height c A height is always one of the sides of a triangle d A height that corresponds to a base must be drawn at an acute angle to the base e A height that corresponds to a base must be drawn at a right angle to the base f Once we choose a base there is only one segment that represents the corresponding height g A segment representing a height must go through a vertex STUDENT RESPONSE Only statements a and e are true 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 109

G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES DISCUSSION GUIDANCE For each statement ask students to indicate whether they think it is true For each true vote ask one or two students to explain how they know Do the same for each false vote Encourage students to use the examples and counterexamples to support their argument e g The last statement is not true because the examples show dashed segments or heights that do not go through a vertex Agree on the truth value of each statement before moving on Record and display the true statements for all to see Students should see that only statements a and e are true that any side of a triangle can be a base and a segment for the corresponding height must be drawn at a right angle to the base What is missing an important gap to fill during discussion is the length of any segment representing a height Ask students How long should a segment that shows a height be If we draw a perpendicular line from the base where do we stop Solicit some ideas from students Explain that the length of each perpendicular segment is the distance between the base and the vertex opposite of it The opposite vertex is the vertex that is not an endpoint of the base Point out the opposite vertex for each base Clarify that the segment does not have to be drawn through the vertex although that would be a natural place to draw it as long as it maintains that distance between the base and the opposite vertex It is helpful to connect this idea to that of heights in a parallelogram Consider duplicating the triangle and use the original and the copy to compose a parallelogram The height for a chosen base in the triangle is also the height of the parallelogram with the same base Students will have many opportunities to make sense of bases and heights in this lesson and an upcoming one so they do not need to know how to draw a height correctly at this point ANTICIPATED MISCONCEPTIONS Some students may struggle to interpret the diagrams Ask them to point out parts of the diagrams that might be unclear and clarify as needed Students may not remember from their experience with parallelograms that a height needs to be perpendicular to a base Consider posting a diagram of a parallelogram with its base and height labeled in a visible place in the room so that it can serve as a reference SUPPORT FOR STUDENTS WITH DISABILITIES Visual Spatial Processing Visual Aids Provide handouts of the representations for students to draw on or highlight 110 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 9 Concept Exploration Activity 1 FINDING THE FORMULA FOR AREA OF A TRIANGLE ZEARN MATH TEACHER LESSON MATERIALS Instructional Routine MLR8 Discussion Supports This task culminates in writing a formula for the area of triangles By now students are likely to have developed the intuition that the area of a triangle is half of that of a parallelogram with the same base and height This activity encapsulates that work in an algebraic expression YOUR NOTES Students first find the areas of several triangles given base and height measurements They then generalize the numerical work to arrive at an expression for finding the area of any triangle If needed remind students how they reasoned about the area of triangles in the previous lesson i e by composing a parallelogram enclosing with one or more rectangles etc Encourage them to refer to their previous work and use tracing paper as needed Students might write b h 2 or b h 12 as the expression for the area of any triangle Any equivalent expression should be celebrated At the end of the activity consider giving students a chance to reason more abstractly and deductively i e to think about why the expression b h 2 would hold true for all triangles See Activity Recap for prompts and diagrams that support such reasoning LAUNCH Arrange students in groups of 2 3 Explain that they will now find the area of some triangles using what they know about base height pairs in triangles and the relationships between triangles and parallelograms Give students time to complete the activity and access to geometry toolkits especially tracing paper Ask them to find the area of at least a couple of triangles independently before discussing with their partner s ACTIVITY 1 TASK 1 2 Use the triangles to complete the table For each triangle label a side that can be used as the base and a segment showing its corresponding height 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 111

G6M1 LESSON 9 Record the measurements for the base and height in the table and find the area of the triangle The side length of each square on the grid is 1 unit In the last row write an expression for the area of any triangle using b and h ZEARN MATH TEACHER LESSON MATERIALS A YOUR NOTES B D C Triangle Base units Height units Area square units b h Triangle Base units Height units Area square units A 10 7 35 B 11 or 6 6 or 11 33 C 10 3 15 D 4 11 22 Any triangle b h b h 2 or equivalent A B C D Any triangle STUDENT RESPONSE 112 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 9 Explanations vary Sample responses We can make a parallelogram from any triangle using the same base and height The triangle will be half of the parallelogram The area of a parallelogram is the length of the base times the length of the height so the area of the triangle will be b h 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES I can cut off the top half of a triangle and rotate it to make a parallelogram That parallelogram has a base of b and a height that is half of the original triangle which is 12 h so its area is b 12 h Since the parallelogram is just the triangle rearranged the area of the triangle is also 12 b h ACTIVITY 1 RECAP Select a few students to share their expression for finding the area of any triangle Record each expression for all to see To give students a chance to reason logically and deductively about their expression ask Can you explain why this expression is true for any triangle Display the following diagrams for all to see Give students time to observe the diagrams Ask them to choose one that makes sense to them and use that diagram to explain or show in writing that the expression b h 2 works for finding the area of any triangle Consider giving each student an index card or a sheet of paper on which to write their reasoning so that their responses could be collected if desired h h b b Elena Jada h b Lin When dealing only with the variables b and h and no numbers students are likely to find Jada s and Lin s diagrams more intuitive to explain Those choosing to use Elena s diagram are likely to suggest moving one of the extra triangles and joining it with the other to form a nonrectangular parallelogram with an area of b h Expect students be less comfortable reasoning in abstract terms than in concrete terms Prepare to support them in piecing together a logical argument using only variables If time permits select students who used different diagrams to share their explanation starting with the most commonly used diagram most likely Jada s Ask other students to support 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 113

G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS refine or disagree with their arguments It time is limited consider collecting students written responses now and discussing them in an upcoming lesson ANTICIPATED MISCONCEPTIONS YOUR NOTES Students may not be inclined to write an expression using the variables b and h and instead replace the variables with numbers of their choice Ask them to reflect on what they did with the numbers for the first four triangles Then encourage them to write the same operations but using the letters b and h rather than numbers SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Eliminate Barriers Chunk this task into more manageable parts e g presenting one question at a time which will aid students who benefit from support with organizational skills in problem solving Executive Functioning Visual Aids Create an anchor chart i e labeled base and height of a triangle and formula for finding area publicly displaying important definitions rules formulas or concepts for future reference SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Use this routine to support group discussion when students share their expressions for finding the area of any triangle Provide students with quiet think time to begin to consider why their expression is true for any triangle before they continue work with a partner to complete their response Select 1 or 2 pairs of students to share with the group then call on students to restate their peers reasoning This will give more students the chance to use language to interpret and describe expressions for the area of triangles Design Principle s Support sense making maximizing meta awareness 114 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 9 Concept Exploration Activity 2 APPLYING THE FORMULA FOR AREA OF TRIANGLES ZEARN MATH TEACHER LESSON MATERIALS Instructional Routine MLR2 Collect and Display In this activity students apply the expression they previously generated to find the areas of various triangles Each diagram is labeled with two or three measurements Before calculating students think about which lengths can be used to find the area of each triangle YOUR NOTES As students work notice students who choose different bases for Triangles B and D Invite them to contribute to the discussion about finding the areas of right triangles later LAUNCH Explain to students that they will now practice using their expression to find the area of triangles without a grid For each triangle ask students to be prepared to explain which measurement they choose for the base and which one for the corresponding height and why Keep students in groups of 2 4 Give students quiet think time followed by a few minutes for discussing their responses in their group ACTIVITY 2 TASK 1 3 For each triangle circle a base measurement that you can use to find the area of the triangle Then find the area of any three triangles Show your reasoning C B A 3 cm 4 cm 7 cm 6 cm 4 cm 5 cm 3 5 cm D E 10 cm 8 73 cm 3 5 cm 8 cm 5 cm 6 cm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 115

G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE In B and D either of the given pair of measurements can be the base A B C 3 cm YOUR NOTES 4 cm 7 cm 6 cm 4 cm 5 cm 3 5 cm D E 10 cm 8 73 cm 3 5 cm 5 cm 8 cm 6 cm Triangle A 15 square cm b 5 h 6 A 5 6 2 15 Triangle B 8 square cm b 4 h 4 A 4 4 2 8 Triangle C 10 5 square cm b 7 h 3 A 7 3 2 10 5 Triangle D 14 square cm b 8 h 3 5 A 8 3 5 2 14 Triangle E 15 square cm b 6 h 5 A 6 5 2 15 DISCUSSION GUIDANCE The aim of this discussion is to deepen students awareness of the base and height of triangles Discuss questions such as For Triangle A can we say that the 6 cm segment is the base and the 5 cm segment is the height Why or why not No the base of a triangle is one of its sides What about for Triangle C Can the 3 cm segment serve as the base Why or why not No that segment is not a side of the triangle Can the 8 73 cm side in Triangle D serve as the base Why or why not Yes it is a side of the triangle but because we don t have the height that corresponds to it it is not helpful for finding the area here More than two measurements are given for Triangles C D and E Which ones are helpful for finding area We need a base and a corresponding height which means the length of one 116 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

side of the triangle and the length of a perpendicular segment between that side and the opposite vertex When it comes to finding area how are right triangles like B and D unique Either of the two sides that form the right angle could be the base or the height In non right triangles like A C and E the height segment is not a side of the triangle a different line segment has to be drawn G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS The extra measurement in Triangles C D and E may confuse some students If they are unsure how to decide the measurement to use ask what they learned must be true about a base and a corresponding height in a triangle Urge them to review the work from the warm up activity SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing MLR 2 Collect and Display While pairs are working circulate and listen to student talk about identifying the bases and corresponding heights for each of the triangles Write down common or important phrases you hear students say about each triangle specifically focusing on how students make sense of the base and height of each triangle Record the words students use to refer to each triangle and display them for all to see during the discussion Design Principle s Support sense making Maximize meta awareness Digital Lesson Circle the triangle that has the height labeled correctly Then explain how to find the area of the triangle height height base base height base STUDENT RESPONSE The third triangle is correct and should be circled Explanations may vary Sample response I can find the area of the triangle by multiplying one half by the base by the height 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 117

G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Wrap Up LESSON SYNTHESIS The area of a parallelogram can be determined using base and height measurements In this lesson you learned that we can do the same with triangles How do we locate the base of a triangle How many possible bases are there Any side of a triangle can be a base There are 3 possible bases How do we locate the height once we know the base Find the length of a perpendicular segment that connects the base and its opposite vertex We can use the base height pair of measurements to find the area of a triangle quite simply What expression works for finding the area of a triangle 1 2 b h or b h 2 Can you explain briefly why this expression or formula works The area of a triangle is always half of the area of a related parallelogram that shares the same base and height You learned that any side of the triangle can be the base but not all sides can be the height Are there cases in which both the base and the height are sides of the triangle When does that happen Yes In a right triangle both the base and height can be the sides of the triangle TERMINOLOGY Opposite Vertex For each side of a triangle there is one vertex that is not on that side This is the opposite vertex For example point A is the opposite vertex to side BC B C A EXIT TICKET Students apply what they learned about the area formula and about the base and height of a triangle in this exit ticket Multiple measurements are given so students need to be attentive in choosing the right pair of measurements that would allow them to calculate the area 118 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS TASK For each triangle identify a base and a corresponding height Use them to find the area Show your reasoning B A YOUR NOTES 5 cm 5 cm 4 cm 4 8 cm 7 2 in 6 in 6 cm 2 5 in 3 in STUDENT RESPONSE Answers vary Possible responses Triangle A 1 2 b 3 h 6 area 9 sq in b 7 2 h 2 5 area 9 sq in 3 6 9 1 2 7 2 2 5 9 Triangle B 1 2 b 6 h 4 area 12 sq cm b 5 h 4 8 area 12 sq cm 6 4 12 1 2 5 4 8 12 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 119

G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 1 LESSON 10 Bases and Heights of Triangles LEARNING GOALS LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING 120 Draw and label the height that corresponds to a given base of a triangle making sure it is perpendicular to the base and the correct length Evaluate orally the usefulness of different base height pairs for finding the area of a given triangle Let s use different base height pairs to find the area of a triangle I can identify corresponding pairs of base and height of any triangle When given information about a base of a triangle I can identify and draw a corresponding height REQUIRED MATERIALS Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles REQUIRED PREPARATION From the geometry toolkit each student especially needs an index card for the Hunting for Heights activity 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

This lesson furthers students ability to identify and work with a base and height in a triangle in two ways G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS 1 By learning to draw not just to recognize a segment to show the corresponding height for any given base and 2 By learning to choose appropriate base height pairs to enable area calculations YOUR NOTES Students have seen that the area of a triangle can be determined in multiple ways Using the base and height measurements and the formula is a handy approach but because there are three possible pairs of bases and heights some care is needed in identifying the right combination of measurements Some base height pairs may be more practical or efficient to use than others so it helps to be strategic in choosing a side to use as a base Warm Up AN AREA OF 12 Instructional Routine Think Pair Share So far students have determined area given a triangle and some measurements In this warmup students are invited to reverse the process They are given an area measure and are asked to create several triangles with that area Expect students to gravitate toward right triangles first or to halve rectangles that have factors of 12 as their side lengths This is a natural and productive starting point Prompting students to create non right triangles encourages them to apply insights from their experiences with non right parallelograms As students work alone and discuss with partners notice the strategies they use to draw their triangles and to verify their areas Identify a few students with different strategies to share later LAUNCH Arrange students in groups of 2 Give students quiet think time and time to share their drawings with their partner afterwards Encourage students to refer to previous work as needed Provide access to their geometry toolkits Tell students to draw a different triangle with the same area if they finish their first one early During partner discussion each partner should convince the other that the triangle drawn is indeed 12 square units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 121

G6M1 LESSON 10 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 YOUR NOTES On the grid draw a triangle with an area of 12 square units Try to draw a non right triangle Be prepared to explain how you know the area of your triangle is 12 square units STUDENT RESPONSE Drawings and explanations vary Sample responses This right triangle has a base of 8 units and a height of 3 units The area is half of 3 8 or half of 24 which is 12 3 8 This triangle has a side of 6 units This can be the base Draw a height segment that is perpendicular to the base and is 4 units long The area of the triangle is b h 2 so it is 6 4 2 which is 12 4 6 Draw a parallelogram with a base of 12 and a height of 2 and then draw a diagonal line to create two identical triangles Each of the triangles has an area of 12 because it is half of a parallelogram with an area of 24 2 12 122 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 10 DISCUSSION GUIDANCE ZEARN MATH TEACHER LESSON MATERIALS Invite a few students to share their drawings and ways of reasoning with the class For each drawing shared ask the creator for the base and height and record them for all to see Ask the class YOUR NOTES Did anyone else draw an identical triangle Did anyone draw a different triangle but with the same base and height measurements To reinforce the relationship between base height and area discuss Which might be a better way to draw a triangle by starting with the base measurement or with the height Why Can you name other base height pairs that would produce an area of 12 square units without drawing How ANTICIPATED MISCONCEPTIONS If students have trouble getting started ask Can you draw a quadrilateral with an area of 12 Can you use what you know about parallelograms to help you Can you use any of the area strategies decomposing rearranging enclosing subtracting to arrive at an area of 12 Students who start by drawing rectangles and other parallelograms may use factors of 12 instead of factors of 24 for the base and height If this happens ask them what the area of the their quadrilateral is and how it relates to the triangle they are trying to draw SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 123

