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Message GRADE8TEACHER EDITIONMission 1Math1 2 3GRADE 84 75 86 9Mission 1 Rigid Transformations and CongruenceMission 2 Dilations, Similarity, and Introducing SlopeMission 3 Linear RelationshipsMission 4Linear Equations and Linear SystemsMission 5Functions and VolumeMission 6Associations in DataMission 7Exponents and Scientific NotationMission 8Pythagorean Theorem and Irrational NumbersMission 9Putting It All TogetherTEACHER EDITIONMathGRADE 8TEACHER EDITIONGrade 8 | Mission 1

GRADE 8Mission 1Rigid Transformations and CongruenceIn this Mission, students learn to understand and use the terms “reection,” “rotation,” “translation,” recognizing what determines each type of transformation, e.g., two points determine a translation. They learn to understand and use the terms “transformation” and “rigid transformation.” They identify and describe translations, rotations, and reections, and sequences of these, using the terms “corresponding sides” and “corresponding angles,” and recognizing that lengths and angle measures are preserved. They draw images of gures under rigid transformations on and o square grids and the coordinate plane. They use rigid transformations to generate shapes and to reason about measurements of gures. They learn to understand congruence of plane gures in terms of rigid transformations. They recognize when one plane gure is congruent or not congruent to another. Students use the denition of “congruent” and properties of congruent gures to justify claims of congruence or non-congruence.

© 2023 ZearnPortions 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: 979-8-88868-957-8

Table of ContentsMISSION OVERVIEW viiiASSESSMENTS xviiTOPIC A: RIGID TRANSFORMATIONSLESSON 1 Moving in the Plane 1LESSON 2 Naming the Moves 13LESSON 3 Grid Moves 28LESSON 4 Making the Moves 38LESSON 5 Coordinate Moves 54LESSON 6 Describing Transformations 68TOPIC B: PROPERTIES OF RIGID TRANSFORMATIONSLESSON 7 No Bending or Stretching 80LESSON 8 Rotation Patterns 92LESSON 9 Moves in Parallel 104LESSON 10 Composing Figures 118TOPIC C: CONGRUENCELESSON 11 What Is the Same? 129LESSON 12 Congruent Polygons 143LESSON 13 Congruence 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. iii

TOPIC D: ANGLES IN A TRIANGLELESSON 14 Alternate Interior Angles 170LESSON 15 Adding the Angles in a Triangle 184LESSON 16 Parallel Lines and the Angles in a Triangle 199TOPIC E: LET’S PUT IT TO WORKLESSON 17 Rotate and Tessellate 213© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.iv

ZEARN MATH MISSION OVERVIEW G8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.M3Comparison of Length, Weight, Capacity, & Numbers to 10WEEK 1 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 36KG1G2G3G4G5G6G7G8M1Numbers to 10M4Number Pairs, Addition, & Subtraction to 10M6Analyzing,Comparing, & Composing ShapesM1Add & Subtract Friendly NumbersM2Meet Place ValueM3Measure LengthM2Explore LengthM3Counting & Place ValueM1Add, Subtract, & RoundM1Multiply & Divide Friendly NumbersM2Measure ItM4Add & Subtract Big NumbersM5Work with ShapesM4Find the AreaM1Area and Surface AreaM1Scale DrawingsM3Measuring CirclesM6Add & Subtract to 100M4Add, Subtract, & SolveM5Add & Subtract Big NumbersM6EqualGroupsM7Length, Money, & DataM3Multiply & Divide Tricky NumbersM5Fractions as NumbersM6Display DataM7Shapes & MeasurementM3Multiply & Divide Big NumbersM5Equivalent FractionsM6Decimal FractionsM1Place Value with Decimal FractionsM7Multiply & MeasureM2Base Ten OperationsM3Add & Subtract FractionsM4Dividing FractionsM5Arithmetic in Base TenM4Multiply and Divide Fractions & DecimalsM5Volume, Area, & ShapesM6The Coordinate PlaneM2Introducing RatiosM2Introducing Proportional RelationshipsM1Rigid Transformations and CongruenceM4Proportional Relationships and PercentagesM4Linear Equations and Linear SystemsM3Rates and PercentagesM6Expressions and EquationsM6Associations in DataM3Linear RelationshipsM5Functions and VolumeM7Exponents and Scientific NotationM7Rational NumbersM8Data Sets and DistributionsM7Angles, Triangles, and PrismsM8Pythagorean Theorem and Irrational NumbersM6Expressions, Equations, and InequalitiesM5Rational Number ArithmeticM8Probability and SamplingM9Putting It ALL TogetherM4Construct Lines, Angles, & ShapesM2Measure & SolveM9Putting It ALL TogetherM9Putting It ALL TogetherM22D & 3D ShapesM5Numbers 10–20; Count to 100 by Ones and TensM8Shapes, Time, & FractionsM2Dilations, Similarity, and Introducing SlopeWhole Numbers and OperationsExpanding Whole Numbers and Operations to Fractions and DecimalsAlgebraic Thinking and Reasoning Leading to FunctionsGeometry Measurement, Statistics and ProbabilityM1Add & Subtract Small NumbersKeyCURRICULUM MAPv

ZEARN MATH MISSION OVERVIEWG8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.Overview of Topics and Lesson ObjectivesEach 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 unnished learnings.ObjectiveTopic A Rigid TransformationsLesson 1 Describe a translation or rotation of a shape using informal language, such as “slide” and “turn.”Lesson 2Describe the movement of shapes using the terms “clockwise,” “counterclockwise,” “translations,” “rotations,” and “reections” of gures.Lesson 3 Draw the image of a gure that results from a translation, rotation, and reection on a grid.Lesson 4Use the terms translation, rotation, and reection to precisely describe transformations and explain a sequence of transformations that takes one gure to its image.Lesson 5Apply transformations to points on a coordinate plane and name the coordinates of points in the image of a transformation.Lesson 6Apply a sequence of transformations to points on a coordinate plane. Determine whether the order of a sequence of transformations has an eect on the image.Topic B Properties of Rigid TransformationsLesson 7 Compare measurements of sides and angles on a shape before and aer rigid transformations.Lesson 8Rotate a line segment 180 degrees around its midpoint, a point on the segment, and a point not on the segment. Generalize the outcomes of rotating a segment 180 degrees around dierent points.Lesson 9Describe the eects of a rigid transformation on a pair of parallel lines and understand that a rotation by 180 degrees about a point of two intersecting lines moves each angle to the angle that is vertical to it.Lesson 10 Identify side lengths and angle measures using properties of rigid transformations.Mid-Mission Assessment: Topics A-BTopic C Congruencevi

ZEARN MATH MISSION OVERVIEW G8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.ObjectiveLesson 11Determine whether two shapes are congruent by considering their area and perimeter and using rigid transformations.Lesson 12 Use rigid transformations to decide whether or not two gures are congruent.Lesson 13 Determine whether shapes are congruent by measuring the distances between corresponding points.Topic D Angles in a TriangleLesson 14 Calculate angle measures using alternate interior, vertical, and supplementary angles to solve problems.Lesson 15 Use the measure of a straight angle to prove that the sum of the angles in a triangle is 180 degrees.Lesson 16Use parallel lines to show that the sum of the angles in any triangle is 180 degrees. Given the measure of two angles in a triangle, nd the measure of the third angle.Topic E Let’s Put It to WorkLesson 17 Create tessellations and designs with rotational symmetry using rigid transformations.End-of-Mission Assessment: Topics C-Evii

ZEARN MATH MISSION OVERVIEWG8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.Mission OverviewWork with transformations of plane gures in grade 8 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. In grade 6, students combined their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to nd surface areas of polyhedra. In grade 7, students worked with scaled copies and scale drawings, learning that angle measures are preserved in scaled copies, but areas increase or decrease proportionally to the square of the scale factor. Their study of scaled copies was limited to pairs of gures with the same rotation and mirror orientation. Viewed from the perspective of grade 8, a scaled copy is a dilation and translation, not a rotation or reection, of another gure.In grade 8, students extend their reasoning to plane gures with dierent rotation and mirror orientations.Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them, students use and extend their knowledge of geometry and geometric measurement. They begin the mission by looking at pairs of cartoons, each of which illustrates a translation, rotation, or reection. Students describe in their own words how to move one cartoon gure onto another. As the mission progresses, they solidify their understanding of these transformations, increase the precision of their descriptions, and begin to use associated terminology, recognizing what determines each type of transformation, e.g., two points determine a translation. They identify and describe translations, rotations, and reections, and sequences of these. In describing images of gures under rigid transformations on and o square grids and the coordinate plane, students use the terms “corresponding points,” “corresponding sides,” and “image.” Students learn that angles and distances are preserved by any sequence of translations, rotations, and reections, and that such a sequence is called a “rigid transformation.” They learn the denition of “congruent”: two gures are said to be congruent if there is a rigid transformation that takes one gure to the other. Students experimentally verify the properties of translations, rotations, and reections, and use these properties to reason about plane gures, understanding informal arguments showing that the alternate interior angles cut by a transversal have the same measure and that the sum of the angles in a triangle is 180°. The latter will be used in a subsequent grade 8 mission on similarity and dilations. Throughout the mission, students discuss their mathematical ideas and respond to the ideas of others.Many of the lessons in this mission ask students to work on geometric gures that are not set in a real-world context. This design choice respects the signicant intellectual work of reasoning about area. Tasks set in real-world contexts are sometimes contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have opportunities in the mission to tackle real-world applications. In the culminating activity of the mission, students examine and create dierent patterns formed by plane gures. This is an opportunity for them to apply what they have learned in the mission.In this mission, several lesson plans suggest that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to develop their abilities to select appropriate tools and use them strategically to solve problems. To support students in their developing understanding, Zearn Math Independent Digital Lessons utilize a viii