G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 1 HUNTING FOR HEIGHTS Instructional Routine MLR2 Collect and Display YOUR NOTES Students may be able to recognize a measurement that can be used for height when they see it but identifying and drawing an appropriate segment is more challenging This activity and the demonstration needed to launch it gives students a concrete strategy for identifying a height accurately When students use a strategy of drawing an auxiliary line to solve problems they are looking for and making use of structure Explicit instruction as in this activity is often needed before students can be expected to use this strategy spontaneously LAUNCH Explain to students that they will try to draw a height that corresponds to each side of a triangle Arrange students in groups of 2 Give each student an index card and time to complete the first question Remind them that there is more than one correct way to draw the corresponding height for a base Ask them to pause after the first question As students work notice how students are using the index cards if at all Afterwards solicit a few quick comments on the exploration Ask questions such as How did you know where to draw the segments How did you draw them Why were you given index cards How might they help Explain that you will now demonstrate a way to draw heights effectively If any students used the index card correctly acknowledge that they were on the right track Remind students that any line we draw to show the height of a triangle must be drawn perpendicular to the base Having a tool with a right angle and with straight edges can help us make sure the line we draw is both straight and perpendicular to the base This is what the index card is for Ask How do we know where to stop this line we are drawing How long should it be Explain that the easiest way is to draw the line so it would pass through the vertex opposite of the chosen base Draw or display a triangle for all to see Demonstrate the following Choose one side of the triangle as the base Identify the opposite vertex Line up one edge of the index card with that base 124 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Slide the card along the base until a perpendicular edge of the card meets the opposite vertex G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS Use that edge to draw a line segment from that vertex to the base The measure of that segment is the height YOUR NOTES opposite vertex opposite vertex base base opposite vertex base opposite vertex base Ask What if the opposite vertex is not directly over the base Explain that sometimes we need to extend the line of the base and demonstrate the process opposite vertex opposite vertex base base opposite vertex base opposite vertex base Demonstrate the process with another example in which the card needs to slide from right to left e g by rotating the obtuse triangle above clockwise Left handed students may find this particularly helpful Prompt students to use this method to check the heights they drew in the first question revise the drawings if they were incorrect and share their revisions with their partners Circulate and support students as they draw Those who finish verifying the heights in the first question can move on to complete the rest of the activity with their partners 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 125

G6M1 LESSON 10 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 YOUR NOTES Here are three copies of the same triangle The triangle is rotated so that the side chosen as the base is at the bottom and is horizontal Draw a height that corresponds to each base Use an index card to help you Side a as the base Side b as the base b c a c a b Side c as the base a b c Pause for your teacher s instructions before moving to the next question STUDENT RESPONSE Drawings vary Sample drawings b b 126 a a c c c a b a b c c a b a b c 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 10 ACTIVITY 1 TASK 2 3 ZEARN MATH TEACHER LESSON MATERIALS Draw a line segment to show the height for the chosen base in each triangle YOUR NOTES A B C base base base D E F base base base STUDENT RESPONSE A B C h base h h base base D E F h h h base base base DISCUSSION GUIDANCE If time permits consider selecting one student to share the height drawing for each triangle or display the solutions in the Student Response for all to see To help students reflect on their work discuss questions such as For which triangles was it easy to find the corresponding height for the given base For which triangles was it harder 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 127

G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES How was the process of finding the height of triangle D different from that of the others The height of a right triangle is already drawn it is the other segment framing the right angle When might we need to extend the line of the base or draw a height line outside of the triangle When dealing with obtuse triangles or when the opposite vertex is not directly over the base ANTICIPATED MISCONCEPTIONS Some students may use the index card simply as a straightedge and therefore draw heights that are not perpendicular to the given base Remind them that a height needs to be perpendicular or at a right angle to the base Students may mistakenly think that a base must be a horizontal side of a triangle or one closest to being horizontal and a height must be drawn inside of the triangle Point to some examples from earlier work to remind students that neither is true Remind them to align their index card to the side labeled base Some students may find it awkward to draw height segments when the base is not horizontal Encourage students to rotate their paper as needed to make drawing easier SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allowing students who struggle with fine motor skills to dictate how to use the index card to draw each height as needed SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing Speaking Listening MLR 2 Collect and Display While students work circulate and collect examples of student drawings of line segments showing the base and height To do this take digital pictures or copy students drawings onto a visual display Look for examples that show the height inside and outside of the triangle as well as bases that are horizontal or vertical During the discussion display the various examples and ask students to compare the diagrams by asking Do any two diagrams have similar methods for determining the base or height Listen for and amplify the mathematical language students use to support their reasoning Design Principle s Support sense making Maximize meta awareness 128 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 10 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES r p q If you needed to find the area of this triangle which side would you choose as the base Explain how you made your choice I would choose side p q r for the base I would choose this side because How would you determine the height of the triangle To determine the height of the triangle first I STUDENT RESPONSE Answers vary depending on which side the student chooses for a base Sample response I would choose side q for the base I would choose this side because it is lined up with the grid lines so I can easily count the side length To determine the height of the triangle first I would draw a line at the top of the triangle Then I would draw a line perpendicular to side q and extending to that line to show the full height of the triangle Lastly I would use the grid to find the length of that line Wrap Up In this lesson we looked closely at the heights of a triangle We located or drew a height for any side of a triangle We also considered which pair of base and height to use to find area 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 129

G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS LESSON SYNTHESIS What must we remember about the relationship between a base of a triangle and its corresponding height The height must be perpendicular to the base YOUR NOTES What tools might help us draw a height segment What is it about an index card or a ruler that helps us A tool with straight edges and a right angle can help us draw perpendicular segments When we have a base and a corresponding height we can find the area quite simply but for every triangle there are multiple base height pairs Does it matter which side we choose as the base How do we decide For the base we need a side with a known length For the height we need a segment that is perpendicular to that base and whose length we can determine EXIT TICKET LAUNCH Provide access to geometry toolkits TASK 1 For each triangle below draw a height segment that corresponds to the given base and label it h Use an index card if needed A B base base 2 Which triangle has the greatest area The least area Explain your reasoning A 130 B C D 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 Answers vary There are many possible locations for a height segment The segments shown are the most straightforward A B base YOUR NOTES h base h 2 All of the triangles have the same area 4 square units They all have a base of 2 units and a height of 4 units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 131

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 1 LESSON 11 Polygons YOUR NOTES Compare and contrast orally different strategies for finding the area of a polygon LEARNING GOALS Describe orally and in writing the defining characteristics of polygons Solve real world and mathematical problems by decomposing polygons into rectangles and triangles to find the area and present the solution method using words and other representations LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS 132 Let s investigate polygons and their areas I can describe the characteristics of a polygon using mathematical vocabulary I can reason about the area of any polygon by decomposing and rearranging it and by using what I know about rectangles and triangles Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Students have worked with polygons in earlier grades and throughout this mission In this lesson students write a definition that characterizes polygons There are many different accurate definitions for a polygon The goal of this lesson is not to find the most succinct definition possible but to articulate the defining characteristics of a polygon that makes sense to students Another key takeaway for this lesson is that the area of any polygon can be found by decomposing it into triangles The proof that all polygons are triangulable not a word students need to know is fairly sophisticated but students can just take it as a fact for now In observing and using this fact students look for and make use of structure Knowing this will be key as students solve real world and mathematical problems involving the area of polygons in the final activity of this lesson G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Warm Up WHICH ONE DOESN T BELONG BASES AND HEIGHTS Instructional Routine Which One Doesn t Belong This warm up prompts students to consolidate what they learned in the past few lessons and make careful observations about triangles Expect students to describe the differences in the triangles in terms of angles acute right or obtuse orientation of sides vertical horizontal the side likely to be chosen as a base length of base or height LAUNCH Arrange students in groups of 2 4 Display the image of triangles for all to see Give students quiet think time and ask them to indicate when they have noticed one triangle that does not belong and can explain why Encourage them to think of more than one possibility When time is up give students more time to share their response with their group and then find together at least one reason if possible that each triangle doesn t belong 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 133

G6M1 LESSON 11 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 Which one doesn t belong YOUR NOTES S T U V STUDENT RESPONSE Answers vary Sample responses S It is the only right triangle It is the only one where two sides can be easily chosen as a base and used to find the area It is the only one where the base and height are both sides of the triangle T It is the only triangle with no vertical side It is the only triangle where the side most likely to be chosen as a base is horizontal U It is the only acute triangle It is the only triangle that is most likely to have its height drawn inside the triangle V It is the only one with a height greater than 7 units DISCUSSION GUIDANCE After students shared their observations in groups invite each group to share one reason why a particular triangle might not belong Record and display the responses for all to see After each response poll the rest of the class to see if others made the same observation Since there is no single correct answer to the question of which pattern does not belong attend to students explanations and ensure the reasons given are correct Prompt students to explain the meaning of any terminology they use parts of triangles types of angles etc and to substantiate their claims 134 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

SUPPORT FOR ENGLISH LANGUAGE LEARNERS Heavier support Review the language used for comparing and contrasting such as All have except What makes different from the others is Only has G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Lighter Support Provide an organizer on the board or overhead Triangle S does not belong because Triangle T does not belong because Triangle U does not belong because Pattern V does not belong because SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Eliminate Barriers Provide an enlarged version of the visual Concept Exploration Activity 1 WHAT ARE POLYGONS Instructional Routine MLR2 Collect and Display Developing a useful and complete definition of a polygon is harder than it seems A formal definition is often very wordy or hard to parse Polygons are often referred to as closed figures but if used this term needs to be defined as the everyday meaning of closed is different than its meaning in a geometric context This activity prompts students to develop a working definition of polygon that makes sense to them but that also captures all of the necessary aspects that makes a figure a polygon Here are some important characteristics of a polygon It is composed of line segments Line segments are always straight Each line segment meets one and only one other line segment at each end The line segments never cross each other except at the end points It is two dimensional 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 135

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES One consequence of the definition of a polygon is that there are always as many vertices as edges Students may observe this and want to include it in their definition although technically it is a result of the definition rather than a defining feature As students work monitor for both correct and incorrect definitions of a polygon Listen for clear and correct descriptions as well as common but inaccurate descriptions so they can be discussed and refined later Notice students with accurate explanations so they could share later LAUNCH Arrange students in groups of 2 4 Give students quiet think time Afterwards ask them to share their responses with their group and complete the second question together If there is a disagreement about whether a figure is a polygon ask them to discuss each point of view and try to come to an agreement Follow with a group discussion ACTIVITY 1 TASK 1 Here are five polygons Here are five figures that are not polygons 2 136 Circle the figures that are polygons Use the examples to help A A B B C C F F G G H H D D E E I I J J 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 11 STUDENT RESPONSE ZEARN MATH TEACHER LESSON MATERIALS B E F and G are polygons YOUR NOTES ACTIVITY 1 TASK 2 3 What do the figures you circled have in common What characteristics helped you decide whether a figure was a polygon STUDENT RESPONSE Answers vary Characteristics that the polygons have in common They are two dimensional composed of line segments that never cross each other and each line segment meets one and only one other line segment at each end DISCUSSION GUIDANCE Display the figures in the first question for all to see For each figure ask at least one student to explain why they think it is or is not a polygon It is fine if students explanations are not precise at this point Then circle the figures that are polygons on the visual display Next ask students to share their ideas about the characteristics of polygons Record them for all to see For each one ask the group if they agree or disagree If they generally agree ask if there is anything they would add or elaborate on to make the description clearer or more precise If they disagree ask for an explanation If a key characteristic listed in the Activity Narrative is not mentioned by students bring it up and revisit it at the end of the lesson Tell students we call the line segments in a polygon the edges or sides and we call the points where the edges meet the vertices Point to the sides and vertices in a few of the identified polygons Point out that polygons always enclose a region but the region is not technically part of the polygon When we talk about finding the area of a polygon we are in fact finding the area of the region it encloses So the area of a triangle for example is really shorthand for area of the region enclosed by the triangle 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 137

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Students may think that Figures C and I are polygons because they can see several triangles or quadrilaterals in each figure Ask students to look closely at the examples and non examples and see there is a figure composed of multiple triangles or quadrilaterals and if so to see in which group it belongs SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing Representing Writing MLR 2 Collect and Display As students work listen for collect and display terms and phrases students use to describe key characteristics of polygons e g polygon edge vertices Remind students to borrow language from the display as needed This will help students use mathematical language when describing polygons Design Principle s Support sense making Maximize meta awareness SUPPORT FOR SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Visual Aids Create an anchor chart i e vertices edges polygons publicly displaying important definitions rules formulas or concepts for future reference Concept Exploration Activity 2 QUADRILATERAL STRATEGIES Instructional Routine Think Pair Share MLR8 Discussion Supports This activity has several aims It prompts students to apply what they learned to find the area of quadrilaterals that are not parallelograms encourages them to plan before jumping into a problem and urges them to reflect on the merits of different methods Students begin by thinking about the moves they would make to find the area of a quadrilateral and explaining their preference to their partners They then consider and discuss the different strategies taken by other students Along the way they may notice that some strategies are more direct or efficient than others Students reflect on these strategies and use their insights to plan the work of finding the area of polygons in this activity and beyond 138 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Note that it is unnecessary for students to take the most efficient path It is more important that they choose an approach that makes sense to them but have the chance to see the pros and cons of various paths G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH YOUR NOTES Ask students to recall the definition of quadrilateral from earlier grades or tell students that a quadrilateral is a polygon with 4 sides Tell students that we will now think about how to find the area of quadrilaterals Arrange students in groups of 4 Display the image of quadrilaterals A F for all to see Direct their attention to Quadrilateral D Give students quiet time to think about the first 2 3 moves they would make to find the area of D Offer some sentence starters First I would Next I would and then I would Encourage them to show their moves on the diagram in their material Emphasize that we are interested only in the plan for finding area and not in the area itself so no calculation is expected Then give them time to share their moves with their group Ask students to indicate what their first move was Did their very first involve decomposing the quadrilateral enclosing the quadrilateral another move Ask the students whose first move is to decompose the figure How many pieces resulted from the decomposition 2 pieces 3 pieces 4 pieces More What is the next move Rearrange Duplicate a piece Calculate the area of a piece Something else Ask the students whose first move is to enclose the figure How many rectangles did you create 1 rectangle 2 rectangle More What is the next move Rearrange the extra pieces Calculate the area of an extra piece Something else For each sequence that students mentioned draw a quick diagram to illustrate it for all to see Once students have a chance to see a variety of approaches ask students to revisit their sequence of moves Give students time to think about the pros and cons of their original plan 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 139

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS and if there was another strategy that they found productive Invite a few students to share their reflections Then give students quiet time to complete the activity and access to their geometry toolkits Ask students to keep in mind the merits of the different strategies they have seen as they plan their work YOUR NOTES ACTIVITY 2 TASK 1 43 Find the area of two quadrilaterals of your choice Show your reasoning A B D E C F STUDENT RESPONSE Reasoning varies Students could decompose the quadrilateral into parallelograms and triangles to find the area decompose and rearrange the pieces into a shape of which they can easily find the area or enclose the figure in a rectangle and subtract the area of the extra pieces Figure A 12 square units Figure B 18 square units Figure C 28 square units Figure D 14 square units Figure E 15 square units Figure F 18 square units 140 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 11 DISCUSSION GUIDANCE To conclude the activity ask students to choose one quadrilateral they worked on other than D and tell their group the first couple of moves they made for finding its area and why Encourage other group members to listen carefully check that the reasoning is valid and offer feedback ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Students may have noticed that all the approaches involved decomposing one or more regions into triangles rectangles or both If not mentioned by students point this out Emphasize that we can decompose any polygon into triangles and rectangles to find its area SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Conceptual Processing Eliminate Barriers Assist students to see the connections between new problems and prior work Students may benefit from a review of different representations to activate prior knowledge SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Use this routine before groups share the first couple of moves they made to find the area of one of the quadrilaterals Ask the class What are some important words or phrases you can use when you describe the moves you made Record and display student responses In addition to the relevant mathematical terms call students attention to the language that helps to communicate order within the approaches Remind students to use the display as a resource during their group discussions Design Principle s Optimize output for explanation Maximize meta awareness Concept Exploration Activity 3 AREA APPLICATIONS This activity provides an opportunity for students to apply the strategies they have learned in this lesson and throughout the mission to solve real world and mathematical problems 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 141

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS involving finding the area of right triangles other triangles special quadrilaterals and polygons As students solve they may notice they need to find the length of missing dimensions Look for students who will be able to share their strategies with the group and also identify the students who have difficulty finding the appropriate dimensions YOUR NOTES LAUNCH Keep students in their groups of 2 4 Begin with some quiet work time Give groups time to compare their answers and strategies then have a discussion ACTIVITY 3 TASK 1 53 In each of the diagrams below find the area of the shaded region Explain or show your reasoning 1 Note Each grid square is 1 square unit 2 13 cm 9 cm 5 cm 20 cm 142 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 11 3 16 cm 11 cm ZEARN MATH TEACHER LESSON MATERIALS 10 cm YOUR NOTES 4 cm STUDENT RESPONSE Reasoning varies Figure A The area is 40 5 square units Sample reasoning Find the area of each of the shaded triangles since their side lengths are easy to count then add the four areas together Figure B The area is 64 square centimeters Sample reasoning Find the area of both triangles and subtract the area of the smaller from the larger leaving only the shaded region Figure C The area is 79 square centimeters Sample reasoning Decompose the shaded region into a rectangle the full width of the figure and a triangle ACTIVITY 3 TASK 2 63 Jessalyn designed a new school flag shown below that she ll make with two different fabric colors purple and orange How much of each color of fabric will she need to construct her flag 2 ft 3 ft 3 4 ft 5 ft STUDENT RESPONSE Jessalyn will need 6 38 ft2 of orange fabric and 8 58 ft2 of purple fabric 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 143

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES DISCUSSION GUIDANCE The discussion should focus on the various strategies we can use to find the area of a polygon both on and off the grid to solve real world and mathematical problems Ask students to share the strategies they used to solve each problem making sure to share a variety of strategies If students have difficulty finding the lengths of missing dimensions have other students share their strategies for all to hear Digital Lesson Clara decomposed the polygon into these two triangles and added their areas to find the area of the polygon Find the area in a different way Explain the strategy you would use and draw on the polygon to help you explain I could use the decompose and add the parts enclose and subtract strategy to find the area First I would Then I would Finally I would STUDENT RESPONSE Answers vary and should include an explanation and drawing on the polygon to show the selected strategy Sample explanation for students who chose the enclose and subtract strategy I could use the enclose and subtract strategy to find the area First I would draw a rectangle around the polygon Then I would find the area of the rectangle and the area of the 4 triangles created when I drew the rectangle Finally I would subtract the area of the triangles from the area of the rectangle to find the area of the polygon 144 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 11 Wrap Up ZEARN MATH TEACHER LESSON MATERIALS LESSON SYNTHESIS YOUR NOTES To review the defining characteristics of a polygon return to the image in the first activity What are Polygons and display the list of defining features students generated in that activity Revisit each figure that is not a polygon and ask students to explain why it is not a polygon Encourage students to use their list to support their explanations as well as to suggest C D E revisionsAto their workingB definition A B C D E F G H I J F G H I J Here is a polygon with 5 sides B A C E D Ask students How do we know this figure is a polygon It is composed of line segments Each segment meets only one other segment at each end The segments do not cross one another It is two dimensional What does it mean to find the area of this polygon It means finding the area of the region inside it How can we find the area of this polygon We can decompose the region inside it into triangles and rectangles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 145

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS TERMINOLOGY B C Edge Each straight side of a polygon is called an edge YOUR NOTES For example the edges of this polygon are segments AB BC CD DE and EA A D B Polygon A polygon is a closed two dimensional shape with straight sides that do not cross each other Figure ABCDE is an example of a polygon E C E A D Quadrilateral A quadrilateral is a type of polygon that has 4 sides A rectangle is an example of a quadrilateral A pentagon is not a quadrilateral because it has 5 sides B Vertex vertices A vertex is a point where two or more edges meet When we have more than one vertex we call them vertices The vertices in this polygon are labeled A B C D and E A C E D EXIT TICKET This Exit Ticket assesses students understanding of the defining characteristics of a polygon and the ways it can be decomposed LAUNCH Give students access to their geometry toolkits Tell students that they need to show only how the area could be found they do not have to actually calculate the area 146 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS TASK 1 Here are two five pointed stars A student said Both figures A and B are polygons They are both composed of line segments and are two dimensional Neither have curves Do you agree with the statement Explain your reasoning AA BB YOUR NOTES 2 Here is a five sided polygon Describe or show the strategy you would use to find its area Mark up and label the diagram to show your reasoning so that it can be followed by others It is not necessary to actually calculate the area STUDENT RESPONSE 1 Disagree Only Figure B is a polygon Explanations vary Sample explanation Every segment in Figure A meets or cross more than two segments at its ends so it is not a polygon Each segment in Figure B meets only one other segment at each end 2 Answers vary Sample diagrams and responses The polygon can be decomposed into three triangles one with a base of 6 units and a height of 3 a second one with a base of 7 and a height of 6 and a third with a base of 4 and a height of 6 All areas can be calculated using the area formula The polygon can be decomposed into two triangles and a rectangle One triangle has a base of 6 and a height 3 and the second has a base of 6 and a height of 1 Their areas can be calculated with the area formula The rectangle is 6 by 4 so its area is the product of 6 and 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 147

G6M1 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 1 LESSON 12 What Is Surface Area YOUR NOTES Calculate the surface area of a rectangular prism and explain orally and in writing the solution method LEARNING GOALS Comprehend that the term surface area in written and spoken language refers to how many square units it takes to cover all the faces of a threedimensional object LEARNING GOALS STUDENT FACING Let s cover the surfaces of some three dimensional objects LEARNING TARGETS STUDENT FACING I know what the surface area of a three dimensional object means Snap cubes REQUIRED MATERIALS REQUIRED PREPARATION 148 Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Prepare 12 cubes per student and extra copies of isometric dot paper for Building with Snap Cubes activity Build several rectangular prisms that are each 2 cubes by 3 cubes by 5 cubes for the Exit Ticket 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

This lesson introduces students to the concept of surface area They use what they learned about area of rectangles to find the surface area of prisms with rectangular faces Students begin exploring surface area in concrete terms by estimating and then calculating the number of square sticky notes it would take to cover a cardboard box Because students are not given specific techniques ahead of time they need to make sense of the problem and persevere in solving it The first activity is meant to be open and exploratory In the second activity they then learn that the surface area in square units is the number of unit squares it takes to cover all the surfaces of a three dimensional figure without gaps or overlaps G6M1 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Later in the lesson students use cubes to build rectangular prisms and then determine their surface areas For this lesson you may want to give the Independent Digital Lesson as homework and do the Warm Up and Concept Exploration as a whole class Warm Up COVERING THE CARDBOARD BOX PART 1 Instructional Routines Poll the Class Notice and Wonder This activity prepares students to think about surface area which they explore in this lesson and upcoming lessons Students watch a video of a box being gradually tiled with nonoverlapping sticky notes The box was left only partially tiled which raises the question of the number of sticky notes it takes to cover the entire rectangular prism Students estimate the answer to this question LAUNCH Arrange students in groups of 2 Show the video of Ms Parker beginning to cover a large box with sticky notes that is in the presentation or display the following still images for all to see Before starting the video or displaying the image ask students to be prepared to share one thing they notice and one thing they wonder 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 149

G6M1 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Give students time to share their observation and question with a partner Invite a few students to share their questions with the class If the question How many sticky notes would it take to cover the entire box is not mentioned ask if anyone wondered how many sticky notes it would take to cover the entire box Give students time to make an estimate WARM UP TASK 1 Estimate an answer How many sticky notes would it take to cover the box excluding the bottom STUDENT RESPONSE Estimates vary The actual number of sticky notes is 432 Good estimates are in the 350 650 range DISCUSSION GUIDANCE Poll the class for students estimates and record them for all to see Invite a couple of students to share how they made their estimate Explain to students that they will now think about how to answer this question 150 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 12 Concept Exploration Activity 1 COVERING THE CARDBOARD BOX PART 2 ZEARN MATH TEACHER LESSON MATERIALS Instructional Routine MLR7 Compare and Connect After making an estimate of the number of sticky notes on a box in the warm up students now brainstorm ways to find that number more accurately and then go about calculating an answer The activity prompts students to transfer their understandings of the area of polygons to find the surface area of a three dimensional object YOUR NOTES Students learn that the surface area of a three dimensional figure is the total area of all its faces Since the area of a region is the number of square units it takes to cover the region without gaps and overlaps surface area can be thought of as the number of square units needed to cover all sides of an object without gaps and overlaps The square sticky notes illustrate this idea in a concrete way As students work notice the varying approaches taken to determine the number of sticky notes needed to tile the faces of the box excluding the bottom Identify students with different strategies to share later LAUNCH Arrange students in groups of 2 4 Give students quiet time to think about the first question and then time to share their responses with their group Ask students to pause afterwards Select some students to share how they might go about finding out the number of sticky notes and what information they would need Students may ask for some measurements The measurements of the box in terms of sticky notes Tell students that the box is 15 by 7 by 6 The measurements of the box in inches or centimeters Tell students that you don t have that information and prompt them to think of another piece of information they could use The measurements of each sticky note Share that it is 3 inches by 3 inches If no students mention needing the edge measurements of the box in terms of sticky notes let them begin working on the second question and provide the information when they realize that it is needed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 151

G6M1 LESSON 12 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 Earlier you learned about a box being covered with sticky notes Answer the questions below about the box YOUR NOTES 1 How could you find the actual number of sticky notes it will take to cover the box excluding the bottom What information would you need to know 2 Use the information you have to find the number of sticky notes needed to cover the box STUDENT RESPONSE 1 Find the area of each side of the box excluding the bottom and add them together Needed information measurements of the box edge lengths in sticky notes 2 Answers vary Strategies may be a combination of the following two strategies Multiply the number of sticky notes along each edge of each side Add all of the products Multiply the edge lengths of each side of the box to find the area of each side Add all of the areas ACTIVITY 1 RECAP Invite previously identified students or groups to share their answer and strategy On a visual display record each answer and each distinct process for determining the surface area i e multiplying the side lengths of each rectangular face and adding up the products After each presentation poll the group on whether others had the same answer or process Play the video that reveals the actual number of sticky notes needed to cover the box If students answers vary from that shown on the video discuss possible reasons for the differences Tell students that the question they have been trying to answer is one about the surface area of the box Explain that the surface area of a three dimensional figure is the total area of all its surfaces We call the flat surfaces on a three dimensional figure its faces 152 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

The surface area of a rectangular prism would then be the combined area of all six of its faces In the context of this problem we excluded the bottom face since it is sitting on the ground and will not be tiled with sticky notes Discuss G6M1 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS What unit of measurement are we using to represent the surface area of the box Square sticky notes YOUR NOTES Would the surface area change if we used larger or smaller sticky notes How Yes if we use larger sticky notes we would need fewer If we use smaller ones we would need more ANTICIPATED MISCONCEPTIONS Students may treat all sides as if they were congruent rectangles That is they find the area of the front of the box and then just multiply by 5 or act as if the top is the only side that is not congruent to the others If there is any large object in the shape of a rectangular prism in the classroom like a cabinet consider showing students that only the sides opposite each other can be presumed identical Students may neglect the fact that the bottom of the box will not be covered Point out that the bottom is inaccessible because of the floor SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Conceptual Processing Visual Aids Use a picture of the cardboard box with sticky notes along the edges to aid students who benefit from access to multiple modalities e g visual kinesthetic etc SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Listening MLR 7 Compare and Connect As students share their strategies for determining the number of sticky notes that cover the box ask students to make connections between the various strategies Some students will calculate the number of sticky notes that will cover each of the five faces of the box and add them together Other students may realize that opposite faces of the box are congruent so it is only necessary to calculate the area of three faces of the box Encourage students to explain why both methods result in the same answer This will promote students use of mathematical language as they make sense of the various methods for finding the surface area of a rectangular prism 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 153

G6M1 LESSON 12 Design Principle s Cultivate conversation maximize meta awareness ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Concept Exploration Activity 2 BUILDING WITH SNAP CUBES Instructional Routines Think Pair Share MLR8 Discussion Supports This activity encourages students to apply strategies for finding the area of polygons to finding the surface area of rectangular prisms Students use 12 cubes to build a prism think about its surface area and use isometric dot paper to draw their prism As students build their prisms notice those with different designs and those with the same design but different approaches to finding surface area e g by counting individual square by multiplying the edge lengths of rectangular faces etc LAUNCH Read the description of the prism in the task statement together Remind students that we refer to the flat surfaces of a three dimensional figure as faces Give students a short time to think about how we know the surface area of the shown prism is 32 square units Ask 1 2 students to explain their reasoning to the group Use students explanations to highlight the meaning of surface area i e that the area of all the faces need to be accounted for including those we cannot see when looking at a two dimensional drawing Tell students they will use 12 cubes to build a different prism draw it and find its surface area Consider doing a quick demonstration on how to draw a simple prism on isometric dot paper Start with one cube and then add a cube in each dimension Tell students that in this activity we call each face of a single cube 1 square unit Give each student 12 cubes to build a prism and quiet work time If students are using snap cubes say that we will pretend all of the faces are completely smooth so they do not need to worry about the innies and outies of the snap cubes As students work consider arranging two students with contrasting designs or strategies as partners Ask partners to share their answers explanations and drawings Stress that each partner should focus their explanation on how they went about finding surface area The listener should think about whether the explanation makes sense or if anything is amiss in the reasoning 154 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 12 ACTIVITY 2 TASK 1 3 Here is a sketch of a rectangular prism built from 12 cubes It has six faces but you can only see three of them in the sketch It has a surface area of 32 square units You have 12 snap cubes from your teacher Use all of your snap cubes to build a different rectangular prism with different edge lengths than shown in the prism here ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 1 How many faces does your figure have 2 What is the surface area of your figure in square units 3 Draw your figure on isometric dot paper Color each face a different color STUDENT RESPONSE 1 There are 6 faces front back left right top and bottom 2 Answers vary based on design Sample responses For a prism that is 12 units by 1 unit by 1 unit the surface area is 50 square units 4 12 2 1 50 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 155

G6M1 LESSON 12 For a prism that is 6 units by 2 units by 1 unit the surface area is 40 square units 2 12 2 6 2 2 40 For a prism that is 4 units by 3 units by 1 unit the surface area is 38 square units 2 12 2 4 2 3 40 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 3 Drawings vary but all prisms should have one edge length that is 1 unit DISCUSSION GUIDANCE After partner discussions select a student to highlight for the group the strategy or strategies for finding surface area methodically Point out that in this activity each face of their prism is a rectangle and that we can find the area of each rectangle by multiplying its side lengths and then add the areas of all the faces Explain that later when we encounter non rectangular prisms we can likewise reason about the area of each face the way we reasoned about the area of a polygon ANTICIPATED MISCONCEPTIONS Students may count the faces of the individual snap cubes rather than faces of the completed prism Help them understand that the faces are the visible ones on the outside of the figure SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allow students who struggle with fine motor skills to dictate how to draw the figure on the isometric dot paper as needed SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Use this routine to support discussion Call on students to use mathematical language e g cubes faces surface area square units etc to restate and or revoice the strategy or strategies presented Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class This will provide more students with an opportunity to produce language that describes strategies for finding surface area Design Principle s Support sense making maximize meta awareness 156 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 12 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 1 What is the surface area of a three dimensional figure The surface area of a three dimensional figure is 2 Explain how you would find the surface area of any prism To find the surface area of a prism first I would Next I would STUDENT RESPONSE Responses vary Sample response The surface area of a three dimensional figure is the total area it takes to cover the entire figure without gaps or overlaps To find the surface area of a prism first I would find the area of each of the 6 faces of the prism Next I would add all of those areas together to get the total surface area Wrap Up LESSON SYNTHESIS In this lesson we found the surface areas of a box and of rectangular prisms built out of cubes What does it mean to find the surface area of a three dimensional figure It means finding the number of unit squares that cover the entire surface of the object without gaps or overlaps 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 157