ZEARN MATH MISSION OVERVIEW G8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.variety of digital manipulatives that mirror what students will see in their geometry toolkit. The combination of physical and digital tools will build a deep understanding of important geometric and measurement concepts.Progression of Disciplinary LanguageIn this mission, teachers can anticipate students using language for mathematical purposes such as describing, generalizing, and justifying. Throughout the mission, students will benet 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:Describe• movements of gures (Lessons 1 and 2)• observations about transforming parallel lines (Lesson 9)• transformations using corresponding points, line segments, and angles (Lesson 10)• observations about angle measurements (Lesson 16)• transformations found in tessellations and in designs with rotational symmetry (Lesson 17)Generalize• about categories for movement (Lesson 2)• about rotating line segments (Lesson 8)• about the relationship between vertical angles (Lesson 9)• about transformations and congruence (Lesson 12)• about corresponding segments and length (Lesson 13)• about alternate interior angles (Lesson 14)• about the sum of angles in a triangle (Lesson 16)Justify• whether or not rigid transformations could produce an image (Lesson 7)• whether or not shapes are congruent (Lesson 11)• whether or not polygons are congruent (Lesson 12)• whether or not ovals are congruent (Lesson 13)• whether or not triangles can be created from given angle measurements (Lesson 15)In addition, students are expected to explain and interpret directions for transforming gures and how to apply transformations to nd specic images. Students are also asked to use language to compare rotations of a line segment and compare perimeters and areas of rectangles. Over the course of the mission, teachers can support students’ ix

ZEARN MATH MISSION OVERVIEWG8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.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 rst 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 the one in which it was rst introduced.New TerminologyLesson Receptive Productive1 planemeasurevertex/verticesdirectionslideturn2 clockwisecorrespondingcounterclockwiseimagereectionrotationtranslationopposite3 angle of rotationcenter (of rotation)line of reectionvertex / vertices4 sequence of transformationsdistanceclockwisecounterclockwisereectrotatetranslate5 pointsegmentcoordinatesx-axisy-axis6 polygon angle of rotationcenter (of rotation)line of reectionx

ZEARN MATH MISSION OVERVIEW G8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.New TerminologyLesson Receptive Productive7 rigid transformationmeasurementspreservereectionrotationtranslationmeasurepoint8 midpoint segment9 vertical anglesparallelintersectdistance10 imagerigid transformationmidpointparallel11 congruentperimeterarea12 x-axisy-axisarearight angle13 corresponding14 alternate interior anglestransversalvertical anglescongruentsupplementary angles15 straight angle16 alternate interior anglestransversalstraight angle17 tessellationsymmetryxi

ZEARN MATH MISSION OVERVIEWG8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.Digital LessonsStudents also learn the concepts from this mission in their Independent Digital Lessons. There are 14 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 rst 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.TerminologyAlternate interior anglesAlternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.This diagram shows two pairs of alternate interior angles. Angles a and d are one pair and angles b and c are another pair.ClockwiseClockwise means to turn in the same direction as the hands of a clock. It is a turn to the right.This diagram shows Figure A turned clockwise to make Figure B.CongruentOne gure is congruent to another if it can be moved with translations, rotations, and reections to t exactly over the other.In the gure, Triangle A is congruent to Triangles B, C, and D. A translation takes Triangle A to Triangle B, a rotation takes Triangle B to Triangle C, and a reection takes Triangle C to Triangle D.CorrespondingWhen part of an original gure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.For example, point B in the rst triangle corresponds to point E in the second triangle.Segment AC corresponds to segment DF.transversala bc dABABCDA C DEFBxii

ZEARN MATH MISSION OVERVIEW G8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.CounterclockwiseCounterclockwise means to turn opposite of the way the hands of a clock turn. It is a turn to the le.This diagram shows Figure A turned counterclockwise to make Figure BImageAn image is the result of translations, rotations, and reections on an object. Every part of the original object moves in the same way to match up with a part of the image.In this diagram, triangle ABC has been translated up and to the right to make triangle DEF. Triangle DEF is the image of the original triangle ABC.ReectionA reection across a line moves every point on a gure to a point directly on the opposite side of the line. The new point is the same distance from the line as it was in the original gure.This diagram shows a reection of A over line ℓ that makes the mirror image B.Rigid transformationA rigid transformation is a move that does not change any measurements of a gure. Translations, rotations, and reections are rigid transformations, as is any sequence of these.RotationA rotation moves every point on a gure around a center by a given angle in a specic direction.This diagram shows Triangle A rotated around center O by 55 degrees clockwise to get Triangle B.ABACBDFEABℓABO55°xiii

ZEARN MATH MISSION OVERVIEWG8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.Sequence of transformationsA sequence of transformations is a set of translations, rotations, reections, and dilations on a gure. The transformations are performed in a given order. This diagram shows a sequence of transformations to move Figure A to Figure C. First, A is translated to the right to make B. Next, B is reected across line ℓ to make C.Straight angleA straight angle is an angle that forms a straight line. It measures 180 degrees.TranslationA translation moves every point in a gure a given distance in a given direction. This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow.TransversalA transversal to two parallel lines is a line that cuts across them, intersecting each one.This diagram shows a transversal line k intersecting parallel lines m and ℓ.Vertical anglesVertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.For example, angles AEC and DEB are vertical angles. If angle AEC measures 120°, then angle DEB must also measure 120°.Angles AED and BEC are another pair of vertical angles.ABCRPQℓstraight angleABℓkmCBADExiv

ZEARN MATH MISSION OVERVIEW G8M1© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.Required MaterialsBlank paperGeometry toolkitsTracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesGraph paperIsometric graph paperTemplatesCopies of templatesPre-printed cards, cut from copies of the templatesL1 Activity 1L2 Activity 2L4 Activity 1L6 Activity 1L15 Activity 1L17 Warm-upToothpicks, pencils, straws, or other objectsxv

© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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 1Name: Date: GRADE 8 / MISSION 1Mid-Mission AssessmentNote: Students may use tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and/or an index card to complete this assessment.1. For each of the following, name the transformation (translation, rotation or reection) that takes Figure A to Figure B. Explain your reasoning.a. b. c. ABℓℓABℓAB

G8M1 Mid-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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. Identify all of the triangles that are translations of Triangle A.3. Triangle ABC is shown on the grid below.a. Translate triangle ABC to the right 4 units and up 5 units and label the image A′B′C′.Original Point Translated PointA (-7, -3) A′B (-7, -7) B′C (-2, -5) C′b. Complete the table to show the coordinates of A′B′C′.A BCDEFG-8-6-4-202468864-8 -6 -4 -2 2yxABCPAGE 2

G8M1 Mid-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.4. Triangle ABC is shown on the grid below.a. Rotate triangle ABC 90° counter clockwise using (0, 0) as the center of rotation and label the image A′B′C′.Original Point Translated PointA (1, 1) A′B (3, 7) B′C (4, 2) C′b. Complete the table to show the coordinates of A′B′C′.5. Quadrilateral QRST is shown on the grid below.a. Reect quadrilateral QRST over the y-axis, and label the image that results as Q′R′S′T′.b. Reect quadrilateral Q′R′S′T′ over the x-axis, and label the image that results as Q′′R′′S′′T′′.Original PointPoints aer reecting QR ST over y-axisPoints aer reecting Q'R'S'T' over x-axisQ (-5, 7) Q′ Q′′R (-2, 7) R′ R′′S (-1, 2) S′ S′′T (-5, 2) T′ T′′c. Complete the table to show the coordinates of Q′R′S′T′ and Q′′R′′S′′T′′.-8-6-4-202468864-8 -6 -4 -2 2yxABC-8-6-4-202468864-8 -6 -4 -2 2yxQ RSTPAGE 3

G8M1 Mid-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.6. Describe a sequence of transformations for which Triangle B is the image of Triangle A.7. Quadrilateral A′B′C′D′ is the image of the quadrilateral ABCD under a rigid transformation.a. Label all vertices on quadrilateral A′B′C′D′.b. On both gures, label all known sides lengths and angle measures.-8-6-4-202468864-8 -6 -4 -2 2yxABBCD3.95.4A155°90°45°70°7.55.2PAGE 4

G8M1 Mid-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.8. Points A′, B′, and C′ are the images of 180 degree rotations of A, B, and C, respectively, around point O. Complete the table to nd the unknown measure of each angle. Show your work and/or explain your reasoning.Angle Angle Measure Work / RationalBOC 30° Provided in the diagramB′OA′ 45° Provided in the diagramBOAC′OB′AOC′COA′A30°45°BCC'B'A'OPAGE 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.PAGE 1Name: Date: GRADE 8 / MISSION 1End-of-Mission AssessmentNote: Students may use tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and/or an index card to complete this assessment.1. Figure B is the image of Figure A when rotated about Point O. Are Figure A and Figure B congruent? Explain your reasoning. 2. Describe a sequence of rigid transformations that shows that shape ABCD is congruent to shape PQRS.AOByxA PBCDSRQ

G8M1 End-of-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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. Describe a sequence of rigid transformations that shows that shape ABCDE is congruent to shape PQRST.EBCDPSTRQyxAPAGE 2

G8M1 End-of-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.4. The diagram shows two parallel lines cut by a transversal. One angle measure is shown. Complete the table to nd the unknown measure of each angle. Show your work and/or explain your reasoning.Angle Angle Measure Work / Rationalbcdwxyzd°b°c°z°w°x°y°a = 36°PAGE 3

G8M1 End-of-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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. Lines AC and DE are parallel, and AB and CB are transversals.a. What is the sum of the measures of angles d, b, and e? Explain your reasoning.b. Describe the relationship between angles a and d and between angles c and e.c. Explain how you can use your answer to part (a) and part (b) to determine the sum of the measures of angles a, b, and c.Ab°c°a°d°e°EBCDPAGE 4

G8M1 End-of-Mission Assessment© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.6. Find the measure of angle c. Show your work.7. In this diagram, lines CB and DE are parallel. The measure of angle BCA is 50° and the measure of angle BAE is 110°. Find the measure of angle BAC. Show your work and/or explain your reasoning. A85°43°BC50°110°CBADEPAGE 5