G6M1 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS How can we find the number of unit squares that cover the entire surface of an object We can count them or we can find the area of each face of the object and add the areas of all faces YOUR NOTES How are finding surface area and finding area alike How are they different They both involve finding the number of unit squares that cover a region entirely without gaps and overlaps Both have to do with two dimensional regions Finding area involves a single polygon Finding surface area means finding the sum of the areas of multiple polygons faces of which a three dimensional figure is composed TERMINOLOGY Face Each flat side of a polyhedron is called a face For example a cube has 6 faces and they are all squares Surface area The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron without any gaps or overlaps For example if the faces of a cube each have an area of 9 cm2 then the surface area of the cube is 6 9 or 54 cm2 EXIT TICKET LAUNCH Prepare several rectangular prisms that are each 2 cubes by 3 cubes by 5 cubes Display one for all to see and pass the rest around for students to examine if needed TASK A rectangular prism made is 3 units high 2 units wide and 5 units long What is its surface area in square units Explain or show your reasoning 158 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 12 STUDENT RESPONSE ZEARN MATH TEACHER LESSON MATERIALS The surface area is 2 3 5 2 5 2 3 62 or 62 square units ANTICIPATED MISCONCEPTIONS YOUR NOTES Students may not include the bottom face as it is not visible when the prism is sitting on a table Students may find the number of cubes instead of the surface area due to their previous volume work with rectangular prisms in grade 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 159

G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 1 LESSON 13 Polyhedra YOUR NOTES Compare and contrast orally and in writing features of prisms and pyramids LEARNING GOALS Comprehend and use the words face edge vertex and base to describe polyhedra in spoken and written language Understand that the word net refers to a two dimensional figure that can be assembled into a polyhedron and create a net for a given polyhedron LEARNING GOALS STUDENT FACING Let s investigate polyhedra I can describe the features of a polyhedron using mathematical vocabulary LEARNING TARGETS STUDENT FACING I can explain the difference between prisms and pyramids I understand the relationship between a polyhedron and its net Nets of polyhedra Scissors Pre assembled or commercially produced polyhedra Tape REQUIRED MATERIALS Glue or gluesticks Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles Template for Warm Up Template for Concept Exploration Activity 1 REQUIRED PREPARATION Assemble collections of geometric figures that each contains at least 2 familiar polyhedra 2 unfamiliar polyhedra and 2 non polyhedra Prepare one collection for each group of 3 4 students If pre made polyhedra are unavailable assemble some from the nets in the template for the warm up Print and pre cut the nets and polygons in the Template for Concept Exploration Activity 1 Prepare 1 set per group of 3 4 students along with tape to join the polygons into a net 160 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

In this lesson students learn about polyhedra and their nets They also study prisms and pyramids as types of polyhedra with certain defining features G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Polyhedra can be thought of as the three dimensional analog of polygons Here are some important aspects of polygons YOUR NOTES They are made out of line segments called edges Edges meet at a vertex The edges only meet at vertices Polygons always enclose a two dimensional region Here is an analogous way to characterize polyhedra They are made out of filled in polygons called faces Faces meet at an edge The faces only meet at edges Polyhedra always enclose a three dimensional region Students do not need to memorize a formal definition of a polyhedron but help them make sense of nets and surface area Warm Up WHAT ARE POLYHEDRA MATERIALS TEMPLATE FOR WARM UP Instructional Routine Notice and Wonder In this warm up students analyze examples and counterexamples of polyhedra observe their defining characteristics and use their insights to sort objects into polyhedra and nonpolyhedra They then start developing a working definition of polyhedra Prepare physical examples of polyhedra and non polyhedra for students to sort These examples should be geometric figures rather than real world objects such as shoe boxes or canisters If such figures are not available make some ahead of time using the nets in the template 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 161

G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS As students work and discuss notice those who can articulate defining features of a polyhedron Invite them to share later LAUNCH YOUR NOTES Arrange students in groups of 3 4 Give students time to study the examples and non examples in the task statement Ask them to be ready to share at least one thing they notice and one thing they wonder Give the class time to share some of their observations and questions Next give each group a physical set of three dimensional figures The set should include some familiar polyhedra some unfamiliar ones and some non polyhedra Ask groups to sort the figures into polyhedra and non polyhedra the first question If group members disagree about whether a figure is a polyhedron discuss the disagreements When the group has come to an agreement give them quiet time to complete the second question WARM UP TASK 1 Study the examples and non examples of polyhedra Use them to answer the questions below Here are pictures that represent polyhedra Here are pictures that do not represent polyhedra 1 Your teacher will give you some figures or objects Sort them into polyhedra and nonpolyhedra 2 What features helped you distinguish the polyhedra from the other figures 162 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 No response required 2 Answers vary Sample responses YOUR NOTES Polyhedra are made from polygons Polyhedra don t have any unattached edges Non polyhedra sometimes have curved or round surfaces Some non polyhedra have a face that is not a polygon DISCUSSION GUIDANCE Invite students to share the features that they believe characterize polyhedra Record their responses for all to see For each one ask the class if they agree or disagree If they generally agree ask if there is anything they would add or elaborate to make the description clearer or more precise If they disagree ask for an explanation or a counterexample Students will have a chance to refine their definition of polyhedra later in the lesson after exploring prisms and pyramids and learning about nets so it is not important to compile a complete or precise set of descriptions or features Use a sample polyhedron or a diagram as shown here to introduce or reinforce the terminology surrounding polyhedra The polygons that make up a polyhedron are called faces The places where the sides of the faces meet are called edges The corners are called vertices Clarify that the singular form is vertex and the plural form is vertices edge face face vertex vertex edge 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 163

G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 1 PRISMS AND PYRAMIDS MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 1 Instructional Routine MLR2 Collect and Display YOUR NOTES This activity serves two goals to uncover the defining features of prisms and pyramids as well as to introduce nets as two dimensional representations of polyhedra Students first analyze prisms and pyramids and try to define their characteristics Next they learn about nets and think about the polygons needed to compose the nets of given prisms and pyramids They then use their experience with the nets of prisms and pyramids to sharpen and refine their definitions of these polyhedra Ask students discuss the features of prisms and pyramids encourage them to use the terms face edge and vertex vertices in their descriptions LAUNCH Arrange students in groups of 3 4 For the first question Tell students that Polyhedra A F are all prisms and Polyhedra P S are all pyramids Display and pass around the prisms and pyramids in the task statement if available Give students quiet time for the first question and time to discuss their observations in their groups Ask them to pause for a discussion before moving on Solicit students ideas about features that distinguish prisms and pyramids Record students responses in a two columns one for prisms and the other for pyramids It is not important that the lists are complete at this point Next tell students that we are going to use nets to better understand prisms and pyramids Explain that a net is a two dimensional representation of a polyhedron Display a cube assembled from the net provided in the template for the warm up as well as a cutout of an unfolded net consider removing the flaps Demonstrate how the net with squares could be folded and assembled into a cube Point out how the number and the shape of the faces on the cube correspond to the number and the shape of the polygons in the net For the second question Give groups quiet think time and time to discuss their response 164 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

To verify their answer give each group one of the three nets from the first page of the template Ask them to try to assemble a triangular pyramid from their net G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Invite groups to share with the small group whether it can be done Discuss why net 3 cannot be assembled into Pyramid P two of the triangles would overlap For the last question tell students that they will create a net of another prism or pyramid YOUR NOTES Assign each group a prism or a pyramid from the task statement except for Prism B and Pyramid P Give each group a set of pre cut polygons from the last two pages of the template Tell students to choose the right kind and number of polygons that make up their polyhedron Then arrange the polygons so that when taped and folded the arrangement is a net and could be assembled into their prism or pyramid Encourage them to think of more than one net if possible ACTIVITY 1 TASK 1 2 Use the images of prisms and pyramids to answer the questions below 1 Here are some polyhedra called prisms 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 165

G6M1 LESSON 13 Here are some polyhedra called pyramids ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES a Look at the prisms What are their characteristics or features b Look at the pyramids What are their characteristics or features 2 Which of the following nets can be folded into Pyramid P Select all that apply STUDENT RESPONSE 1 Answers vary Sample responses a A prism has rectangular faces Some of the faces are parallel to one another A prism may have two faces that are not rectangles b A pyramid has a triangle for all but one of its faces That one face might be a different polygon and all triangles share an edge with it All triangles also meet at a single vertex 2 Nets 1 and 2 can be assembled into Pyramid P but net 3 cannot ACTIVITY 1 TASK 2 3 Your teacher will give your group a set of polygons and assign a polyhedron a Decide which polygons are needed to compose your assigned polyhedron List the polygons and how many of each are needed 166 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

b Arrange the cut outs into a net that if taped and folded can be assembled into the polyhedron Sketch the net If possible find more than one way to arrange the polygons show a different net for the same polyhedron G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE YOUR NOTES a Answers vary Possible responses Prism A 2 triangles 3 squares Prism C 2 triangles 3 rectangles Prism D 2 pentagons 5 rectangles Prism E 2 squares 4 rectangles Prism F 2 hexagons 6 rectangles Pyramid Q 1 square 4 triangles Pyramid R 1 pentagon 5 triangles Pyramid S 1 hexagon 6 triangles b Drawings vary DISCUSSION GUIDANCE Select groups to share their arrangements of polygons If time permits and if possible have students tape their polygons and fold the net to verify that it could be assembled into the intended polyhedron Discuss What do the nets of prisms have in common They all have rectangles They have a pair of polygons that may not be rectangles What do the nets of pyramids have in common They all have triangles They have one polygon that may not be a triangle Is there only one possible net for a prism or a pyramid No the polygons can be arranged in different ways and still be assembled into the same prism or pyramid Explain the following points about prisms and pyramids A prism has two parallel identical faces called bases and a set of rectangles connecting the bases 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 167

G6M1 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Prisms are named for the shape of the bases For example if the base of a prism is a pentagon then the prism is called a pentagonal prism A pyramid has one face called the base that can be any polygon and a set of faces that are all triangles Each edge of the base is shared with an edge of a triangle All of these triangles meet at a single vertex YOUR NOTES Pyramids are named for the shape of their base For example if the base of a pyramid is a square then the pyramid is called a square pyramid SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives Provide manipulatives i e prisms and pyramids to aid students who benefit from hands on activities Executive Functioning Visual Aids Create an anchor chart i e polyhedra prism pyramid base face vertex edge etc publicly displaying important definitions rules formulas or concepts for future reference SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Conversing MLR 2 Collect and Display As students work on creating a net of their assigned polyhedron circulate and listen to the language students use when talking about the polygons that make up their polyhedra as well as the characteristics of their polyhedra e g triangle rectangle square hexagon pentagon vertex edge face Collect this language with corresponding drawings and display it for all students to see Remind students to borrow language from the display as they describe the features of prisms and pyramids This will help students produce mathematical language to describe and define characteristics of polyhedra Design Principle s Support sense making 168 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 13 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS Is this polyhedron a prism a pyramid or neither Explain how you know How many faces does it have YOUR NOTES STUDENT RESPONSE Answers vary Sample response This polyhedron is a prism I know because it has two identical polygon sides that are connected by rectangles This prism has 10 faces Wrap Up LESSON SYNTHESIS Review the features of prisms and pyramids by selecting 1 2 polyhedra used in the warm up Ask students to explain using the terminology they learned if possible why each one is or is not a prism or a pyramid If it is a prism or pyramid ask students to name it Revisit the working definition of polyhedra generated earlier in the lesson and ask students to see if or how it might be refined Ask if there is anything they should add remove or adjust given their work with prisms pyramids and nets Highlight the following points about polyhedra Ask students to illustrate each point using a figure or a net A polyhedron is a three dimensional figure built from filled in polygons We call the polygons faces The plural of polyhedron is polyhedra All edges of polygons meet another polygon along a complete edge 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 169

G6M1 LESSON 13 Each polygon meets one and only one polygon on each of the edges ZEARN MATH TEACHER LESSON MATERIALS The polygons enclose a three dimensional region Consider displaying in a visible place the key ideas from students list and from this discussion so that they can serve as a reference later YOUR NOTES TERMINOLOGY Base of a prism or pyramid The word base can also refer to a face of a polyhedron A prism has two identical bases that are parallel A pyramid has one base base base base A prism or pyramid is named for the shape of its base pentagonal prism hexagonal pyramid Face Each flat side of a polyhedron is called a face For example a cube has 6 faces and they are all squares Net A net is a two dimensional figure that can be folded to make a polyhedron Here is a net for a cube Polyhedron Polyhedra A polyhedron is a closed threedimensional shape with flat sides When we have more than one polyhedron we call them polyhedra Here are some drawings of polyhedra Prism A prism is a type of polyhedron that has two bases that are identical copies of each other The bases are connected by rectangles or parallelograms Here are some drawings of some prisms 170 triangular prism pentagonal prism rectangular prism 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 13 Pyramid A pyramid is a type of polyhedron that has one base All the other faces are triangles and they all meet at a single vertex Here are some drawings of pyramids ZEARN MATH TEACHER LESSON MATERIALS rectangular pyramid hexagonal pyramid heptagonal pyramid YOUR NOTES EXIT TICKET 1 Write your best definition or description of a polyhedron If possible use the terms you learned in this lesson 2 Which of these five polyhedra are prisms Which are pyramids STUDENT RESPONSE 1 Answers vary but might include one or more of these elements A polyhedron is a threedimensional figure made from faces that are filled in polygons Each face meets one and only one other face along a complete edge The points where edges meet are called vertices 2 A C and D are prisms B and E are pyramids ANTICIPATED MISCONCEPTIONS Some students may mistake a triangular prism especially one that is not sitting on one of its bases as a pyramid because it has triangular faces 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 171

Template for Lesson 13 Warm Up page 1 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra A Download for free at openupresources org 172 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 13 Warm Up page 2 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra B Download for free at openupresources org 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 173

Template for Lesson 13 Warm Up page 3 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra C Download for free at openupresources org 174 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Template for Lesson 13 Warm Up page 5 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra E Download for free at openupresources org 176 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 13 Warm Up page 6 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra F Download for free at openupresources org 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 177

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Template for Lesson 13 Warm Up page 8 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra H Download for free at openupresources org 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 179

Template for Lesson 13 Warm Up page 9 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra J Download for free at openupresources org 180 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 13 Warm Up page 10 of 10 Blackline Master for Classroom Activity 6 1 13 1 What are Polyhedra K Download for free at openupresources org 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 181

Template for Lesson 13 Concept Exploration Activity 1 page 1 of 3 182 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 13 Concept Exploration Activity 1 page 2 of 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 183

Template for Lesson 13 Concept Exploration Activity 1 page 3 of 3 184 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 14 Nets and Surface Area G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Match polyhedra with their nets and justify orally that they match LEARNING GOALS Use a net with gridlines to calculate the surface area of a prism or pyramid and explain in writing the solution method Visualize and identify the polyhedron that can be assembled from a given net LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Let s use nets to find the surface area of polyhedra I can match polyhedra to their nets and explain how I know When given a net of a prism or a pyramid I can calculate its surface area Template for Warm Up Template for Concept Exploration Activity 1 REQUIRED MATERIALS Nets of polyhedra Tape Geometry toolkits Tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles REQUIRED PREPARATION Prepare physical copies of the nets in the warm up in case needed to support students with visualization The template contains a larger version of these nets Make copies of the nets in the Template for Concept Exploration Activity 1 Prepare one set of 3 nets A B and C and some glue or tape for each group of 3 students 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 185

G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS Previously students learned about polyhedra analyzed and defined their features and investigated their physical representations Students also identified the polygons that compose a polyhedron they recognized a net as an arrangement of these polygons and as a twodimensional representation of a three dimensional figure YOUR NOTES This lesson extends students understanding of polyhedra and their nets They practice visualizing the polyhedra that could be assembled from given nets and use nets to find the surface area of polyhedra Warm Up MATCHING NETS MATERIALS TEMPLATE FOR WARM UP Instructional Routine Think Pair Share This warm up prompts students to match nets to polyhedra It invites them to think about the polygons that make up a polyhedron and to mentally manipulate nets which helps develop their visualization skills LAUNCH Give students quiet think time to match nets to polyhedra and then time to discuss their response and reasoning with a partner Encourage students to use the terminology they learned in prior lessons To support students who need more time or help in visualization prepare physical models of the polyhedra and copies of the nets from the template Pre cut the nets or have scissors available so that students can assemble the nets and test their ideas 186 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 14 WARM UP TASK 1 ZEARN MATH TEACHER LESSON MATERIALS Match each net with its corresponding polyhedron and name the polyhedron YOUR NOTES Be prepared to explain how you know the net and polyhedron go together B A 1 2 C 3 D E 4 5 STUDENT RESPONSE Net A is a square pyramid 3 It has five faces one square and four triangles just like the square pyramid similar reasoning for each figure Net B is a rectangular prism 2 Net C is a triangular pyramid 4 Net D is a triangular prism 5 Net E is a cube or square prism 1 DISCUSSION GUIDANCE Invite a few students to share their matching decisions and reasoning with the class Ask students What clues did you use to help you match How did you check if you were right If there is not unanimous agreement on any of the nets ask students with differing opinions to explain their reasoning Discuss to come to an agreement 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 187