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 1Moving in the PlaneLEARNING GOALSDescribe (orally and in writing) a translation or rotation of a shape using informal language, e.g., “slide,” “turn left,” etc.Identify angles and rays that do not belong in a group and justify (orally) why the object does not belong.LEARNING GOALS(STUDENT FACING)Let’s describe ways figures can move in the plane.LEARNING TARGETS(STUDENT FACING)I can describe how a figure moves and turns to get from one position to another.REQUIRED MATERIALSTemplate for Concept Exploration: Activity 1Copies of templateGeometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor and an index card to use as a straightedge or to mark right anglesREQUIRED PREPARATIONYou will need the Triangle Square Dance template for this lesson. Make 1 copy of all 3 pages for every 2 students.Assemble geometry toolkits. It would be best if students had access to these toolkits at all times throughout the mission. Toolkits include tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles. Access to tracing paper is particularly important in this mission. Tracing paper cut to a small-ish size (roughly 5” by 5”) is best—commercially available “patty paper” is ideal for this. If using larger sheets of tracing paper, such as 8.5” by 11”, cut each sheet into fourths.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:The purpose of this lesson is to introduce students to translations and rotations of plane gures and to have them describe these movements in everyday language. Expect students to use words like “slide” and “turn.” In the next lesson, they will be introduced to the mathematical terms. The term “transformation” is not yet used and will be introduced in a later lesson.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, ruler, protractor, and an index card. For this mission, access to tracing paper and a straight edge are particularly important. 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.Warm-UpWHICH ONE DOESN’T BELONG: DIAGRAMSInstructional Routines: Think Pair Share, Which One Doesn’t Belong?This warm-up prompts students to compare four images. It encourages students to explain their reasoning and hold mathematical conversations. It gives you the opportunity to hear how they use terminology and talk about characteristics of the images in comparison to one another. To allow all students to access the activity, each image has one obvious reason it does not belong. Encourage students to nd reasons based on mathematical properties (e.g., Figure B is the only right angle). During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the mission. The activity also gives students an opportunity to nd useful tools in their geometry toolkit.Before students begin, consider establishing a small, discreet hand signal students can display to indicate they have an answer they can support with reasoning. This signal could be a thumbs up, or students could show the number of ngers that indicate the number of responses they have for the problem. This is a quick way to see if students have had enough time to think about the problem and keeps them from being distracted or rushed by hands being raised around the class.As students share their responses, listen for important ideas and terminology that will be helpful in upcoming work of the mission, such as references to angles and their measures.LAUNCHArrange students in groups of 2–4, and provide access to geometry toolkits. Display the gures for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students quiet think time and then time to share their thinking with their © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:small group. In their small groups, tell each student to share their reasoning why a particular image does not belong. Together, nd at least one reason each image doesn’t belong.WARM-UP TASKWhich one doesn’t belong? STUDENT RESPONSEAnswers vary. Sample responses:A doesn’t belong because:• The rays point in opposite directions.• It is not possible to make a triangle by joining points on the rays.B doesn’t belong because:• They make a right angle.• Both rays are to the right of the vertex.C doesn’t belong because:• It is an acute angle.• Both rays point downward.D doesn’t belong because:• It is an obtuse angle (measuring less than 180 degrees).• The long ray points to the le of the short ray.1A 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. 3

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP RECAPAsk each group to share one reason why a particular image does not belong. Record and display the responses for all to see. Aer each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.During the discussion, prompt students to explain the meaning of any terminology they use, such as ray, degree, or acute angle. Also, press students on unsubstantiated claims. For example, a student may make claims about the angle measures. Ask how they know the measure and demonstrate how the tracing paper or the ruler from the toolkit could be used to check.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the gures to students who benet from extra processing time to review prior to implementation of this activity.Expressive Language: Eliminate BarriersProvide sentence frames for students to explain their reasoning (i.e., doesn’t belong because .)Concept Exploration: Activity 1TRIANGLE SQUARE DANCEMATERIALS: TEMPLATE FOR CONCEPT EXPLORATION: ACTIVITY 1Instructional Routines: MLR2: Collect and Display, MLR8: Discussion SupportsThe purpose of this activity is for students to begin to observe and describe translations and rotations. In groups of 2, they describe one of 3 possible dances, presented in cartoon form, and the partner identies which dance is being described. Identify students who use specic and detailed language to describe the dance and select them to share during class discussion.While students are not expected to use precise language yet, this activity both provides the intellectual need for agreeing upon common language and gives students a chance to experiment with dierent ways of describing some moves in the plane.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.4

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHArrange students in groups of 2, and give a copy of all 3 templates to each group. Explain that each sheet is a cartoon with 6 frames showing the moves made by the dancing gures. Instruct students to place all three sheets face up, and tell them to take turns selecting a dance and describing it to their partner, without revealing which dance they have selected. The other student identies which dance is being described. Use MLR 2 (Collect and Display) to record language students use to describe the movement of shapes to later be grouped and connected to more formal language such as “rotation,” “translation,” etc. Give students time to work in their groups followed by a whole-class discussion.ACTIVITY 1 TASK 1 Your teacher will give you three pictures. Each shows a dierent set of dance moves.1. Arrange the three pictures so you and your partner can both see them the right way up. Choose who will start the game.• The starting player mentally chooses A, B, or C and describes the dance to the other player.• The other player identies which dance is being talked about: A, B, or C.2. Aer one round, trade roles. When you have described all three dances, come to an agreement on the words you use to describe the moves in each dance.3. With your partner, write a description of the moves in each dance.STUDENT RESPONSEAnswers vary. Sample response:A: Move right, turn 90º clockwise, move up, move le, and turn 90º counterclockwise.B: Move right, turn 90º clockwise, move le, move up, and turn 90º counterclockwise.C: Move right, turn 90º counterclockwise, move le, move up, and turn 90º clockwise.The terms le, right, and up in this answer are from the point of view of an observer watching the dance. Alternatively students might put themselves in the shoes of the triangles and describe 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. 5

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:things in terms of the triangle’s le, right, up, and down. Students might use other words, such as “shi” and “step” for translations, and “spin” and “rotate” for the turns. They might describe the 90º turns as “quarter turns.”ACTIVITY 1 RECAP Select one student to share their description for each of the pages, and use MLR 2 (Collect and Display) to display their words and other language you observed during the activity for dierent types of moves. Arrange the words in two groups, those that describe translations and those that describe rotations (but do not use these terms). Come to an agreement on a word for each type, and discuss what extra words are needed to specify the transformation exactly (e.g., move right, turn clockwise 90º).Consider asking students what they found most challenging about describing the dances. Expected responses include being as precise as possible about the dierent motions (for example, describing whether the shape is rotating clockwise or counterclockwise). Also consider asking students if they were sometimes able to identify the dance before their partner nished describing all of the moves. All three dances begin by moving to the right, but in the second step, Dances A and B rotate 90 degrees clockwise while Dance C rotates 90 degrees counterclockwise. (So if the second move was to rotate 90 degrees counterclockwise, this must be Dance C.) Dances A and B diverge at slide 4.ANTICIPATED MISCONCEPTIONSStudents oen confuse or are unsure about the meaning of the terms clockwise and counterclockwise. Discuss with them (and demonstrate, if possible) how the hands on a clock rotate, emphasizing the direction of the rotation. Students may also be unsure of how to measure the rotation in terms of degrees. Consider asking a student who expresses angle measures in terms of degrees to explain how they see it.SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversing:This activity uses MLR 2 Collect and Display to gather informal language students use to describe movements (e.g., “clockwise,” “slide,” “step right,” “turn to right,” etc.), prior to introducing more formal language such as “rotation,” “translation,” etc. in a future lesson.Design Principle(s): Optimize output (for explanation); 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.6

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Eliminate BarriersDemonstrate the steps for the activity or game by having a group of students and sta play an example round while the rest of the class observes.Receptive/Expressive Language: Processing TimeStudents who benet from extra processing time would also be aided by MLR 8 (Discussion Supports).Digital LessonThe six frames show a shape’s dierent positions. Describe how the shape moves to get from its position in each frame to the next.To get from frame 1 to frame 2,To get from frame 2 to frame 3,To get from frame 3 to frame 4,To get from frame 4 to frame 5,To get from frame 5 to frame 6,Frame 1 Frame 2 Frame 3Frame 4 Frame 5 Frame 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. 7

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSETo get from frame 1 to frame 2, the shape slides right. To get from frame 2 to frame 3, the shape turns clockwise 90 degrees. To get from frame 3 to frame 4, the shape slides down. To get from frame 4 to frame 5, the shape turns clockwise 90 degrees. To get from frame 5 to frame 6, the shape slides le.Wrap-UpLESSON SYNTHESISWe have started to reason about what it means to move a gure in the plane. Display two gures that clearly show a slide and two gures that clearly show a turn. Example of a slide:Example of a turn:Invite students to share the language they would use to describe them: for example, “moving” or “sliding” for translations and “turning” for rotations. Consider asking students how they might quantify each move, for example with a distance and direction for the slides and a center and angle of rotation for the turns. Tell them that we will continue to look at these moves in more detail in future lessons.1 25 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.8

G8M1 | LESSON 1ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:EXIT TICKETHere are successive positions of a shape:Describe how the shape moves from:1. Frame 1 to Frame 2.2. Frame 2 to Frame 3.3. Frame 3 to Frame 4.STUDENT RESPONSE1. Slide down.2. Turn counterclockwise 90 degrees (or one quarter of a full turn).3. Slide up.Frame 1 Frame 2 Frame 4Frame 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. 9

TEMPLATE FOR LESSON 1 CONCEPT EXPLORATION: ACTIVITY 1PAGE 1 OF 3A123456© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.10

TEMPLATE FOR LESSON 1 CONCEPT EXPLORATION: ACTIVITY 1PAGE 2 OF 3B123456© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

TEMPLATE FOR LESSON 1 CONCEPT EXPLORATION: ACTIVITY 1PAGE 3 OF 3C123456© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.12

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 2Naming the MovesLEARNING GOALSComprehend that the term “image” refers to a figure after a transformation has occurred (e.g., Figure B is the image of Figure A) and the term “corresponding points” refers to a point that is in the same part of the figure in both the original figure and the image.Describe (orally and in writing) the movement of shapes informally and formally using the terms “clockwise,” “counterclockwise,” “translations,” “rotations,” and “reflections” of figures.LEARNING GOALS(STUDENT FACING)Let’s be more precise about describing moves of figures in the plane.LEARNING TARGETS(STUDENT FACING)I can identify corresponding points before and after a transformation.I know the difference between translations, rotations, and reflections.REQUIRED MATERIALSTemplate for Concept Exploration: Activity 2Geometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesPre-printed cards, cut from copies of the templateREQUIRED PREPARATIONPrint and cut up cards from the Translations, Rotations, and Reflections template. Prepare 1 copy for every 3 students. Make sure students have access to items in their geometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles.Access to tracing paper is particularly important. Each student will need about 10 small sheets of tracing paper (commercially available “patty paper” is ideal). If using large sheets of tracing paper, such as 8.5 inches by 11 inches, cut each sheet into fourths.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, students begin to describe a given translation, rotation, or reection with greater precision and are introduced to the terms translation, rotation, and reection. The collective terms “transformation” and “rigid transformation” are not used until later lessons. Students are introduced to the terms image and corresponding points. They use the terms clockwise and counterclockwise. Students then use this language to identify the individual moves on various gures.Students experiment with ways to describe moves precisely enough for another to understand their meaning.Warm-UpA PAIR OF QUADRILATERALSInstructional Routines: Think Pair ShareStudents estimate an angle of rotation. While they do not need to use a protractor, a protractor is an ideal tool and allows them to estimate the angle measure more accurately. Monitor for how students report the measure of the angle: do they round to the nearest degree, to the nearest 5 degrees?LAUNCHArrange students in groups of 2–4. Provide access to geometry toolkits. Display the two quadrilateral gures for all to see. (They should also look at the Warm-up in their notes.) Ask students to give a discreet hand signal when they have an estimate for the angle of rotation. Give students quiet think time and then time to share their thinking with their group before a whole-class discussion. WARM-UP TASKQuadrilateral A can be rotated into the position of Quadrilateral B. Estimate the angle of rotation.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.14

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAnswers vary. Sample response: About 120 degrees (counterclockwise)This gure doesn’t need to be part of students’ responses but is provided as an example of an angle between two segments that could be measured to nd the angle of rotation from A to B.DISCUSSION GUIDANCEInvite students to share their estimates for the angle of rotation. Ask students how they knew, for example, that the angle is more than 90 degrees (because the angle is obtuse) but less than 180 degrees (because the angle is less than a straight line).Introduce or reiterate the language of clockwise (for rotating in the direction the hands on a clock move) and counterclockwise (for rotating in the opposite direction). In this case, the direction of rotation is not specied but it is natural to view Figure A being rotated counterclockwise onto Figure B. Make sure to introduce the language of the center of rotation (the vertex shared by A and B is the center of rotation) and tell students that we call Figure B the image of Figure A for the rotation. It may be helpful to display the picture from the task statement to support this discussion, and if possible, show the 120° counterclockwise turn dynamically.AB120°© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ANTICIPATED MISCONCEPTIONSStudents may not be sure which angle to measure. They may measure the acute angle between Shape A and Shape B. Ask these students to trace Shape A on tracing paper and rotate it by that angle to see that this does not give Shape B.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the picture to students who benet from extra processing time to review prior to implementation of this activity.Concept Exploration: Activity 1HOW DID YOU MAKE THAT MOVE?Instructional Routines: Think Pair Share; MLR8: Discussion SupportsThis activity informally introduces reections, which appear in addition to some translations and rotations (that were introduced informally in the previous lesson). Students are given a 6-frame cartoon showing the change in position of a polygon. As in the previous lesson, they describe the moves, but this time there are reections, which may seem impossible as physical moves unless you allow the shape to leave the plane. Students identify the new moves and try to describe them.Aer the end of this activity, the three basic moves have been introduced and the next activity will introduce their names (translations, rotations, and reections).LAUNCHKeep students in the same groups, and maintain access to geometry toolkits. Give students quiet work time, and then invite them to share their responses with their group. Follow with a discussion. Tell students that they will be describing moves as they did in the previous lesson, but this time there is a new move to look out for. Recall the words the class used to describe slides and turns.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.16

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 1Here is another set of dance moves. 1. Describe each move or say if it is a new move.a) Frame 1 to Frame 2.b) Frame 2 to Frame 3.c) Frame 3 to Frame 4.d) Frame 4 to Frame 5.e) Frame 5 to Frame 6.2. How would you describe the new move?STUDENT RESPONSE1. Answers vary. Sample response:a) Frame 1 to Frame 2: Shi to the rightb) Frame 2 to Frame 3: New movec) Frame 3 to Frame 4: Turn 90º clockwised) Frame 4 to Frame 5: Shi up2© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:e) Frame 5 to Frame 6: New move2. The new move is like becoming your mirror image through a mirror placed at the center of the frame. For the second move, the mirror is vertical and for the last move it is horizontal.ACTIVITY RECAPThe purpose of this discussion is an initial understanding that there is a third type of move that is fundamentally dierent from the moves encountered in the previous lesson, because it reverses directions. Some possible discussion questions to help them identify these are:• “How is the motion from panel 2 to panel 3 dierent than the ones we discussed yesterday?”• “Is there anywhere else that happens in this cartoon?”• “What features of the image help us to see that this move is happening?”To help answer these questions, tell students to pay attention to the direction that the “beak” of the polygon is pointing, le or right. Draw a dotted vertical line in the middle of Frame 2, and say, “Here is a mirror. The polygon in Frame 3 is what the polygon in Frame 2 sees when it looks in the mirror.”Demonstrate using tracing paper or transparencies to show they are mirror images. Then ask students if there are any other mirror lines in other frames. For the second reection, compare Frame 5 to Frame 6.ANTICIPATED MISCONCEPTIONSStudents may see a reection as a translation especially since the gures are not on the same frame. Ask these students to trace Frame 2 on tracing paper. Is there any way to turn it into Frame 3 by sliding it? What do they have to do to turn it into Frame 3? (They have to ip the tracing paper over, so, this is a new kind of move.) © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.18

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In describing reections, students may confuse the terms horizontal and vertical. Consider posting the terms horizontal and vertical with examples in your room.SUPPORT FOR ENGLISH LANGUAGE LEARNERSRepresenting, Conversing: MLR 8 Discussion SupportsUse this to routine to invite students to act out or show the movements of the polygon in order to help them describe its movement. Aer students have completed the two problems, they should describe the dierences in dance moves with a partner. Encourage students to show what they know by doing one of the following: create a simple dance move and show with your body how it would change from frame to frame, draw the shape on a small whiteboard and show with arrows how it turns or moves, or draw the shape on paper and cut it out. If students draw and cut out the shape, they should physically slide, turn, or rotate the shape. Remind students to use terms like: horizontal and vertical, slide to the le/right, or turn.Design Principles: Support sense-making; Cultivate conversationSUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Eliminate BarriersDemonstrate the steps for the activity or game, in which a group of students and sta play an example round while the rest of the group observes.Executive Functioning: Graphic OrganizersProvide a Venn diagram with which to compare the similarities and dierences between rotations and reections.Concept Exploration: Activity 2MOVE CARD SORTMATERIALS: TEMPLATE FOR CONCEPT EXPLORATION: ACTIVITY 2Instructional Routines: Take Turns; MLR8: Discussion SupportsThe purpose of this card sort activity is to give students further practice identifying translations, rotations, and reections, and in the discussion aer they have completed the task, introduce those terms. In groups of 3 they sort 9 cards into categories. There are 3 translations, 3 rotations, and 3 reections. Students explain their categories and come to agreement on 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. 19

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:On the template, there are actually 12 cards. The last three show slightly more complicated moves than the rst 9. These can be withheld, at rst, and used if time permits.Students might identify only 2 categories, putting the reections with the translations (in the case of Card 3) or the rotations (in the case of Card 5). As students work, monitor for groups who have sorted the cards into translations, rotations, and reections (though not necessarily using those words). Also monitor for descriptions of corresponding points such as “these points go together” or “here are before and aer points.”LAUNCHArrange students into groups of 3, and provide access to geometry toolkits. Give each group the rst 9 cards. Reserve the last 3 cards for use if time permits.Tell students that their job is to sort the cards into categories by the type of move that they show. Aer they come to consensus about which categories to use, they take turns placing a card into a category and explaining why they think their card goes in that category. When it is not their turn, their job is to listen to their partner’s reasoning and make sure they understand. Consider conducting a short demonstration with a student of productive ways to communicate during this activity. For example, show what it looks like to take turns, explain your thinking, and listen to your partner’s thinking.Give students time to sort the cards. Do not explicitly instruct students at the beginning to use the words translations, rotations, and reections. Monitor for a group who uses these categories, even if they use dierent names for them. If time permits, distribute the remaining 3 cards. Follow with discussion.ACTIVITY 2 TASK 1Your teacher will give you a set of cards. Sort the cards into categories according to the type of move they show. Be prepared to describe each category and why it is dierent from the others.STUDENT RESPONSETranslations: 1, 7, 8, 10Rotations: 2, 6, 9, 12Reections: 3, 4, 5, 1133© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.20

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:To detect if one gure is a translation of another, look to see if it is still sitting in exactly the same way, e.g., the two gures have the same orientation and are sitting on the same base. To detect if one gure is a rotation of another, look to see if one gure is not standing up in exactly the same way as the other but appears to be turned. Reections can be confused with both translations (if the two gures are still on the same base) and rotations (if they appear to be turned). The way to detect a reection in these examples is to choose a feature of the gure that exists on one side of it but not the other (e.g., the sharp “rabbit ears” in this activity) and see if it is pointing to the le in one gure and to the right in the other. (Alternatively, up and down if the line of reection is horizontal.)ACTIVITY RECAPSelect one or more groups to share the names of their categories. Select one or more groups to share how they sorted the cards into the categories. Ask the whole group if they disagree with any of the choices, and give students opportunities to justify their reasoning.Introduce the terms translation, rotation, and reection. It may be helpful to display an example of each to facilitate discussion:Point out ways to identify which type of move it is. Translations are a slide with no turning. Rotations are a turn. Reections face the opposite direction. Introduce the term corresponding points. If we see the gures as rabbits, then the ear tips in the original gure and the ear tips in its image are corresponding points, for example. Also introduce the term image for a gure aer a transformation is applied: for each of the cards, one gure is the image of the other gure aer a translation, rotation, or reection has been applied.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion SupportsUse this support to amplify mathematical uses of language to communicate about rotations, reections, and translations. As groups share how they categorized and sorted the shapes, revoice their ideas using the terms “translation,” “rotation,” “reection,” “image,” and “corresponding points.” Then, invite students to use the terms when describing their categories and strategies for sorting. Some students may benet from chorally repeating the phrases that include the terms “translation,” “rotation,” “reection,” “image,” and “corresponding points” in context.Design Principle: Optimize output (for explanation)SUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer TutorsPair students with their previously identied peer tutors.Digital LessonLabel each set of 2 frames as a translation, rotation, or reection and explain how you decided.Frame 1 to Frame 2 shows a , becauseFrame 3 to Frame 4 shows a , becauseFrame 5 to Frame 6 shows a , because© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.22