G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS If students have trouble distinguishing between figures A C and D remind them that prisms and pyramids can both contain faces that are triangles In a pyramid all triangular faces that are not the base meet at a one vertex and have shared edges In a prism there can be a triangular base but the other faces are quadrilaterals SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Processing Time Provide the image to students who benefit from extra processing time to review prior to implementation of this activity Concept Exploration Activity 1 USING NETS TO FIND SURFACE AREA MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 1 Instructional Routines Anticipate Monitor Select Sequence Connect MLR2 Collect and Display In this activity students cut and assemble nets into polyhedra and learn to use nets to find surface area The presence of a grid supports students in their calculations It also reinforces the idea of area as the number of unit squares in a region and the connection between area and surface area Students apply what they learned earlier about areas of triangles and parallelograms to find surface area As students make calculations monitor their processes Note those who work systematically to find surface area e g by organizing the measurements of each face calculating the area of each face and adding the areas together and those who don t Encourage students with disorganized or scattered work to take a more systematic approach Demonstrate strategies such as labeling both the polygons on the net and portions of their work that pertain to those faces Also notice students who look for and use structure for instance by grouping certain polygons together and finding the area of the composite shape e g a group of rectangles that have a common side length or by identifying multiple copies of the same polygon and calculating the area once Select them to share their work later 188 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Arrange students in groups of 3 Give each group one of each net A B and C tape and access to their geometry toolkits especially scissors Explain to students that they will cut some nets assemble them into polyhedra and calculate their surface areas Remind students that the surface area of a three dimensional figure is the sum of the areas of all of its faces Ask students to complete the first question before cutting anything YOUR NOTES Point out that the net has shaded and unshaded polygons Explain that only the shaded polygons in the nets will show once the net is assembled The unshaded polygons are flaps to make it easier to glue or tape the polygons together They will get tucked behind the shaded polygons and are not really part of the polyhedron Tell students that creasing along all of the lines first will make it easier to fold the net up and attach the various polygons together A straightedge can be very helpful for making the creases Tell students that it is easy to miss or double count the area of a face when finding surface area Ask them to think carefully about how to record their calculations to ensure that all faces are accounted for correct measurements are used and errors are minimized When students have completed their calculations ask them to compare and discuss their work with another student with the same polyhedron ACTIVITY 1 TASK 1 2 You will be given the nets of three polyhedra Use the nets to answer the questions in your notes A B C 1 Name the polyhedron that each net would form when assembled A 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 189

G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS B C 2 Cut out your nets and use them to create three dimensional shapes YOUR NOTES 3 Find the surface area of each polyhedron Explain your reasoning clearly STUDENT RESPONSE 1 A rectangular prism B square pyramid C triangular prism 2 No answer required 3 Explanations vary Sample responses A The surface area is 82 square units 2 6 1 2 5 1 2 6 5 82 B The surface area is 48 square units 4 4 4 C The surface area is 48 square units 3 5 3 3 3 4 2 C The combined area of the three rectangular faces is 36 square units 3 12 36 The combined area of the two right triangles is 12 square units 2 12 3 4 12 The surface area is 48 square units because 36 12 48 1 2 4 4 48 1 2 3 4 48 DISCUSSION GUIDANCE For each polyhedron select at least 2 students with correct calculations but different approaches to share their work if possible For Polyhedron A select students who took the following approaches in this sequence Found the area of each rectangle separately Found the areas of pairs of identical rectangles 3 pairs total Calculated the area of a group of connected rectangles with the same length or width e g the four rectangles on the net with side length 6 units For Polyhedron B select students who Found the area of each of the 5 polygons separately Found the area of the square rearranged the 4 triangles into 2 parallelograms and calculated the area of each parallelogram 190 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Calculated the area of the square and the area of 1 triangle and multiplying the area of the triangle by 4 G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS For Polyhedron C select students who Found the area of each of the 5 polygons separately YOUR NOTES Rearranged the 2 right triangles into a rectangle and then found the area of each rectangle separately Calculated the area of each right triangle and doubled it and found the area of the group of connected rectangles with a width of 4 units Point out that the reasoning strategies we used earlier in the mission still apply here Even though we are working with three dimensional figures surface area is a two dimensional measure Highlight the benefits of approaching the problems systematically e g by labeling parts listing measurements and computations in order etc ANTICIPATED MISCONCEPTIONS If students do not identity the specific type of prism or pyramid remind them that they should also name each figure by the shape of their base SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Eliminate Barriers Assist students to see the connections between new problems and prior work Students may benefit from a review of different representations to activate prior knowledge SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Representing MLR 2 Collect and Display While students are working circulate and note how students record their surface area calculations Display common strategies for recording calculations using words visuals and symbols Remind students that they can borrow language from the display as they describe their approach This will help students connect the representations of the net to the words and calculations used in finding the surface area Design Principle s Maximize meta awareness 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 191

G6M1 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS Digital Lesson How can you use a net to find the surface area of a polyhedron Consider drawing a picture of a net to help you explain your answer YOUR NOTES STUDENT RESPONSE A net allows you to see all the faces of a polyhedron at once To find the surface area find the area of each face and then add all the areas together Wrap Up In this lesson we matched nets to the polyhedra assembled polyhedra from nets and used nets to find surface area LESSON SYNTHESIS Discuss How do we use a net to find surface area We calculate the area of each polygon on the net and add all the areas How is finding surface area using a net different from finding surface area by looking at a picture of a polyhedron as we had done with the filing cabinet or by studying the actual object as we had done with the snap cubes A net allows us to see all the faces of a polyhedron at once When working from a picture or drawing we need to visualize the hidden faces Working with an actual polyhedron could help but again we are not looking at all the faces at once we have to rotate the object and might miss or double count a face When using a net how do we keep track of your calculations or make sure all faces are accounted for We can label all the polygons and the calculations Are there ways to simplify the calculations Or is it best to find the area of each polygon one at a time Sometimes we can simplify the process by combining polygons and finding the area of the combined region e g a group of rectangles with the same side length If there are several polygons that are identical we can find the area of one polygon and multiply it by the number of identical polygons in the net 192 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 14 EXIT TICKET ZEARN MATH TEACHER LESSON MATERIALS 1 What kind of polyhedron can be assembled from this net 2 Find the surface area in square units of the polyhedron Show your reasoning YOUR NOTES STUDENT RESPONSE 1 It would assemble into a rectangular prism 2 The surface area would be 52 square units 2 3 4 2 2 4 2 2 3 52 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 193

Template for Lesson 14 Warm Up page 1 of 1 194 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 14 Concept Exploration Activity 1 page 1 of 3 A 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 195

Template for Lesson 14 Concept Exploration Activity 1 page 2 of 3 B 196 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Lesson 14 Concept Exploration Activity 1 page 3 of 3 C 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 197

G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 1 LESSON 15 More Nets More Surface Area Draw and assemble a net for the prism or pyramid shown in a given drawing LEARNING GOALS Interpret using words and other representations two dimensional representations of prisms and pyramids Solve real world and mathematical problems by using a net without gridlines to calculate the surface area of a prism or pyramid and explain in writing the solution method LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Let s draw nets and find the surface area of polyhedra I can draw the nets of prisms and pyramids I can solve real world and mathematical problems by calculating the surface area of prisms and pyramids Tape Demonstration nets with and without flaps REQUIRED MATERIALS Pre printed slips cut from copies of the Template for Concept Exploration Activity 1 Glue or gluesticks Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles REQUIRED PREPARATION 198 Copy and cut the Template for Concept Exploration Activity 1 Make one copy for every 9 students so that each student gets one drawing of a polyhedron Consider assignments of polyhedra in advance 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

This lesson further develops students ability to visualize the relationship between nets and polyhedra and their capacity to reason about surface area Previously students started with nets and visualized the polyhedra that could be assembled from the nets Here they go in the other direction from polyhedra to nets They practice mentally unfolding three dimensional shapes drawing two dimensional nets and using them to calculate surface area This sets them up to solve real world and mathematical problems involving surface area in the final activity of this lesson In the following lesson students will have a chance to compare and contrast surface area and volume as measures of two distinct attributes of a three dimensional figure G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Warm Up NOTICE AND WONDER WRAPPING PAPER Instructional Routine Notice and Wonder This warm up prompts students to think about a prism and its measurements in context and to consider potential questions that could be asked and answered Given their recent work students are likely to notice and wonder about surface area nets and the missing height of the box Students may also wonder about the volume of the box given their geometric work in grade 5 When prompted about how to find the surface area or the volume of the box students will likely ask about the missing measurement This is an opportunity for them to practice making a reasonable estimate LAUNCH Arrange students in groups of 2 Give quiet time to observe the image Ask students to be prepared to share at least one thing they notice and one thing they wonder about the picture Ask them to give a signal when they have noticed or wondered about something 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 199

G6M1 LESSON 15 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 Kiran is wrapping this box of sports cards as a present for a friend What do you notice What do you wonder YOUR NOTES STUDENT RESPONSE Answers vary Possible responses Notice the given side lengths of the box the box being covered in wrapping paper the box being a rectangular prism the area of the top and bottom faces being 10 square inches each Wonder the missing side length how many cards are in the box the volume of the box the surface area of the box how much wrapping paper it will take to cover the box DISCUSSION GUIDANCE Invite students to share their observations and questions Record the responses for all to see If no students wonder about the surface area the amount of wrapping paper needed or the volume of the box bring these questions up Tell students to choose either a question about surface area or one about volume and give them a minute to discuss with a partner how they would find the answer to the question If students suggest that it cannot be done because of missing information ask them to estimate the missing information Select a couple of students to share how they would find the surface area or the volume of the given box After each response poll the class on whether they agree or disagree 200 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Graphic Organizers Provide a t chart for students to record what they notice and wonder prior to being expected to share these ideas with others G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Concept Exploration Activity 1 BUILDING PRISMS AND PYRAMIDS MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 1 Instructional Routine MLR7 Compare and Connect Previously students used a given net of a polyhedron to find its surface area Here they use a given polyhedron to draw a net and then calculate its surface area Use the provided polyhedra to differentiate the work for students with varying degrees of visualization skills Rectangular prisms A and C triangular prisms B and D and square pyramids F and G can be managed by most students Triangular Prism E requires a little more interpretive work i e the measurements of some sides may not be immediately apparent to students Trapezoidal Prism H and Polyhedron I a composite of a cube and a square pyramid require additional interpretation and reasoning As students work remind them of the organizational strategies discussed in previous lessons i e labeling polygons showing measurements on the net etc LAUNCH Arrange students in groups of 2 3 Give each student in the group a different polyhedron from the template and access to their geometry toolkits Students need graph paper and a straightedge from their toolkits Explain to students that they will draw a net find its surface area and have their work reviewed by a peer Give students quiet time to draw their net on graph paper and then time to share their net with their group and get feedback When the group is sure that each net makes sense and all polygons of each polyhedron are accounted for students can proceed and use the net to help calculate surface area If time permits prompt students to cut and assemble their net into a polyhedron Demonstrate how to add flaps to their net to accommodate gluing or taping There should be as many flaps as there are edges in the polyhedron Remind students that this is different than the number of edges in the polygons of the net 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 201

G6M1 LESSON 15 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 You will be given a drawing of a polyhedron Follow the directions in your notes to draw its net and calculate its surface area YOUR NOTES 1 What polyhedron do you have 2 Study your polyhedron Then draw its net on graph paper Use the side length of a grid square as the unit 3 Label each polygon on the net with a name or number 4 Find the surface area of your polyhedron Show your thinking in an organized manner so that it can be followed by others 202 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 15 STUDENT RESPONSE ZEARN MATH TEACHER LESSON MATERIALS 1 Answers vary depending on polyhedron received A and C are rectangular prisms B D and E are triangular prisms F and G are square pyramids H is a trapezoidal prism I is a composite of a cube and a square pyramid YOUR NOTES 2 Net drawings vary A and C should have 6 rectangles B D and E should have 5 polygons 2 right triangles and 3 rectangles F and G should have 5 polygons 1 square and 4 triangles H should have 6 polygons 2 trapezoids and 4 rectangles I should have 9 polygons 5 squares and 4 triangles 3 Answers vary 4 Answers vary A 340 square units 2 5 8 2 5 10 2 8 10 340 B 408 square units 2 12 6 8 6 15 8 15 10 15 408 C 274 square units 2 13 4 2 13 5 2 4 5 274 D 300 square units 2 12 5 12 5 8 12 8 13 8 300 E 216 square units 2 12 6 4 6 12 2 5 12 216 F 240 square units 4 12 8 11 8 8 240 G 156 square units 4 12 6 10 6 6 156 H 316 square units The trapezoidal base can be decomposed into a 5 by 4 rectangle and a right triangle with a base of 3 units and a height of 4 2 5 4 2 12 3 4 8 12 2 5 12 4 12 316 I 205 square units 5 5 5 4 12 5 8 205 ACTIVITY 1 TASK 2 Do your calculations match Should they Do your nets result in the same polyhedra Should they Do your models match the picture you were given Why or why not 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 203

G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS DISCUSSION GUIDANCE Ask students who finish their calculation to find another person in the class with the same polyhedron and discuss the questions above displayed for all to see YOUR NOTES If time is limited consider having the answer key posted somewhere in the classroom so students can quickly check their surface area calculations Reconvene briefly for a discussion Invite students to reflect on the process of drawing a net and finding surface area based on a picture of a polyhedron Ask questions such as How did you know that your net shows all the faces of your polyhedron How did you know where to put each polygon or how to arrange all polygons so that if folded they can be assembled into the polyhedron in the drawing How did the net help you find surface area ANTICIPATED MISCONCEPTIONS Students may know what polygons make up the net of a polyhedron but arrange them incorrectly on the net i e when cut and assembled the faces overlap instead of meeting at shared edges or the faces are oriented incorrectly or are in the wrong places Suggest that students label some faces of the polyhedron drawing and transfer the adjacencies they see to the net If needed demonstrate the reasoning e g Face 1 and face 5 both share the edge that is 7 units long so I can draw them as two attached rectangles sharing a side that is 7 units long It may not occur to students to draw each face of the polyhedron to scale Remind them to use the grid squares on their graph paper as units of measurement If a net is inaccurate this becomes more evident when it is being folded This may help students see which parts need to be adjusted and decide the best locations for the flaps Reassure students that a few drafts of a net may be necessary before all the details are worked out and encourage them to persevere SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allow students who struggle with fine motor skills to dictate how to draw the net for the given polyhedron as needed 204 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Conceptual Processing Eliminate Barriers Begin with a physical demonstration of the actions that occur in a situation G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS SUPPORT FOR ENGLISH LANGUAGE LEARNERS YOUR NOTES Representing Conversing MLR 7 Compare and Connect Use this routine to help students consider their audience when preparing a visual display of their work Ask students to prepare a visual display that shows their net drawings and surface area calculations Students should consider how to display their calculations so that another student can interpret them Some students may wish to add notes or details to their drawings to help communicate their thinking When students find another person in the class with the same polyhedron provide quiet think time for students to read and interpret each other s drawings before they discuss the questions on display Design Principle s Cultivate conversation Maximize meta awareness Concept Exploration Activity 2 CAN YOU COVER IT This activity provides students an opportunity to extend and apply their learning to solve realworld and mathematical problems involving surface area The problems allow for multiple solution methods and students should be encouraged to engage with each problem in a way that makes sense to them LAUNCH Keep students in the same groups Students will still need access to their geometry toolkits including graph paper and a straightedge Begin with some quiet work time Give groups time to compare their answers and strategies then have a discussion 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 205

G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS ACTIVITY 2 TASK 1 3 Solve the following problems Show your reasoning YOUR NOTES 1 Jamica has a 12 inch by 12 inch sheet of paper and needs to create the net of the polyhedron below Is the sheet of paper large enough 5 in 6 in 3 in 4 in 2 Sophie has a box she uses to store personal items and she wants to decorate the outside of the box by wrapping it in decorative paper The box is 12 in wide 6 in deep and 4 in tall and she has a single sheet of decorative paper that is 22 in by 22 in Does she have enough paper to wrap her box STUDENT RESPONSE Strategies and reasoning vary Sample responses 1 No my net is a total of 12 in tall and 14 in long so it is too long for the sheet of paper 5 in 3 in 4 in 3 in 3 in 6 in 6 in 3 in 6 in 4 in 5 in 4 in 6 in 3 in 6 in 4 in 3 in 2 Yes the box has a total surface area of 288 sq in while the paper has an area of 484 sq in 206 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS DISCUSSION GUIDANCE Invite students to share their strategies and reasoning Include students who drew a net to help them solve one or both problems as well as students who reasoned their way through the problems without drawing a net t YOUR NOTES Digital Lesson 4 units 6 units 3 units 1 Circle the net that matches the polyhedron above The side length of a grid square is 1 unit 2 What s the surface area of the polyhedron Show your reasoning STUDENT RESPONSE 1 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 207

G6M1 LESSON 15 2 The surface area of the prism is 108 square units ZEARN MATH TEACHER LESSON MATERIALS The surface area is the sum of the areas of the 6 faces There are 2 faces that are 3 units by 4 units so that s 2 3 4 24 square units YOUR NOTES There are 2 faces that are 3 units by 6 units so that s 2 3 6 36 square units There are 2 faces that are 6 units by 4 units so that s 2 6 4 48 square units 24 36 48 108 square units total Wrap Up LESSON SYNTHESIS To highlight some key points from the lesson display a picture of a prism or a pyramid and a drawing of its net Discuss these questions Can you find the surface area of a simple prism or pyramid from a picture if all the necessary measurements are given Can you find the surface area from a net if all the measurements are given Which might be more helpful for calculating surface area a picture of a polyhedron or a net If the polyhedron is simple e g a cube a square pyramid etc and does not involve hidden faces with different measurements or require a lot of visualizing either a picture or a net can work Otherwise a net may be more helpful because we can see all of the faces at once and can find the area of each polygon more easily A net may also help us keep track of our calculations and notice missing or extra areas 208 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 15 EXIT TICKET ZEARN MATH TEACHER LESSON MATERIALS 1 In this net the two triangles are right triangles All quadrilaterals are rectangles What is its surface area in square units Show your reasoning YOUR NOTES 2 If the net is assembled which of the following polyhedra would it make 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 209

G6M1 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES STUDENT RESPONSE 1 The surface area is 168 square units Explanations vary Sample response There are two triangular faces with area of 24 square units each 12 6 8 24 There is a rectangular face with area of 50 square units 10 5 50 There is one rectangular face with area of 40 square units 5 8 40 There is one rectangular face with area 5 6 30 square units 2 24 50 40 30 168 2 Prism C 210 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Template for Concept Exploration 15 Activity 1 page 1 of 2 Blackline Master for Classroom Activity 6 1 15 2 Building Prisms and Pyramids A B C D E 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 211

Template for Concept Exploration 15 Activity 1 page 2 of 2 F G H I 212 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 16 G6M1 LESSON 16 OPTIONAL LESSON Distinguishing Between Surface Area and Volume ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES OPTIONAL LESSON ZEARN MATH TIP This lesson is optional and there is no Independent Digital Lesson included If you choose to use this lesson we recommend teaching this whole group with your students Distinguish among measures and units of one two and threedimensional attributes LEARNING GOALS See that volume and surface area are different attributes of a threedimensional figure Explain in writing or in words how two figures can have the same surface area but different volumes and vice versa LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Let s contrast surface area and volume I know how one two and three dimensional measurements and units are different I can explain how it is possible for two polyhedra to have the same surface area but different volumes or to have different surface areas but the same volume Snap cubes REQUIRED MATERIALS Sticky notes Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles REQUIRED PREPARATION Prepare sets of 16 snap cubes and two sticky notes for each student 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 213

G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES In this optional lesson students distinguish among measures of one two and threedimensional attributes and take a closer look at the distinction between surface area and volume building on students work in earlier grades Use this lesson to reinforce the idea that length is a one dimensional attribute of geometric figures surface area is a two dimensional attribute and volume is a three dimensional attribute By building polyhedra drawing representations of them and calculating both surface area and volume students see that different three dimensional figures can have the same volume but different surface areas and vice versa This is analogous to the fact that two dimensional figures can have the same area but different perimeters and vice versa Students must attend to units of measure throughout the lesson Note Students will need to bring in a personal collection of 10 50 small objects ahead of time for the first lesson of the next mission Examples include rocks seashells trading cards or coins Warm Up ATTRIBUTES AND THEIR MEASURES Instructional Routine Think Pair Share This activity strengthens students awareness of one two and three dimensional attributes and the units commonly used to measure them Students decide on the units based on the attributes being measured and the size of the units and how appropriate they would be for describing given quantities As students work select a few students to share their responses to the last two questions of the activity on the quantities that could be measured in miles and in cubic meters LAUNCH Consider a quick review of metric and standard units of measurement before students begin work Include some concrete examples that could help illustrate the size of each unit Then pick an object in the classroom for which surface area and volume could be measured e g a desk Ask students What units might we use to measure the surface area of the desktop What units might we use to measure the volume of a drawer Clarify the relative sizes of the different units that come up in the conversation For instance discuss how a meter is a little over three feet a yard is three feet a kilometer is about twothirds of a mile a millimeter is one tenth of a centimeter etc 214 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Give students quiet think time and then time to share their responses with a partner Prepare to display the answers to the first six questions for all to see WARM UP TASK 1 G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES For each quantity choose one or more appropriate units of measure For the last two quantities think of a quantity that could be appropriately measured with the given units 1 Perimeter of a parking lot millimeters mm 2 Volume of a semi truck feet ft 3 Surface area of a refrigerator meters m 4 Length of an eyelash square inches sq in 5 Area of a state square feet sq ft 6 Volume of an ocean square miles sq mi 7 miles cubic kilometers cu km 8 cubic meters cubic yards cu yd STUDENT RESPONSE 1 Meters feet 2 Cubic yards 3 Square inches square feet 4 Millimeters 5 Square miles 6 Cubic kilometers cubic yards 7 Answers vary Sample responses distance between home and school length of a river 8 Answers vary Sample responses volume of a room volume of a swimming pool 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 215

G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES WARM UP RECAP Display the solutions to the first six questions for all to see and to use for checking Then select a few students to share their responses to the last two questions Ask students what they notice about the units for area and the units for volume If not already mentioned by students highlight that area is always measured in square units and volume in cubic units ANTICIPATED MISCONCEPTIONS Depending on the students familiarity with metric and standard units there may be some confusion about the size of each unit Consider displaying measuring tools or a reference sheet that shows concrete examples of items measured in different sized units SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Visual Aids Include images depicting some of the described quantities which would aid students who benefit from multiple pathways for language processing Activity 1 BUILDING WITH 8 CUBES Instructional Routines Anticipate Monitor Select Sequence Connect Notice and Wonder This activity clarifies the distinction between volume and surface area and illustrates that two polyhedra can have the same volume but different surface areas Students build shapes using two sets of eight cubes and determine their volumes and surface areas Since all of the designs are made of the same number of cubes they all have the same volume Students then examine all of the designs and discuss what distinguishes shapes with smaller surface areas from those with greater ones As students work monitor the range of surface areas for the shapes that students built Select several students whose designs collectively represent that range 216 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 16 OPTIONAL LESSON LAUNCH Give each student or group of 2 students 16 snap cubes two sticky notes and time to complete the activity Explain that their job is to design and build two figures using 8 cubes for each and find the volume and surface area of each figure Ask them to give each figure a name or a label and then record the name surface area and volume on a sticky note ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ACTIVITY 1 TASK 1 2 Your teacher will give you 16 cubes Build two different shapes using 8 cubes for each For each shape Give a name or a label e g Mae s First Shape or Eric s Steps Determine the volume Determine the surface area Record the name volume and surface area on a sticky note STUDENT RESPONSE Designs vary Here are two possible shapes They both have a volume of 8 cubic units The first has a surface area of 24 square units The second has a surface area of 28 square units The smallest possible surface area for an 8 cube construction is 24 square units and the largest is 34 square units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 217

G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ACTIVITY 1 RECAP Ask all students to display their designs and their sticky notes and give them time to circulate and view one another s work Then ask previously identified students to arrange their designs in the order of their surface area from least to greatest and display their designs for all to see Record the information about the designs in a table in the same sequence Display the table for all to see Here is an example Shapes Volume Surface Area Andre s cube 8 24 Lin s steps 8 28 Jada s first shape 8 28 8 34 Noah s tower Give students time to notice and wonder about the designs and the information in the table Ask students to give a signal when they have noticed and wondered about at least one thing Invite a few students to share their observations and questions Then discuss the following questions if not already mentioned by students What do all of the shapes have in common Their volume Why are all the volumes the same Volume measures the number of unit cubes that can be packed into a figure All the designs are built using the same number of cubes Why do some shapes have larger surface areas than others What do shapes with larger surface areas look like The cubes are more spread out and have more of their faces exposed What about those with smaller surface areas They are more compact and have more of their faces hidden or shared with another cube Is it possible to build a shape with a different volume How Yes but it would involve using fewer or more cubes If students have trouble visualizing how surface area changes when the design changes demonstrate the following Make a cube made of 8 smaller cubes Point to one cube and ask how many of its faces are exposed 3 218 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Pop that cube off and move it to another place Point out that in the hole left by the cube that was moved 3 previously interior faces now contribute to the surface area At the same time the relocated cube now has 5 faces exposed G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Even though students are dealing with only 8 cubes at a time they may make counting errors by inadvertently omitting or double counting squares or faces This is especially likely for designs that are non prisms Encourage students to think of a systematic way to track the number of square units they are counting Some students may associate volume only with prisms and claim that the volume of non prism designs cannot be determined Remind them of the definition of volume SUPPORT FOR ENGLISH LANGUAGE LEARNERS Heavier Support Provide students with language for making the relevant comparisons spread out vs compact exposed visible vs hidden covered etc by using the cube designs to demonstrate Activity 2 COMPARING PRISMS WITHOUT BUILDING THEM Previously students studied shapes with the same volume but different surface areas Here they see that it is also possible for shapes to have the same surface area but different volumes Students think about how the appearance of these shapes might compare visually Students are given the side lengths of three rectangular prisms and asked to find the surface area and the volume of each Some students can visualize these but others may need to draw nets sketch the figures on isometric grid paper or build physical prisms Prepare cubes for students to use Each of the three prisms can be built with 15 or fewer cubes but 40 cubes are needed to build all three simultaneously If the cubes are not centimeter cubes ask students to treat them as if the edge length of each cube was 1 cm As students work look out for errors in students calculations in the first question which will affect the observations they make in the second question Select a few students who notice that the volumes of the prisms are all different but the surface areas are the same 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 219

G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Arrange students in groups of 2 Provide access to snap cubes and geometry toolkits Give students quiet think time and then time to discuss their responses with their partner Ask partners to agree upon one key observation to share with the whole class YOUR NOTES ACTIVITY 2 TASK 1 3 Three rectangular prisms each have a height of 1 cm Prism A has a base that is 1 cm by 11 cm Prism B has a base that is 2 cm by 7 cm Prism C has a base that is 3 cm by 5 cm 1 Find the surface area and volume of each prism Use the dot paper to draw the prisms if needed 2 Analyze the volumes and surface areas of the prisms What do you notice Write 1 2 observations about them 220 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 16 OPTIONAL LESSON STUDENT RESPONSE 1 ZEARN MATH TEACHER LESSON MATERIALS Surface areas Prism A 4 11 1 2 1 1 46 square centimeters Prism B 2 7 2 2 7 1 2 2 1 46 square centimeters Prism C 2 5 3 2 5 1 2 3 1 46 square centimeters YOUR NOTES Volumes Prism A 11 cubic centimeters 11 1 1 11 Prism B 14 cubic centimeters 7 2 1 14 Prism C 15 cubic centimeters 5 3 1 15 2 Answers vary Sample responses The surface areas of the prisms are all the same but the volumes are all different The polygons that make up the faces of each prism are different sized rectangles but their areas all add up to the same total of square centimeters Prism C can fit the most centimeter cubes but because the cubes would fit together in a compact way some of the cubes would only have two square centimeters of exposed faces Prism A can fit the fewest centimeter cubes but because the cubes would be more spread out more of their faces would be exposed DISCUSSION GUIDANCE Ask students to share their observations in response to the second question Record them for all to see For each unique observation poll the class to see if others noticed the same thing Highlight the following observations or point them out if not already mentioned by students 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 221

G6M1 LESSON 16 OPTIONAL LESSON The volumes of the prisms are all different but the surface areas are the same ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Volume is described in terms of unit cubes and surface area in terms of the exposed faces of those unit cubes Explain that in an earlier activity we saw how different shapes could have the same volume i e being made up of the same number of unit cubes but different surface areas Now we see that it is also possible for shapes with different volumes i e consisting of different numbers of unit cubes to have the same surface area If students have trouble conceptualizing the idea of figures with different volume having the same surface area refer to the filing cabinet activity in the first lesson on surface area The number of square sticky notes needed to cover all of the faces of the filing cabinet was its surface area If we use all of those square notes no more no less to completely cover without overlapping sticky notes a cabinet that has a different volume we can say that the two pieces of furniture have the same surface area and different volumes ANTICIPATED MISCONCEPTIONS Students may miss or double count one or more faces of the prisms and miscalculate surface areas Encourage students to be systematic in their calculations and to use organizational strategies they learned when finding surface area from nets Students may need reminders to use square units for area and cubic units for volume SUPPORT FOR ENGLISH LANGUAGE LEARNERS Heavier Support Provide specific guidance for question 2 For example As you observe your prisms think about the faces of the prisms the number of unit cubic centimeters similarities and differences among the three figures using the comparison language already provided compact vs spread out exposed vs hidden SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Conceptual Processing Manipulatives Provide manipulatives i e snap cubes to aid students who benefit from hands on activities 222 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Wrap Up In this lesson we refreshed our memory of measures of one two and three dimensional attributes Reiterate that length is a one dimensional attribute of geometric figures area is a two dimensional attribute and volume is a three dimensional attribute Revisit a few examples of units for length area and volume G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES We also explored the surface areas and volumes of polyhedra and noticed that two shapes can have the same volumes but different surface areas and vice versa LESSON SYNTHESIS How could two figures with a volume of 4 cubic units have a surface area of 16 square units and 18 square units Surface area and volume measure different attributes of a three dimensional shape What kind of measure is surface area What kind of measure is volume Surface area is a two dimensional attribute we measure it in square units Volume is a three dimensional attribute we measure it in cubic units Are the two measures related Does a greater volume necessarily mean a greater surface area and vice versa No one measure does not affect the other A figure that has a greater volume than another may not necessarily have a greater surface area EXIT TICKET LAUNCH Encourage students to refer to the class list of observations from Activity 2 TASK Choose two figures that have the same surface area but different volumes Show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 223

G6M1 LESSON 16 OPTIONAL LESSON ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES STUDENT RESPONSE Figure D and E both have a surface area of 26 square units but D has a volume of 6 cubic units and E has a volume of 7 cubic units 224 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

GRADE 6 MISSION 1 LESSON 17 Squares and Cubes G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Generalize a process for finding the volume of a cube and justify orally why this can be abstracted as s3 LEARNING GOALS Include appropriate units orally and in writing when reporting lengths areas and volumes e g cm cm2 cm3 Interpret and write expressions with exponents 2 and 3 to represent the area of a square or the volume of a cube LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS REQUIRED PREPARATION Let s investigate perfect squares and perfect cubes I can write and explain the formula for the volume of a cube including the meaning of the exponent When I know the edge length of a cube I can find the volume and express it using appropriate units Snap cubes Prepare sets of 32 snap cubes for each group of 2 students 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 225