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAnswers vary. Sample response: Frame 1 to Frame 2 shows a reection, because the shape in Frame 2 looks like a mirror image of the shape in Frame 1. Frame 3 to Frame 4 shows a rotation because the shape looks like it turned. Frame 5 to Frame 6 shows a translation, because the shape slid down.Wrap-UpLESSON SYNTHESISQuestions for discussion:• “We encountered a new type of move that was dierent from yesterday. What can you tell me about it?” (It’s like a mirror image, you can’t make the move by sliding or turning, the gure faces the opposite direction.)• “We gave mathematical names to the three types of moves we have seen. What are they called?” (The “slide” is called a translation, the “turn” is called a rotation, and the mirror image is called a reection.)• “What do we mean by corresponding points?” (A point that is in the same part of the gure in both the original gure and the image.)• “What do we mean by a gure’s image?” (The resulting gure aer a move has been performed.)Consider creating a semi-permanent display that shows these three terms and their denitions for reference throughout the mission.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:A translation slides a gure without turning it. Every point in the gure goes the same distance in the same direction. For example, Figure A was translated down and to the le, as shown by the arrows. Figure B is a translation of Figure A.A rotation turns a gure about a point, called the center of the rotation. Every point on the gure goes in a circle around the center and makes the same angle. The rotation can be clockwise, going in the same direction as the hands of a clock, or counterclockwise, going in the other direction. For example, Figure A was rotated 45° clockwise around its bottom vertex. Figure C is a rotation of Figure A.A reection places points on the opposite side of a reection line. The mirror image is a backwards copy of the original gure. The reection line shows where the mirror should stand. For example, Figure A was reected across the dotted line. Figure D is a reection of Figure A.BAAC45°A 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.24

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TERMINOLOGYClockwise Clockwise means to turn in the same direction as the hands of a clock. It is a turn to the right. This diagram shows Figure A turned clockwise to make Figure B.Corresponding When part of an original gure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.For example, point B in the rst triangle corresponds to point E in the second triangle.Segment AC corresponds to segment DF.Counterclockwise Counterclockwise means to turn opposite of the way the hands of a clock turn. It is a turn to the le.This diagram shows Figure A turned counterclockwise to make Figure B.Image An image is the result of translations, rotations, and reections on an object. Every part of the original object moves in the same way to match up with a part of the image.In this diagram, triangle ABC has been translated up and to the right to make triangle DEF. Triangle DEF is the image of the original triangle ABC.Reection A reection across a line moves every point on a gure to a point directly on the opposite side of the line. The new point is the same distance from the line as it was in the original gure.This diagram shows a reection of A over line 𝓁 that makes the mirror image B.ABA CBD FEABACBDFEAB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 2ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Rotation A rotation moves every point on a gure around a center by a given angle in a specic direction.This diagram shows Triangle A rotated around center O by 55 degrees clockwise to get Triangle B.Translation A translation moves every point in a gure a given distance in a given direction. This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow.EXIT TICKETWhat type of move takes Figure A to Figure B?STUDENT RESPONSEAnswers vary. Sample response: It is a rotation. If Figure A is turned around the point shared by Figures A and B, it can land on Figure B.ABO55°ABAB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.26

TEMPLATE FOR LESSON 2 CONCEPT EXPLORATION: ACTIVITY 2PAGE 1 OF 11 2 34 5 67 8 910 11 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. 27

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 3Grid MovesLEARNING GOALSDescribe (orally) the moves needed to perform a transformation.Draw and label the image and “corresponding points” of figures that result from translations, rotations, and reflections.Draw the “image” of a figure that results from a translation, rotation, and reflection in square and isometric grids and justify (orally) that the image is a transformation of the original figure.LEARNING GOALS(STUDENT FACING)Let’s transform some figures on grids.LEARNING TARGETS(STUDENT FACING)I can decide which type of transformations will work to move one figure to another.I can use grids to carry out transformations of figures.REQUIRED MATERIALSGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesREQUIRED PREPARATIONMake sure students have access to items in their geometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles.Access to tracing paper is particularly important. Each student will need about 10 small sheets of tracing paper (commercially available “patty paper” is ideal). If using large sheets of tracing paper, such as 8.5 inches by 11 inches, cut each sheet into fourths.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.28

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Prior to this lesson, students have learned the names for the basic moves (translation, rotation, and reection) and have learned how to identify them in pictures. In this lesson, they apply translations, rotations, and reections to gures. They also label the image of a point P as P′. While not essential, this practice helps show the structural relationship between a gure and its image. Students also encounter the isometric grid (one made of equilateral triangles with 6 meeting at each vertex). They perform translations, rotations, and reections both on a square grid and on an isometric grid. Expect a variety of approaches, mainly making use of tracing paper but students may also begin to notice how the structure of the dierent grids helps draw images resulting from certain moves.Warm-UpNOTICE AND WONDER: THE ISOMETRIC GRIDThe purpose of this warm-up is to familiarize students with an isometric grid. While students may notice and wonder many things, characteristics such as the measures of the angles in the grid and the diagonal parallel lines will be important properties for students to notice in their future work performing transformations on the isometric grid. Students are not expected to know each angle in an equilateral triangle is 60 degrees, but aer previous experience with supplementary angles, circles and rotations, they may be able to explain why each smaller angle is 60 degrees. Many things they notice may be in comparison to the square grid paper which is likely more familiar.LAUNCHArrange students in groups of 2. Tell students that they will look at an image. Their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students quiet think time, and then time to discuss the things they notice with their partner, followed by a whole-class 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. 29

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASK What do you notice? What do you wonder? STUDENT RESPONSEThings students may notice:• There are three sets of parallel grid lines.• The line segments form equilateral triangles.• The individual angles in the equilateral triangles are 60 degrees.• There are vertical lines but no horizontal lines.• There are no 90 degree angles made by the grid lines.• Each vertex has 6 line segments coming from it.• The grid is made out of equilateral triangles instead of squares.Things students may wonder:• Why are there no 90 degree angles?• Why are there no squares?• Are we going to use this kind of grid?• Why would we use this grid instead of the square grid?• Why are there no horizontal lines?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.30

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEAsk students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image, and show where each of the features students notice is located on the actual grid itself, such as triangles, angles, and line segments. Aer each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If angle measures do not come up during the conversation, ask them to think about how they could gure out the measure of each angle. Some may measure with a protractor, and some may argue that since 6 angles share a vertex where each angle is identical, each angle measures 60° because 360 ÷ 6 = 60. Establish that each angle measures 60°.SUPPORT FOR STUDENTS WITH DISABILITIESExecutive Functioning: Graphic OrganizersProvide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.Concept Exploration: Activity 1TRANSFORMATION INFORMATIONInstructional Routines: MLR8: Discussion SupportsThe purpose of this activity is for students to interpret the information needed to perform a transformation and draw an image resulting from the transformation.Through hands-on experience with transformations, students prepare for the more precise denitions they will learn in later grades. This activity is the rst time students start to use A′, B′, etc. to denote points in the image that correspond to A, B, etc. in the original gure. This is also a good activity to use the word “image” to describe the transformed gure—this can happen before, as, or aer students work.If students exploit the mathematical properties of the grid lines to draw transformed gures, they are making use of structure. In order to draw the transformed gures correctly, students must attend to the details of the given information.Watch for students who use tracing paper and those who use properties of the grids to help decide where to place the transformed gures. Tracing paper may be particularly useful for the isometric grid which may be unfamiliar to some 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. 31

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHPoint out A′ in the rst question. Tell students we call point A′ “A prime” and that, aer a transformation, it corresponds to A in the original.Optionally, before students start working, demonstrate the mechanics of performing each type of transformation using tracing paper. Distribute about 10 small sheets of tracing paper to each student (or ensure they can nd it in their geometry toolkits). Give students quiet work time followed by discussion.ACTIVITY 1 TASK 1 Your teacher will give you tracing paper to carry out the moves specied. Use A′, B′, and C′ to indicate vertices in the new gure that correspond to the points A, B, and C in the original gure.1. In Figure 1, translate triangle ABC so that A goes to A′.2. In Figure 2, translate triangle ABC so that C goes to C′.3. In Figure 3, rotate triangle ABC 90° counterclockwise using center O.4. In Figure 4, reect triangle ABC using line ℓ.2ℓAFigure 2Figure 4Figure 1Figure 3BCCʹABCABCABCOAʹ© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.32

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 2 Your teacher will give you tracing paper to carry out the moves specied. Use A′, B′, C′, and D′ to indicate vertices in the new gure that correspond to the points A, B, C, and D in the original gure.1. In Figure 5, rotate quadrilateral ABCD 60° counterclockwise using center B.2. In Figure 6, rotate quadrilateral ABCD 60° clockwise using center C.3. In Figure 7, reect quadrilateral ABCD using line ℓ.4. In Figure 8, translate quadrilateral ABCD so that A goes to C.STUDENT RESPONSETask 133ℓABDABDABDABDFigure 6Figure 8Figure 5Figure 7CCC CℓAFigure 2Figure 4Figure 1Figure 3BCʹAʹABCABCABCCO© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Task 2Sample strategy:• Trace the gure onto tracing paper, and then move the tracing paper according to the description of each move. Observe where the tracing paper ends up, and draw a copy of the gure at that location.• Use the structure of the grid to move each vertex of the original gure according to the description of each move. For example in Figure 1, point A moves up 2 and right 6 to A′. A translation is a slide, so each vertex makes this same move along the grid from its original location.DISCUSSION GUIDANCEAsk students to share how they found the images, and highlight the information they needed in each to perform the transformation. Invite students who used tracing paper to share how they found the images and also ask students what mathematical patterns they found. For example, for the reection in Figure 4, ask where some intersections of grid lines go (they stay on the same horizontal line and go to the other side of ℓ, the same distance away). How can this be used to identify the image of △ABC?Ask students how working on the isometric grid is similar to working on a regular grid and how it is dierent. Possible responses include:• Translations work the same way, identifying how far and in which direction to move the shape.• Rotations also work the same way but the isometric grid works well for multiples of 60 degrees (with center at a grid point), while the regular grid works well for multiples of 90 degrees (also with center at a grid point).• Reections on the isometric grid require looking carefully at the triangular pattern to place the reection in the right place. Like for the regular grid, these reections are diicult to visualize if the line of reection is not a grid line.ℓABDABDABDABDFigure 6Figure 8Figure 5Figure 7CCC 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.34