G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES In this lesson students learn about perfect squares and perfect cubes They see that these names come from the areas of squares and the volumes of cubes with whole number side lengths Students find unknown side lengths of a square given the area or unknown edge lengths of a cube given the volume To do this they make use of the structure in expressions for area and volume Students also use exponents of 2 and 3 and see that in this geometric context exponents help to efficiently express multiplication of the side lengths of squares and cubes Students learn that expressions with exponents of 2 and 3 are called squares and cubes and see the geometric motivation for this terminology The term exponent is deliberately not defined more generally at this time Students will work with exponents in more depth in a later mission In working with length area and volume throughout the lesson students must attend to units In order to write the formula for the volume of a cube students look for and express regularity in repeated reasoning Note Students will need to bring in a personal collection of 10 50 small objects ahead of time for the first lesson of the next mission Examples include rocks seashells trading cards or coins Warm Up PERFECT SQUARES This activity introduces the concept of perfect squares It also includes opportunities to practice using units of measurement which offers insights on students knowledge from preceding lessons Provide access to square tiles if available Some students may benefit from using physical tiles to reason about perfect squares As students work notice whether they use appropriate units for the second and third questions LAUNCH Tell students Some numbers are called perfect squares For example 9 is a perfect square Nine copies of a small square can be arranged into a large square Display a square like this for all to see 226 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Explain that 10 however is not a perfect square Display images such as these below emphasizing that 10 small squares can not be arranged into a large square the way 9 small squares can G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Tell students that in this warm up they will find more numbers that are perfect squares Give students quiet think time to complete the activity WARM UP TASK 1 Answer the questions about squares 1 The number 9 is a perfect square Find four numbers that are perfect squares and two numbers that are not perfect squares 2 A square has side length 7 km What is its area 3 The area of a square is 64 sq cm What is its side length STUDENT RESPONSE 1 Answers vary For example here are some squares 9 25 4 49 100 and non squares 12 2 3 10 2 The square has an area of 49 square kilometers 3 The side length is 8 centimeters DISCUSSION GUIDANCE Invite students to share the examples and non examples they found for perfect squares Solicit some ideas on how they decided if a number is or is not a perfect square 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 227

G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES If a student asks about 0 being a perfect square wait until the end of the lesson when the exponent notation is introduced 0 is a perfect square because 02 0 Briefly discuss students responses to the last two questions the last one in particular If not already uncovered in discussion highlight that because the area of a square is found by multiplying side lengths to each other finding the side lengths of a square with a known area means figuring out if that area measure is a product of two of the same number ANTICIPATED MISCONCEPTIONS If students do not recall what the abbreviations km cm and sq stand for provide that information Students may divide 64 by 2 for the third question If students are having trouble with this ask them to check by working backwards i e by multiplying the side lengths to see if the product yields the given area measure SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives Provide manipulatives i e one inch tiles to aid students who benefit from hands on activities Concept Exploration Activity 1 PERFECT CUBES Instructional Routines Think Pair Share MLR8 Discussion Supports Earlier students looked at examples and non examples of perfect squares In this activity they think about examples and non examples of perfect cubes and find the volumes of cubes given their edge lengths Students see that the edge length of a cube determines its volume notice the numerical expressions that can be written when calculating volumes and write a general expression for finding the volume of a cube Some students may feel uncomfortable writing the answer to the last question symbolically because it involves a variable and may prefer writing a verbal explanation This is fine the exponential notation that follows will help greatly 228 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 17 LAUNCH ZEARN MATH TEACHER LESSON MATERIALS Tell students Some numbers are called perfect cubes For example 27 is a perfect cube Display a cube like this for all to see YOUR NOTES Arrange students in groups of 2 Give students quiet think time and then time to discuss their responses with their partner ACTIVITY 1 TASK 1 2 Answer the questions about cubes 1 The number 27 is a perfect cube Find four other numbers that are perfect cubes and two numbers that are not perfect cubes 2 A cube has an edge length of 4 cm What is its volume 3 A cube has an edge length of 10 cm What is its volume 4 A cube has an edge length of s cm What is its volume STUDENT RESPONSE 1 Answers vary 1 8 64 125 216 1 000 Non cubes 2 3 4 2 4 4 4 or 64 cubic cm 3 10 10 10 or 1 000 cubic inches 4 s s s cubic units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 229

G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ACTIVITY 1 RECAP After partner discussions invite students to share how they thought about the first question and decided if a number is or is not a perfect cube Highlight the idea that multiplying three edge lengths allows us to determine volume efficiently and that determining if a number is a perfect cube involves thinking about whether it is a product of three of the same number If a student asks about 0 being a perfect cube wait until the end of the lesson when exponent notation is introduced 0 is a perfect cube because 03 0 Make sure students see the answers to the last three questions written as expressions 4 4 4 10 10 10 s s s ANTICIPATED MISCONCEPTIONS Watch for students using square units instead of cubic units Remind them that volume is a measure of the space inside the cube and is measured in cubic units Students may multiply by 3 when finding the volume of a cube instead of multiplying three edge lengths which happen to be the same number Likewise they may think a perfect cube is a number times 3 Suggest that they sketch or build a cube with that edge length and count the number of unit cubes Or ask them to think about how to find the volume of a prism when the edge lengths are different e g a prism that is 1 unit by 2 units by 3 units SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing Representing MLR 8 Discussion Supports As students work in pairs to make sense of perfect cubes encourage thinking aloud revoicing and pressing for details e g How do you know _____ is or is not a perfect cube Tell me more about Hands on use of snap cubes will help students see examples and non examples of perfect cubes and recognize that the edge length of a cube determines its volume Design Principle s Support sense making Maximize meta awareness 230 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives Provide manipulatives i e snap cubes to aid students who benefit from hands on activities G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Concept Exploration Activity 2 INTRODUCING EXPONENTS Instructional Routine MLR3 Clarify Critique Correct This activity introduces students to the exponents of 2 and 3 and the language we use to talk about them Students use and interpret this notation in the context of geometric squares and their areas and geometric cubes and their volumes Students are likely to have seen exponent notation for 103 in their work on place values in grade 5 That experience would be helpful but is not necessary Note that the term exponent is deliberately not defined more generally at this time Students will work with exponents in more depth in a later mission As students work observe how they approach the last two questions Identify a couple of students who approach the fourth question differently so they can share later Also notice whether students include appropriate units written using exponents in their answers LAUNCH Ask students if they have seen an expression such as 103 before Tell students that in this expression the 3 is called an exponent Explain the use of exponents of 2 and 3 When we multiply two of the same number together such as 5 5 we say we are squaring the number We can write the expression as 52 Because 5 5 is 25 we can write 52 25 and we say 5 squared is 25 We can also say that 25 is a perfect square The raised 2 in 52 is called an exponent When we multiply three of the same number together like 4 4 4 we say we are cubing the number We can write it like this 43 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 231

G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS Because 4 4 4 is 64 we can write 43 64 and we say 4 cubed is 64 We also say that 64 is a perfect cube The raised 3 in 43 is called an exponent Explain that we can also use exponents as a shorthand for the units used for area and volume YOUR NOTES A square with side length 5 inches has area of 25 square inches which we can write as 25 in2 A cube with edge length 4 centimeters has a volume of 64 cubic centimeters which we can write as 64 cm3 Ask students to read a few areas and volumes in different units e g 100 ft2 is read 100 square feet and 125 yd3 is read 125 cubic yards Keep students in groups of 2 Give students quiet time to complete the activity and then time to discuss their response with their partner Ask partners to note any disagreements so they can be discussed ACTIVITY 2 TASK 1 3 Answer the questions about exponents Make sure to include correct units of measure as part of each answer 1 A square has side length 10 cm Use an exponent to express its area 2 The area of a square is 72 sq in What is its side length 3 The area of a square is 81 m2 Use an exponent to express this area 4 A cube has edge length 5 in Use an exponent to express its volume 5 The volume of a cube is 63 cm3 What is its edge length 6 A cube has edge length s units Use an exponent to write an expression for its volume STUDENT RESPONSE 1 102 cm2 2 7 inches 3 92 m2 232 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 17 4 53 in3 ZEARN MATH TEACHER LESSON MATERIALS 5 6 cm 6 s3 units3 or s3 cubic units YOUR NOTES DISCUSSION GUIDANCE Ask partners to share disagreements in their responses if any Then focus the discussion on the last two questions Select a couple of previously identified students to share their interpretations of the fourth question Highlight that a cube with a volume of 63 cubic units has an edge length of 6 units because we know there are 6 6 6 unit cubes in a cube with that edge length In other words we can express the volume of a cube using a number 216 a product of three numbers 6 6 6 or an expression with exponent 63 This idea can be extended to all cubes The volume of a cube with edge length s is s s s s3 Students will have more opportunities to generalize the expressions for the volume of a cube in the next lesson ANTICIPATED MISCONCEPTIONS Upon seeing 63 in the fourth question some students may neglect to interpret the question automatically calculate 6 6 6 and conclude that the edge length is 216 cm Ask them to check their answer by finding the volume of a cube with edge length 216 cm SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 3 Clarify Critique Correct Present an incorrect response such as If the volume of a cube is 63 cm3 then the edge length is 216 cm because 6 x 6 x 6 is 216 Ask students to identify the error and to offer a correct argument to determine the edge length of the cube This will help students to use mathematical language and reflect on their work as well as the work of others Design Principle s Optimize output for justification Maximize meta awareness 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 233

G6M1 LESSON 17 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Executive Functioning Visual Aids Create an anchor chart i e squaring cubing exponent publicly displaying important definitions rules formulas or concepts for future reference Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Digital Lesson What does it mean to square a number What does it mean to cube a number Consider using pictures notations and words to help you answer STUDENT RESPONSE Answers vary Sample response Squaring a number is when we multiply two of the same number together like 10 10 Cubing a number is when we multiply three of the same number together like 10 10 10 Wrap Up LESSON SYNTHESIS Review the language and notation for squaring and cubing a number Remind students we use this notation for square and cubic units too 234 When we multiply two of the same number together like 10 10 we say we are squaring the number We write for example 102 100 and say Ten squared is one hundred When we multiply three of the same number together like 10 10 10 we say we are cubing the number We write for example 103 1 000 and say Ten cubed is one thousand 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Exponents are used to write square and cubic units The area of a square with side length 7 km is 72 km2 The volume of a cube with side length 2 millimeters is 23 mm3 G6M1 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS TERMINOLOGY YOUR NOTES Cubed We use the word cubed to mean to the third power This is because a cube with side length s has a volume of s s s or s3 Exponent In expressions like 53 and 82 the 3 and the 2 are called exponents They tell you how many factors to multiply For example 53 5 5 5 and 82 8 8 Squared We use the word squared to mean to the second power This is because a square with side length s has an area of s s or s2 EXIT TICKET 1 Which is larger 52 or 33 2 A cube has an edge length of 21 cm Use an exponent to express its volume STUDENT RESPONSE 1 33 27 and 52 25 so 33 is larger than 52 2 213 cm3 or 213 cubic centimeters ANTICIPATED MISCONCEPTIONS Students may perform calculations on the second question which is not necessary since the target is an expression with an exponent 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 235

G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 1 LESSON 18 Surface Area of a Cube YOUR NOTES Generalize a process for finding the surface area of a cube and justify orally why this can be abstracted as 6 s LEARNING GOALS Interpret orally expressions that include repeated addition multiplication repeated multiplication or exponents Write expressions with or without exponents to represent the surface area of a given cube LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS 236 Let s write a formula to find the surface area of a cube I can write and explain the formula for the surface area of a cube When I know the edge length of a cube I can find its surface area and express it using appropriate units Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

In this lesson students practice using exponents of 2 and 3 to express products and to write square and cubic units Along the way they look for and make use of structure in numerical expressions They also look for and express regularity in repeated reasoning to write the formula for the surface area of a cube Students will continue this work later in the course in the unit on expressions and equations Note Students will need to bring in a personal collection of 10 50 small objects ahead of time for the first lesson of the next unit Examples include rocks seashells trading cards or coins G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Warm Up EXPONENT REVIEW In this warm up students compare pairs of numerical expressions and identify the expression with the greater value The task allows students to review what they learned about exponents and prompts them to look for and make use of structure in numerical expressions Students should do these without calculators and without calculating although it is fine for them to check their answers with a calculator LAUNCH Give students quiet think time Ask them to answer the questions without multiplying anything or using a calculator and to give a signal when they have an answer for each question and can explain their reasoning WARM UP TASK 1 Select the greater expression of each pair without calculating the value of each expression Be prepared to explain your choices a 10 3 or 10 b 13 or 12 12 c 97 97 97 97 97 97 or 5 97 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 237

G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE a 10 is greater because it is 1 000 b 13 is greater because it is 13 13 and this will be greater than 12 12 YOUR NOTES c 97 97 97 97 97 97 is greater because it is 6 97 which is greater than 5 97 DISCUSSION GUIDANCE Ask one or more students to explain their reasoning for each choice If not mentioned in students explanations highlight the structures in the expressions that enable us to evaluate each one without performing any calculations Point out for example that since we know that 10 means 10 10 10 we can tell that it is much larger than 10 3 For the last question remind students that we can think of repeated addition in terms of multiple groups i e that the sum of six 97s can be seen as six groups of 97 or 6 97 The idea of using groups to write equivalent expressions will support students as they write expressions for the surface area of a cube later in the lesson i e writing the areas of all square faces of a cube as 6s ANTICIPATED MISCONCEPTIONS When given an expression with an exponent students may misinterpret the base and the exponent as factors and multiply the two numbers Remind them about the meaning of the exponent notation For example show that 5 3 15 which is much smaller than 5 5 5 which equals 125 SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Language Processing Time Read all statements aloud Students who both listen to and read the information will benefit from extra processing time 238 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

G6M1 LESSON 18 Concept Exploration Activity 1 THE NET OF A CUBE ZEARN MATH TEACHER LESSON MATERIALS Instructional Routines MLR7 Compare and Connect Anticipate Monitor Select Sequence Connect Think Pair Share YOUR NOTES This activity contains two sets of problems The first set involves computations with simple numbers and should be solved numerically Use students work here to check that they are drawing a net correctly The second set encourages students to write expressions rather than to simplify them through calculations The goal is to prepare students for the general rules s and 6s which are more easily understood through an intermediate step involving numbers Note that students will be introduced to the idea that 5 x means the same as 5x in a later unit so expect them to write 6 17 instead of 6 17 It is not critical that they understand that a number and a variable or a number and an expression in parentheses placed next to each other means they are being multiplied As students work on the second set monitor the ways in which they write their expressions for surface area and volume Identify those whose expressions include products e g 17 17 or 17 17 17 sums of products e g 17 17 17 17 combination of like terms e g 6 17 17 exponents e g 17 17 or 17 and completed calculation e g 289 Select these students to share their work later Notice the lengths of the expressions and sequence their explanations in order from the longest expression to the most succinct LAUNCH Give students access to their geometry toolkits and quiet think time Tell students to try to answer the questions without using a calculator Ask them to share their responses with their partner afterwards 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 239

G6M1 LESSON 18 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 Answer the following questions about a cube with edge length 5 inches YOUR NOTES a Draw a net for this cube and label its sides with measurements b What is the shape of each face c What is the area of each face d What is the surface area of this cube e What is the volume of this cube STUDENT RESPONSE a Drawings vary 11 unique nets are possible b Square c 25 square inches d 150 square inches e 125 cubic inches ACTIVITY 1 TASK 2 3 240 Answer the following questions about a cube with edge length 17 units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

a Draw a net for this cube and label its sides with measurements b Explain why the area of each face of this cube is 17 square units G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS c Write an expression for the surface area in square units d Write an expression for the volume in cubic units YOUR NOTES STUDENT RESPONSE a Drawings vary but should be one of the 11 nets shown in the previous problem b Answers vary Sample explanation The side length of each square face is 17 units so its area is 17 17 or 17 square units c 6 17 or equivalent d 17 or equivalent DISCUSSION GUIDANCE After partner discussions select a couple of students to present the solutions to the first set of questions which should be straightforward Then invite previously identified students to share their expressions for the last two questions If possible sequence their presentation in the following order If any expressions are missing but needed to illustrate the idea of writing succinct expressions add them to the lists Surface area 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 6 17 17 6 17 6 289 1 734 Volume 17 17 17 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 241