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ANTICIPATED MISCONCEPTIONSStudents may struggle to understand the descriptions of the transformations to carry out. For these students, explain the transformations using the words they used in earlier activities, such as “slide,” “turn,” and “mirror image” to help them get started. Students may also struggle with reections that are not over horizontal or vertical lines.Some students may need to see an actual mirror to understand what reections do, and the role of the reection line. If you have access to rectangular plastic mirrors, you may want to have students check their work by placing the mirror along the proposed mirror line.Working with the isometric grid may be challenging, especially rotations and reections across lines that are not horizontal or vertical. For the rotations, you may want to ask students what they know about the angle measures in an equilateral triangle. For reections, the approach of using a mirror can work or students can look at individual triangles in the grid, especially those with a side on the line of reection, and see what happens to them. Aer checking several triangles, they develop a sense of how these reections behave.SUPPORT FOR ENGLISH LANGUAGE LEARNERSListening, Representing: MLR 8 Discussion SupportsTo help develop students’ meta-awareness and understanding of the task expectations, think aloud as you transform the quadrilateral in the question about rotating quadrilateral ABCD 60° counterclockwise using center B. As you talk, model mathematical language use and highlight the relationship between quadrilateral ABCD, the image (i.e., quadrilateral A′B′C′D′), and the steps taken to rotate quadrilateral ABCD.Design Principle(s): Maximize meta-awarenessSUPPORT FOR STUDENTS WITH DISABILITIESFine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allow students who struggle with ne motor skills to dictate transformations 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. 35

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Digital LessonReect gure A across line ℓ.STUDENT RESPONSEWrap-UpLESSON SYNTHESIS Display one example each of a translation, a rotation, and a reection. Choose one of each from the activity, or create new examples. Ask students:• “What are important things to keep in mind when we want to do a [translation, rotation, reection]?”ℓFGAD EℓFGAD EF' G'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.36

G8M1 | LESSON 3ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• “What is something new that you learned today about [translations, rotations, reections]?”• “Describe the two dierent kinds of grids we saw. What was the same and what was dierent about them?”EXIT TICKET This is a quick check to see if students can distinguish translations from rotations and reections.TASKWhich of these triangles are translations of Triangle A? Select all that apply.STUDENT RESPONSETriangle B and Triangle D. Triangles C and E are reections of Triangle A, while Triangle F is rotated.ABDFEC© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 4Making the MovesLEARNING GOALSComprehend that a “transformation” is a translation, rotation, reflection, or a combination of these.Draw a transformation of a figure using information given orally.Explain (orally) the “sequence of transformations” that “takes” one figure to its image.Identify (orally and in writing) the features that determine a translation, rotation, or reflection.LEARNING GOALS(STUDENT FACING)Let’s draw and describe translations, rotations, and reflections.LEARNING TARGETS(STUDENT FACING)I can use the terms translation, rotation, and reflection to precisely describe transformations.REQUIRED MATERIALSTemplate for Concept Exploration: Activity 1Geometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesPre-printed cards, cut from copies of the templateREQUIRED PREPARATIONPrint and cut up cards from the Make that Move template. Prepare 1 set of cards for every 4 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.38

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In the previous lesson, students were introduced to the terms “translation,” “rotation,” and “reection.” In this lesson, students understand that:• A translation is determined by two points that specify the distance and direction of the translation.• A rotation is determined by a point, angle of rotation, and a direction of rotation.• A reection is determined by a line.These moves are called transformations for the rst time and students draw images of gures under these transformations. They also study where shapes go under sequences of these transformations and identify the steps in a sequence of transformations that takes one gure to another. Note the subtle shi in language. In the previous lesson, one shape “moves” to the other shape—it is as if the original shape has agency and does the moving. In this lesson, the transformation “takes” one shape to the other shape—this language choice centers the transformation itself as an object of study.Students may make use of tracing paper to experiment with moving shapes. Encourage students to choose to make use of an appropriate tool. Students are also likely starting to begin thinking strategically about which transformations will take one gure to another, identifying properties of the shapes that indicate whether a translation, rotation, reection or sequence of these will achieve this goal.Warm-UpREFLECTION QUICK IMAGEInstructional Routines: MLR7: Compare and Connect; MLR1: Stronger and Clearer Each TimeIn this warm-up, students are asked to sketch a reection of a given triangle and explain the strategies they used. The goal is to prompt students to notice and articulate that they can use the location of a single point and the fact that the image is a reection of the triangle to sketch the image. To encourage students to use what they know about reections and not count every grid line, this gure is ashed for a few seconds and then hidden. It is ashed once more for students to check 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. 39

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHBefore beginning, make sure students have their notes open to the correct page. Tell students you will show them an image of a reection of triangle ABD for 3 seconds. Their job is to draw the image and explain any strategies they used.Display the completed image for a few seconds and then hide it. Do this twice. Give students quiet work time aer each ash of the image. Encourage students to think about any shortcuts they used to draw the reected image.WARM-UP TASKHere is triangle ABD. Your teacher will ash a completed image of triangle ABD twice. Your job is to complete the image in your notes. AD BA'B'D'1ADB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.40

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAnswers vary. Possible response: The rst ash showed where to put B′ and the second ash where to put A′. Once these were in place, there was only one place D′ could go, underneath the segment A′B′, so that A′B′D′ is a reected image of ABD.DISCUSSION GUIDANCESelect a few students to share strategies they used in sketching their gure. Consider asking some of the following questions:• “What was important in creating your sketch (what did you need)?”• “What did you look for in the rst ash? The second?”• “What stayed the same and what is dierent in the shape and its image?”• “How did you decide where to place the vertices of the image?”• “How did you decide how long to make the sides?”ANTICIPATED MISCONCEPTIONSStudents may struggle drawing the image under transformation from the quick ashes of the image because they are trying to count the number of spaces each vertex moves. Encourage these students to use the line in the diagram to help them draw the reected image.AD BA'B'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. 41

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESMemory: Processing TimeThis instructional routine can be very taxing to student’s working memory. For students with challenges in this area, show the image for a longer period of time or repeat the image ash as needed. Students also benet from being explicitly told to look for helpful structure within the image.Receptive/Expressive Language: Processing TimeStudents who benet from extra processing time would also be aided by MLR 1 (Stronger and Clearer Each Time).Concept Exploration: Activity 1MAKE THAT MOVEMATERIALS: TEMPLATE FOR CONCEPT EXPLORATION: ACTIVITY 1Instructional Routine: MLR2: Collect and DisplayThe purpose of this activity is for students to give precise descriptions of translations, rotations, and reections. By the end of the previous lesson, students have identied and sketched these transformations from written directions, however they have not used this more precise language themselves to give descriptions of the three motions. The gures in this activity are given on grids to allow and to encourage students to describe the transformation in terms of specic points, lines or angles. Students need to describe the moves precisely and clearly.There are four dierent transformation cards students use in this activity: 1A, 1B, 2A, and 2B. Each card has the original gure and the image under a transformation. Students are arranged in groups of 2 and each one gets a dierent transformation card: some pairs are given cards 1A and 1B while other pairs get 2A and 2B. Each A card is a translation, while the B cards show either a rotation or reection.As students are describing their transformations and sketching their images under transformation, monitor for students using precise descriptions to their partner in terms of specic points, lines, or angles. LAUNCHIntroduce the word transformation: This is a word for a move (like the reection in the warm-up) or for a sequence of moves. It’s a mathematical word which will be used in place of “move,” which has been used up to this point.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.42

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Arrange students in groups of 2. Display the original gure that each student also has in front of them. Give each student one of the four transformation cards and tracing paper—make sure students know not to show their card to their partner! Tell students they will sketch a transformation based on directions given to them by a partner.Partners have Cards 1A and 1B or Cards 2A and 2B. Tell students in the rst round, those with the A cards give a precise description of the transformation displayed on their card to their partner. Their partners may use tracing paper to produce the image under transformation on the grid with the original image. When the sketch is complete, the student describing the transformation reveals their card, and together, students decide if the sketch is correct. In the second round, the roles are reversed. The students with the B cards describe their transformation while their partner sketches.The student describing the transformation is allowed to repeat, revise, or add any important information as their partner sketches, however, they are not allowed to tell them to x anything until they are nished. The student sketching should not speak, just sketch. (This is to encourage the describer to use mathematical language.) Remind students to use the geometric language for describing reections, rotations, and translations that was used in the previous lesson.ACTIVITY 1 TASK 1Your partner will describe the image of this triangle aer a certain transformation. Sketch it in your notes.STUDENT RESPONSEThe correct transformations are shown on the cards.2BAC© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES: ACTIVITY 1 RECAP Display the following questions for all to see and give groups time to discuss:• What pieces of your partner’s description were helpful when you were sketching?• What pieces did you nd diicult to explain to your partner? Point to specic examples on your cards.• When you were sketching, what questions would have been helpful to be able to ask the describer?Ask selected students who were observed using precise descriptions and sketching based on those descriptions to explain why they used the information they did and how it was helpful in sketching. Focus on:• The direction and distance of a translation.• The center and the measure of a rotation.• The line of a reection.If there is time, ask students who were both using and not using tracing paper to explain their process.Reinforce the term transformation as a term that encompasses translations, rotations, and reections. Tell them that there are other types of transformations, but for now, we will focus on these three types.ANTICIPATED MISCONCEPTIONSStudents may get stuck thinking they need to use the precise terms for the transformation in their description. Encourage these students to describe it in a way that makes sense to them and to look for things they know about the specic points, lines, or angles on their card to help 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.44

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSListening, Speaking: Math Language Routine 2 Collect and DisplayThis is the rst time Math Language Routine 2 is suggested as a support in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing students use. The language collected is displayed visually for the whole class to use throughout the lesson and mission. Generally, the display contains dierent 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): Optimize output (for explanation); Maximize meta-awarenessHow It Happens:1. As students describe the transformation of triangle ABC to their partner, listen for and collect vocabulary and phrases students use to describe the moves. Focus on capturing students using geometric language for describing reections, rotations, and translations. If the speaker is stuck, consider asking these questions: “How did point A transform to A′?”, “Choose one of the points, lines, or angles and describe how it changed.”, and “Overall, does it look like the new triangle is a translation, rotation, or reection of the original?”.2. Write students’ words on a visual display. Divide the display into 3 sections. Group language about Cards 1A and 2A on the le side of the display, language about Card 1B in the middle, and language about Card 2B on the right side. Record all language (whether precise, ambiguous, correct, or incorrect) in the appropriate column as described by the students.3. Arrange students in groups of 2, and invite partners to discuss which words or phrases stand out to them. Prompt students by asking, “Are there any words or phrases that stand out to you or don’t belong in a specic column? Why?” Again, circulate around the room, collecting any additional words, phrases, and sketches onto the display. Students should notice that the le side consists of language describing translations, the middle consists of language describing reections, and the right side consists of language describing rotations.4. Select 3–4 groups to share their ideas with the small group. Invite students to demonstrate their reasoning with tracing paper and be sure to modify the display accordingly. Use this discussion to clarify, revise, and improve how ideas are communicated and represented. If students are still using vague words (e.g., move, ip, mirror image, etc.), reinforce the precise geometric terms (e.g., transformation, translation, rotation, reection, etc.). Ask students, “Is there another way we can say this?” or “Can someone help clarify this language?”5. Close this conversation by posting the display in the front of the classroom for students to reference for the remainder of the lesson, and be sure to update the display throughout the remainder of the lesson.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer TutorsPair students with their previously identied peer tutors.Executive Functioning: Visual AidsCreate an anchor chart (i.e., word wall) publicly displaying important denitions, rules, formulas or concepts for future reference.Concept Exploration: Activity 2A TO B TO CInstructional Routines: MLR7: Compare and Connect; Anticipate, Monitor, Select, Sequence, ConnectStudents have seen images showing a sequence of transformations in the rst lesson of this mission, however they have not heard the term sequence of transformations. They have also not been asked to describe the moves in the sequence using precise language. The launch of this activity introduces this term and gives students an opportunity to describe the sequence of more than one transformation.For the second problem, encourage students to nd a dierent sequence of transformations than the one shown in the image. Each time a reection is mentioned, ask students where the line of reection is located and when a rotation is mentioned, ask for the center of the rotation and the number of degrees. Monitor for students who apply dierent transformations (or apply transformations in a dierent order).LAUNCHArrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reection that would take one bird to another. Aer a minute, verify that this is not possible.Ask students to describe how we could use translations, rotations, and reections to take one bird to another. Collect a few dierent responses. (One way would be to take the bird on the le, translate it up, and then reect it over a vertical line.) Tell students when we do one or more transformations in a row to take one gure to another, it is called a sequence of transformations.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.46

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Give students quiet work time to engage in the task followed by time to discuss their responses with a partner and complete any unnished questions. Follow with a discussion.ACTIVITY 2 TASK 1 Here are some gures on an isometric grid. 1. Name a transformation that takes Figure A to Figure B. Name a transformation that takes Figure B to Figure C.2. What is one sequence of transformations that takes Figure A to Figure C? Explain how you know.STUDENT RESPONSE1. There are a lot of ways to describe the translation that takes A to B: Any pair of corresponding points works. In the gure, two corresponding points P and Q are shown. 33ABCABCRPQ© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:There are two ways to take B to C with a single transformation. One is a reection with line of reection ℓ (shown). The other is a rotation 60 clockwise around point R on line ℓ.2. Answers vary. Sample response: Using the transformations from problem 1, rst apply a translation so that A goes to B and then a reection taking B to C.DISCUSSION GUIDANCESelect students with dierent correct responses to show their solutions. Be sure to highlight at least one rotation. If no students mention that, demonstrate a way to take A to C that involves a rotation.• Emphasize that there are many ways to describe the translation that takes gure A to gure B. All one needs is to identify a pair of corresponding points and name them in the correct order (and to use the word “translate”).• For students who used a reection to take B to C, emphasize that reections are determined by lines and we should name the line when we want to communicate about it.• Aer a student or the teacher uses a rotation, emphasize that a rotation is dened by a center point and an angle (with a direction). The center point needs to be named and the angle measure or an angle with the correct measure needs to be named as well (as does the direction). Reinforce to students that when we do more than one transformation in a row, we call this a sequence of transformations.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: Math Language Routine 7 Compare and ConnectThis is the rst 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.), comparing approaches, and identifying correspondences between dierent representations. A typical discussion prompt is: “What is the same and what is dierent?” 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: Support sense-making© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.48

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:How It Happens:1. Use this routine to compare and contrast dierent strategies for transforming Figure A to Figure C. Invite students to create a visual display showing how they made sense of the problem and why their solution makes sense for transforming Figure A to Figure C. Students should include these features on their display:• a sketch of the gures• a sketch of the gures aer each transformation• a written sequence of transformations with an explanation2. Before selecting students to show their solutions to the whole group, rst give students an opportunity to do this in a group of 3–4. Ask students to exchange and investigate each other’s work. Allow time for each display and signal when it is time to switch. While investigating each other’s work, ask students to consider what is the same and what is dierent about each approach. Next, give each student the opportunity to add detail to their own display.3. As groups are presenting, circulate the room and select 2–3 students to share their sequence of transformations taking Figure A to Figure C. Be sure to select a variety of approaches, including one that involves a rotation. Draw students’ attention to the dierent ways the gures were transformed (e.g., rotations, reections, and translations) and how the sequence of transformation is expressed in their explanation. Also, use the bullet points in the Discussion Guidance to emphasize specic features of translations, reections, and rotations.4. Aer the selected students have nished sharing with the small group, lead a discussion comparing, contrasting, and connecting the dierent approaches. Consider using these prompts to amplify student language while comparing and contrasting the dierent approaches: “Why did the approaches lead to the same outcome?”, “What worked well in ’s approach? What did not work well?”, and “What would make ’s strategy more complete or easy to understand?”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.Digital LessonDescribe a sequence of translations, rotations, and reections that takes Polygon A to Polygon J.AEDBCMNJKL© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAnswers vary. Sample response: A rotation 90 degrees clockwise around point C, followed by a translation 5 units down and 3 units right takes Polygon A to Polygon J.Wrap-UpLESSON SYNTHESIS The goal for this lesson is for students to begin to identify the features that determine a translation, rotation, or reection. Refer to the permanent display produced in a previous lesson as you discuss. To highlight the features specic to each type of transformation, consider asking the following questions:• “If you want to describe a translation, what important information do you need to include?” (A translation is determined by two points that specify the distance and direction of the translation.)• “If you want to describe a rotation, what important information do you need to include?” (A rotation is determined by a center point and an angle with a direction.)• “If you want to describe a reection, what important information do you need to include?” (A reection is determined by a line.)• “What does the word transformation mean?” (Translations, rotations, and reections, or any combination of these.)• “What does sequence of transformations mean?” (More than one applied one aer the other.)© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.50

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TERMINOLOGYSequence of transformations A sequence of transformations is a set of translations, rotations, reections, and dilations on a gure. The transformations are performed in a given order.This diagram shows a sequence of transformations to move Figure A to Figure C. First, A is translated to the right to make B. Next, B is reected across line ℓ to make C.Transformation A transformation is a translation, rotation, reection, or dilation, or a combination of these.EXIT TICKET1. If you were to describe a translation of triangle ABC, what information would you need to include in your description?2. If you were to describe a rotation of triangle ABC, what information would you need to include in your description?3. If you were to describe a reection of triangle ABC, what information would you need to include in your description?ABCRPQABC© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 4ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. The distance and direction of the translation. One way to do this would be by picking a point on the triangle (A, for example) and then showing where this point goes (A′ on the translated triangle).2. A center point, an angle, and a direction (clockwise or counterclockwise).3. A line.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.52

TEMPLATE FOR LESSON 4 CONCEPT EXPLORATION: ACTIVITY 1PAGE 1 OF 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. 53

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 5Coordinate MovesLEARNING GOALSDraw and label a diagram of a line segment rotated 90 degrees clockwise or counterclockwise about a given center.Generalize (orally and in writing) the process to reflect any point in the coordinate plane.Identify (orally and in writing) coordinates that represent a transformation of one figure to another.LEARNING GOALS(STUDENT FACING)Let’s transform some figures and see what happens to the coordinates of points.LEARNING TARGETS(STUDENT FACING)I can apply transformations to points on a grid if I know their coordinates.REQUIRED MATERIALSGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, 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.54

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Students continue to investigate the eects of transformations. The new feature of this lesson is the coordinate plane. In this lesson, students use coordinates to describe gures and their images under transformations in the coordinate plane. Reections over the x-axis and y-axis have a very nice structure captured by coordinates. When we reect a point like (2, 5) over the x-axis, the distance from the x-axis stays the same but instead of lying 5 units above the x-axis the image lies 5 units below the x-axis. That means the image of (2, 5) when reected over the x-axis is (2, -5). Similarly, when reected over the y-axis, (2, 5) goes to (-2, 5), the point 2 units to the le of the y-axis. When using the coordinates to help understand transformations, students discover the patterns that coordinates obey when transformations are applied.Warm-UpTRANSLATING COORDINATESThe purpose of this warm-up is to remind students how the coordinate plane works and to give them an opportunity to see how one might describe a translation when the gure is plotted on the coordinate plane.There are many ways to express a translation because a translation is determined by two points P and Q once we know that P is translated to Q. There are many pairs of points that express the same translation. This is dierent from reections which are determined by a unique line and rotations which have a unique center and a specic angle of rotation.LAUNCHAsk students how they describe a translation. Is there more than one way to describe the same translation? Aer they have had time to think about this, give them a few minutes of quiet work time followed by a whole-class 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. 55

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASK Select all of the translations in your notes that take Triangle T to Triangle U. There may be more than one correct answer.1. Translate (−3, 0) to (1, 2).2. Translate (2, 1) to (−2, −1).3. Translate (−4, −3) to (0, −1).4. Translate (1, 2) to (2, 1).STUDENT RESPONSEThese are both correct: (−3, 0) to (1, 2) and (−4, −3) to (0, −1)DISCUSSION GUIDANCERemind students that once you name a starting point and an ending point, that completely determines a translation because it species a distance and direction for all points in the plane. Appealing to their experiences with tracing paper may help. In this case, we might describe that distance and direction by saying “all points go up 2 units and to the right 4 units.” Draw the arrow for the two correct descriptions and a third one not in the list, like this:1-5 -4 -3 -2 -1 1 2 3 4 5yTUx54321-1-2-3-4-500(1, 2)(2, 1)(0, -1)(-4, -3)(-2, -1)(-3, 0)-5 -4 -3 -2 -1 1 2 3 4 5yTU(1, 2)(2, 1)(0, -1)(-4, -3)(-3, 0)x54321-1-2-3-4-500(-2, -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.56