G6M1 LESSON 18 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 4 913 Discuss how multiplication can simplify expressions involving repeated addition and exponents can do the same for repeated multiplication While the last expression in each set above is the simplest to write getting there requires quite a bit of computation Highlight 6 17 and 17 as efficient ways to express the surface area and volume of the cube As the group discusses the different expressions consider directing students attention to the units of measurements Remind students that rather than writing 6 17 square units we can write 6 17 units and instead of 17 cubic units we can write 17 units3 Unit notations will appear again later in the course so it can also be reinforced later If students are not yet ready for the general formula which comes next offer another example For instance say A cube has edge length 38 cm How can we express its surface area and volume Help students see that its surface area is 6 38 cm and its volume is 38 cm The large number will discourage calculation and focus students on the form of the expressions they are building and the use of exponents ANTICIPATED MISCONCEPTIONS Students might think the surface area is 17 17 6 Prompt students to write down how they would compute surface area step by step before trying to encapsulate their steps in an expression Dissuade students from using calculators in the last two problems and assure them that building an expression does not require extensive computation Students may think that refraining from using a calculator meant performing all calculations including those of larger numbers on paper or mentally especially if they are unclear about the meaning of the term expression Ask them to refer to the expressions in the warm up or share examples of expressions in a few different forms to help them see how surface area and volume can be expressed without computation SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allowing students who struggle with fine motor skills to dictate how to draw each net as needed 242 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing Conversing MLR 7 Compare and Connect While students are participating in their partner discussions of the two cubes ask What is the same and what is different in the calculations of surface area and volume for each respective cube This helps students make and describe the connections between the calculations no matter the edge length of the cube G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Design Principle s Cultivate conversation Maximize meta awareness Concept Exploration Activity 2 EVERY CUBE IN THE WHOLE WORLD Instructional Routines MLR3 Clarify Critique Correct Anticipate Monitor Select Sequence Connect In this activity students build on what they learned earlier and develop the formulas for the surface area and the volume of a cube in terms of a variable edge length s Encourage students to refer to their work in the preceding activity as much as possible and to generalize from it As before monitor for different ways of writing expressions for surface area and volume Identify students whose work includes the following products e g s s or s s s sums of products e g s s s s combination of like terms e g 6 s s and exponents e g s s or s Select these students to share their work later Again notice the lengths of the expressions and sequence their explanations in order from the longest expression to the most succinct LAUNCH Give students access to their geometry toolkits and quiet think time Tell students they will be answering the same questions as before but with a variable for the side length Encourage them to use the work they did earlier to help them here 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 243

G6M1 LESSON 18 ACTIVITY 2 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 43 Answer the following questions about a cube with edge length s YOUR NOTES 1 Draw a net for the cube 2 Write an expression for the area of each face Label each face with its area 3 Write an expression for the surface area 4 Write an expression for the volume STUDENT RESPONSE 1 Drawings vary Here is one possible labeled net each face is a square whose side lengths are s S S S S S S 2 The area of each face is s 3 The surface area is 6 s 4 The volume is s DISCUSSION GUIDANCE Discuss the problems in as similar a fashion as was done in the earlier activity involving a cube with edge length 17 units Doing so enables students to see structure in the expressions and to generalize through repeated reasoning Select previously identified students to share their responses with the group If possible sequence their presentation in the following order to help students see how the expressions 6 s and s come about If any expressions are missing but needed to illustrate the idea of writing succinct expressions add them to the lists 244 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Surface area s s s s s s s s s s s s G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS s s s s s s 6 s s YOUR NOTES 6 s or 6 s Volume s s s s Refer back to the example involving numerical side length a cube with edge length 17 units if students have trouble understanding where the most concise expression of surface area comes from Present the surface area as 6 s You can choose to also write it as 6s ANTICIPATED MISCONCEPTIONS If students are unclear or unsure about using the variable s explain that we are looking for an expression that would work for any edge length and that a variable such as s can represent any number The s could be replaced with any edge length in finding surface area and volume To connect students work to earlier examples point to the cube with edge length 17 units from the previous activity Ask If you wrote the surface area as 6 17 before what should it be now As students work encourage those who may be more comfortable using multiplication symbols to instead use exponents whenever possible SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allow students who struggle with fine motor skills to dictate how to draw the net as needed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 245

G6M1 LESSON 18 SUPPORT FOR ENGLISH LANGUAGE LEARNERS ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Conversing MLR 3 Clarify Critique Correct Present an incorrect response such as If the cube has edge length s then the area of each face is 2s because s s 2 s Ask students to identify the error and to offer a correct argument to write an expression for the area of each face This will help students to use symbolic representations while generalizing calculations related to surface area Design Principle s Optimize output for generalization Maximize metaawareness Digital Lesson The side length of the cube is labeled in the net below 2 in 3 Describe what surface area is using this net of a cube as an example Without calculating write an expression that represents the surface area of this cube Consider using an exponent in your expression Describe what volume is using this cube as an example Then without calculating write an expression that represents the volume of this cube Consider using an exponent in your expression STUDENT RESPONSE Answers vary Examples of expressions for the surface area of the cube are 246 2 3 2 3 6 2 3 6 2 3 2 3 2 3 2 3 2 3 2 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Examples of expressions for the volume of the cube are 2 3 2 3 G6M1 LESSON 18 ZEARN MATH TEACHER LESSON MATERIALS 2 3 2 3 YOUR NOTES Wrap Up LESSON SYNTHESIS Review the formulas for volume and surface area of a cube The volume of a cube with edge length s is s A cube has 6 faces that are all identical squares The surface area of a cube with edge length s is 6 s EXIT TICKET 1 A cube has edge length 11 inches Write an expression for its volume and an expression for its surface area 2 A cube has a volume of 73 cubic centimeters What is its surface area STUDENT RESPONSE 1 Volume 11 or 11 11 11 Surface area 6 11 11 or equivalent 2 The surface area is 6 7 which is 294 square centimeters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 247

G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 1 LESSON 19 Designing a Tent YOUR NOTES TEACHER INSTRUCTION ONLY ZEARN MATH TIP There is no Independent Digital Lesson for this Lesson We recommend teaching this lesson whole group with your students Apply understanding of surface area to estimate the amount of fabric in a tent and explain orally and in writing the estimation strategy LEARNING GOALS Compare and contrast orally different tent designs Interpret information presented in writing and through other representations about tents and sleeping bags LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS 248 Let s design some tents I can apply what I know about the area of polygons to find the surface area of three dimensional objects I can use surface area to reason about real world objects Geometry toolkits tracing paper graph paper colored pencils scissors and an index card to use as a straightedge or to mark right angles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

In this culminating lesson students use what they learned in this mission to design a tent and determine how much fabric is needed for the tent The task prompts students to model a situation with the mathematics they know make assumptions and plan a path to solve a problem It also allows students to choose tools strategically and to make a logical argument to support their reasoning The lesson has two parts In the first part students learn about the task gather information and begin designing The introduction is important to ensure all students understand the context Then after answering some preparatory questions in groups and as a class students work individually to design and draw their tents They use their knowledge of area and surface area to calculate and justify an estimate of the amount of fabric needed for their design G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES The second part involves reflection and discussion on students work Students explain their work to a partner or small group discuss and compare their designs and consider the impact of design decisions on the surface areas of their tents Depending on instructional choices made this lesson could take one or more class meetings The time needed will vary based on instructional decisions made It may depend on whether students use the provided information about tents and sleeping bags or research this information expectations around drafting revising and the final product how student work is ultimately shared with the class not at all informally or with formal presentations Note Students will need to bring in a personal collection of 10 50 small objects ahead of time for the first lesson of the next mission Examples include rocks seashells trading cards or coins SUPPORT FOR STUDENTS WITH DISABILITIES Peer Tutors This activity integrates multiple skills and understandings and student may need extra help accessing learning and organizing it in a clear manner in order to accomplish this task Reduce Barriers Strength Based learning Many students who may struggle in the design will shine in the construction of the tent Tablets Fine motor Skills Visual Spatial Processing Use stylus for design and zoom for clarity and reduction of extraneous distractions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 249

G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Visual Aids Conceptual Processing Have height specs on board for permanent support and reference while creating tent Group Work Progressions Expressive and Receptive Language For second part organize sharing the work so that the student with disability can add to the discussion either to share first or have a clear add on to another students idea so their idea is not taken up by another losing their opportunity to participate Activity 1 TENT DESIGN PART 1 Instructional Routine MLR5 Co Craft Questions and Problems This activity has two parts an introduction to the task and individual work time In the first part students read the design problem and ask clarifying questions and then work with a partner or two to look at tent designs and specifications Then they work individually to design a tent create necessary representations of it calculate its surface area and estimate of the amount of fabric needed to construct it As students work individually circulate and focus your observations on two main goals 1 Notice the strategies and mathematical ideas students use to complete the task Are students decomposing or rearranging parts of their tent design to find the area How drawing a net of their design labeling their drawings with measurements calculating area precisely using formulas they learned in this mission How accounting for the areas of all surfaces of their tent design using square units for area measures 2 To record the sizes in terms of numbers of people accommodated and shapes of individual tent designs Use this information to arrange students into groups by tent size in the next activity 250 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Collect student work at the end of the session Arrange for 2 3 students who have tents that accommodate the same number of people but different designs to work together e g two students design tents for three people but one designed a triangular prism and the other a pentagonal prism Put their papers together to begin the second session LAUNCH G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Give students a few minutes to read the task statement individually and ask any clarifying questions At this point students only need to understand that the tents need to accommodate same sized sleeping bags and that there is not one right way to design them Next arrange students in groups of 2 Give groups time to look at and discuss potential tent designs tent specifications and sleeping bag information Tell students that the designs are provided for inspiration and reference but students are not limited to them Give students quiet think time to sketch out their tent design create necessary drawings calculate surface area and justify their estimate Provide blank paper for students to use to draw their designs and access to their geometry toolkits Note that a scale drawing is not an expectation ACTIVITY 1 TASK 1 1 Use the information below to help you design some tents Have you ever been camping You might know that sleeping bags are all about the same size but tents come in a variety of shapes and sizes Your task is to design a tent to accommodate up to four people and estimate the amount of fabric needed to make your tent Your design and estimate must be based on the information given and have mathematical justification First look at these examples of tents the average specifications of a camping tent and standard sleeping bag measurements Talk to a partner about Similarities and differences among the tents Information that will be important in your designing process The pros and cons of the various designs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 251

G6M1 LESSON 19 Tent Styles ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Tent Height Specifications Height Description Height of Tent Notes Sitting Height 3 feet Campers are able to sit lie or crawl inside tent Kneeling Height 4 feet Campers are able to kneel inside tent Found mainly in 3 4 person tents Stooping Height 5 feet Campers are able to move around on their feet inside tent but most campers will not be able to stand upright Standing Height 6 feet Most adult campers are able to stand upright inside tent Roaming Height 7 feet Adult campers are able to stand upright and walk around inside tent Sleeping Bag Measurements Standard 34 74 How many people can sleep in your tent What is the height of your tent 252 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Sketch the bottom panel of your tent and the locations where sleeping bags will go Sketch the overall design of your tent What decisions are important when choosing a tent design G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES What decisions are important when choosing a tent design Use the remaining space to show any work sketches of side panels calculations etc needed to estimate the amount of fabric that will be necessary to make your tent STUDENT RESPONSE Answers vary DISCUSSION GUIDANCE After students complete the task engage students in a whole class discussion Ask students What were important things you had to think about in your design Collect student work at the end of the session Tell students they will continue to think about the problem and their proposed solution in the next activity Arrange for 2 3 students who have tents that accommodate the same number of people but have different designs to work together e g two students designed tents for three people but one designed a triangular prism and the other a pentagonal prism Put their papers together to begin the second session ANTICIPATED MISCONCEPTIONS Some students may find it challenging to develop and represent a three dimensional object on paper Ask them what might help them create or convey their design Some may find it useful to 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 253

G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES think in two dimensional terms and start by drawing a net Others may wish to build a physical model of their design from paper or other flexible material Encourage students to consider the tools at their disposal and choose those that would enable them to complete the task SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Listening Conversing MLR 5 Co Craft Questions Use this routine to help students identify what will be important to consider when designing their tents Display only the images of the tents without the remaining information of the task Ask students to work with a partner to create a list of mathematical questions they might ask about the tents Invite pairs to share their questions with the class This provides students with an opportunity to engage with the context of the task before they consider what is asked Design Principle s Optimize output for explanation Cultivate conversation Maximize meta awareness SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors Receptive Expressive Language Processing Time Students who benefit from extra processing time would also be aided by MLR 6 Activity 2 TENT DESIGN PART 2 Instructional Routine MLR7 Compare and Connect This activity gives students a chance to explain and reflect on their work In groups of 2 3 they share drawings of their tent design an estimate of the amount of fabric needed and the justification They compare their creations with one or more peers Students discuss not only the amount of fabric required but also the effects that different designs have on that amount Prior to the session identify 2 3 students who have tents that accommodate the same number of people but different designs e g two students each design a 3 person tent but one designed a triangular prism and the other a pentagonal prism Put their papers from Part 1 together 254 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

As students discuss in groups notice how they reason about and communicate their work Do they G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS provide justification for their measurements and choices explain clearly their process of calculating surface area YOUR NOTES see how the type of design affects the amount of fabric compare their tents in terms of the differences in the measurements at the base and the height of tent LAUNCH Tell students that they will now reflect on and discuss their tents with another student who designed a tent for the same number of people but in a different way Arrange students in the predetermined groups of 2 3 and return the presorted sets of papers to them ACTIVITY 2 TASK 1 Discuss with a partner 1 Explain your tent design and fabric estimate to your partner or partners Be sure to explain why you chose this design and how you found your fabric estimate 2 Compare the estimated fabric necessary for each tent in your group Discuss the following questions Which tent design used the least fabric Why Which tent design used the most fabric Why Which change in design most impacted the amount of fabric needed for the tent Why STUDENT RESPONSE Answers vary 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 255

G6M1 LESSON 19 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES DISCUSSION GUIDANCE Much of the discussion will take place within the groups Once groups have had an opportunity to share their designs reconvene as a class One idea would be to display tent designs that used the most and the least amount of fabric Also consider asking students to reflect on the following prompts What design choices lead to using less fabric What design choices lead to using more fabric What are some ways that tents designed to accommodate the same number of people could use very different amounts of fabric When calculating the surface area of your tent what kinds of techniques from this unit did you find useful SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 7 Compare and Connect As students compare designs that fit the same number of people but differ in surface area call their attention to the way that utilizing different polygons to create each tent impacts the surface area Ask students to compare their design with the design that uses the least amount of fabric Discuss whether and how their knowledge of the area of polygons could inform their design decisions and the amount of fabric used Design Principle s Optimize output for justification Cultivate conversation Maximize meta awareness Wrap Up LESSON SYNTHESIS This culminating lesson could be wrapped up in a number of ways depending on the time available and your goals and expectations You could choose a simple wrap up discussion or assign students to develop a more elaborate presentation of their tent design involving posters or three dimensional models of their tents 256 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Math Math TEACHER EDITION GRADE 6 TEACHER EDITION Mission 1 1 Mission 1 Area and Surface Area Mission 2 Introducing Ratios Mission 3 Unit Rates and Percentages Mission 4 Dividing Fractions Mission 5 Arithmetic in Base Ten Mission 7 Rational Numbers Zearnmath_TE_Grade6_M1 indd 1 Grade 6 Mission 1 Mission 9 Putting It All Together 4 5 6 7 8 9 6 GRADE Mission 6 Expressions and Equations Mission 8 Data Sets and Distributions 3 TEACHER EDITION GRADE 6 2 1 19 23 12 15 PM