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Point out that each arrow does, in fact, go up 2 and 4 to the right.ANTICIPATED MISCONCEPTIONSStudents may think that they need more information to determine the translation. Remind them that specifying one point tells you the distance and direction all of the other points move in a translation.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the graph image to students who benet from extra processing time to review prior to implementation of this activity.Concept Exploration: Activity 1REFLECTING POINTS ON THE COORDINATE PLANEInstructional Routines: MLR7: Compare and ConnectWhile the warm-up focuses on studying translations using a coordinate grid, the goal of this activity is for students to work through multiple examples of specic points reected over the x-axis and then generalize to describe where a reection takes any point. They also consider reections over the y-axis with slightly less scaolding. In the next activity, students will study 90 degree rotations on a coordinate grid, rounding out this preliminary investigation of how transformations work on the coordinate grid.Watch for students who identify early the pattern for how reections over the x-axis or y-axis inuence the coordinates of a point. Make sure that they focus on explaining why the pattern holds as the goal here is to understand reections better using the coordinate grid. The rule is less important than understanding how it is essential to see the coordinate grid and state the rule. LAUNCHTell students that they will have a few minutes of quiet think time to work on the activity, and tell them to pause aer question 2.Select 2–3 students to share their strategies for the rst 2 questions. You may wish to start with students who are measuring distances of points from the x-axis or counting the number of © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:squares a point is from the x-axis and then counting out the same amount to nd the reected point. These strategies work, but overlook the structure of the coordinate plane. To help point out the role of the coordinate plane, select a student who noticed the pattern of changing the sign of the y-coordinate when reecting over the x-axis.Aer this initial discussion, give a few minutes of quiet work time for the remaining questions, which ask them to generalize how to reect a point over the y-axis.ACTIVITY 1 TASK 1Below is a list of points and a coordinate plane. Follow the directions in your notes.Here is a list of points:2-10 -8 -6 -4 -2 2 4 6 8 10yx108642-2-4-6-8-1000A = (0.5, 4) B = (-4, 5) C = (7, -2) D = (6, 0) E = (0, -3)1. On the coordinate plane:a) Plot each point and label each with its coordinates.b) Using the x-axis as the line of reection, plot the image of each point.c) Label the image of each point with its coordinates.d) Include a label using a letter. For example, the image of point A should be labeled A'.2. If the point (13, 10) were reected using the x-axis as the line of reection, what would be the coordinates of the image? What about (13, -20)? (13, 570)? Explain how you know.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.58

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. The picture shows the points A, B, C, D, E and also their reections over the x-axis: A' = (0.5, -4), B' = (-4, -5), C' = (7, 2), D' = (6, 0), E' = (0, 3).2. Using the x-axis as line of reection, the reection of (13, 10) is (13, -10), the reection of (13, -20) is (13, 20) and the reection of (13, 570) is (13, -570). Using the x-axis as line of reection does not move points horizontally but it does move points which are not on the x-axis vertically. In coordinates, the x-coordinate of the point stays the same while the y-coordinate changes sign.ACTIVITY 1 TASK 2The point R has coordinates (3, 2). Answer the questions in your notes.a) Without graphing, predict the coordinates of the image of point R if point R were reected using the y-axis as the line of reection.-4 -2 2 4 6 8 10yx6B = (-4, 5)A = (0.5, 4)A' = (0.5, -4)B' = (-4, -5)E' = (0, 3)E = (0, -3)D and D' = (6, 0)C' = (7, 2)C = (7, -2)42-2-4-6003354321-1-2-3-4-50yx-5 -4 -3 -2 -101 2 3 4 5R© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:b) Check your answer by nding the image of R on the graph.c) Label the image of point R as R'.d) What are the coordinates of R'?e) Suppose you reect a point using the y-axis as the line of reection. How would you describe its image?STUDENT RESPONSEUsing the y-axis as line of reection does not move points vertically but it does move points that are not on the y-axis horizontally. In coordinates, the y-coordinate of the point stays the same while the x-coordinate changes sign. The point R has coordinates (3, 2). When I reect it over the y-axis it will go to (-3, 2): the x-coordinate changes sign but the y-coordinate remains the same. The point will have the same y-coordinate but the x-coordinate will change signs. The distance from the y-axis does not change and the y-coordinate does not change.ACTIVITY 1 RECAPTo facilitate discussion, display a blank coordinate grid.Questions for discussion:• "When you have a point and an axis of reection, how do you nd the reection of the point?"• "How can you use the coordinates of a point to help nd the reection?"• "Are some points easier to reect than others? Why?"• "What patterns have you seen in these reections of points on the coordinate grid?"-10 -8 -6 -4 -2 2 4 6 8 10yx108642-2-4-6-8-1000© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.60

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:The goal of the activity is not to create a rule that students memorize. The goal is for students to notice the pattern of reecting over an axis changing the sign of the coordinate (without having to graph). The coordinate grid can sometimes be a powerful tool for understanding and expressing structure and this is true for reections over both the x-axis and y-axis.ANTICIPATED MISCONCEPTIONSIf any students struggle getting started because they are confused about where to plot the points, refer them back to the warm-up activity and practice plotting a few example points with them.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 7 Compare and ConnectUse this routine when students present their strategies for reecting points using the x-axis as the line of reection before continuing on. Ask students to consider what is the same and what is dierent about the strategies. Draw students’ attention to the dierent ways students reasoned to nd the reected coordinates. These exchanges strengthen students’ mathematical language use and reasoning of reections along the x-axis and y-axis.Design Principle(s): Maximize meta-awarenessSUPPORT FOR STUDENTS WITH DISABILITIESFine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allowing students who struggle with ne motor skills to dictate transformations as needed.Executive Functioning: Eliminate BarriersChunk this activity into more manageable parts (e.g., presenting one question at a time), which will aid students who benet from support with organizational skills in problem solving.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Concept Exploration: Activity 2TRANSFORMATIONS OF A SEGMENTInstructional Routine: MLR8: Discussion SupportsThis activity concludes looking at how the dierent basic transformations (translations, rotations, and reections) behave when applied to points on a coordinate grid. In general, it is diicult to use coordinates to describe rotations. But when the center of the rotation is (0, 0) and the rotation is 90 degrees (clockwise or counterclockwise), there is a straightforward description of rotations using coordinates. Unlike translations and reections over the x or y axis, it is more diicult to visualize where a 90 degree rotation takes a point. Tracing paper is a helpful tool, as is an index card.LAUNCHDemonstrate how to use tracing paper in order to perform a 90 degree rotation. It is helpful to put a small set of perpendicular axes (a + sign) on the piece of tracing paper and place their intersection point at the center of rotation. One of the small axes can be lined up with the segment being rotated and then the rotation is complete when the other small axis lines up with the segment. An alternative method to perform rotations would be with the corner of an index card, which is part of the geometry toolkit.ACTIVITY 2 TASK 1Apply each of the transformations listed in your notes to segment AB, shown below.34-6 -4 -2 2 4 6 8 10yx8642-2-400A = (0, 3)B = (4, 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.62

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. Rotate segment AB 90 degrees counterclockwise around center B. Label the image of A as C. What are the coordinates of C?2. Rotate segment AB 90 degrees counterclockwise around center A. Label the image of B as D. What are the coordinates of D?3. Rotate segment AB 90 degrees clockwise around (0, 0). Label the image of A as E and the image of B as F. What are the coordinates of E and F?4. Compare the two 90-degree counterclockwise rotations of segment AB. What is the same about the images of these rotations? What is dierent?STUDENT RESPONSE1. C = (3, −2)2. D = (1, 7)3. E = (3, 0), F = (2, −4)4. Answers vary. Sample response. The two counterclockwise rotations of AB are in dierent locations. The points A and B move dierent distances with the dierent rotations. One rotation can be mapped to the other by a translation.DISCUSSION GUIDANCEAsk students to describe or demonstrate how they found the rotations of segment AB. Make sure to highlight these strategies:• Using tracing paper to enact a rotation through a 90 degree angle.• Using an index card: Place the corner of the card at the center of rotation, align one side with the point to be rotated, and nd the location of the rotated point along an adjacent side of the card. (Each point's distance from the corner needs to be equal.)-6 -4 -2 2 4 6 8 10yx8642-2-400ADBECF© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• Using the structure of the coordinate grid: All grid lines are perpendicular, so a 90 degree rotation with center at the intersection of two grid lines will take horizontal grid lines to vertical grid lines and vertical grid lines to horizontal grid lines.The third strategy should only be highlighted if students notice or use this in order to execute the rotation, with or without tracing paper. This last method is the most accurate because it does not require any technology in order to execute, relying instead on the structure of the coordinate grid.If some students notice that the three rotations of segment AB are all parallel, this should also be highlighted.SUPPORT FOR STUDENTS WITH DISABILITIESFine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allowing students who struggle with ne motor skills to dictate rotations as needed.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion SupportsTo support students in explaining the similarities and dierences of the segment rotations for the last question, provide sentence frames for students to use when they are comparing segments, points, and rotations. For example, “ is similar to because ” or “ is dierent than because ” Revoice student ideas using mathematical language use as needed.Design Principle(s): Support sense-making; Optimize output for (comparison)Digital LessonAnswer the questions about the points on both coordinate planes below.1. What are the coordinates of A, B, and C aer a translation to the right 5 units and down 3 units? Plot these on the grid, and label them A', B', and 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.64

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:2. What are the coordinates of D, E, and F aer a reection over the y-axis? Plot these points on the grid, and label them D', E', and F'. STUDENT RESPONSE1. The coordinates are: A' = (7, −1), B' = (2, −3), C' = (−4, −1).2. The coordinates are D' = (4, −4), E' = (8, 8), and F' = (−6, 2). Wrap-UpLESSON SYNTHESISBy this point, students should start to feel condent applying translations, reections over either axis, and rotations of 90 degrees clockwise or counterclockwise to a point or shape in the coordinate plane.-10 -8 -6 -4 -2 2 4 6 8 10yx108642-2-4-6-8-1000ABC-10 -8 -6 -4 -2 2 4 6 8 10yx108642-2-4-6-8-1000FDE-10 -8 -6 -4 -2 2 4 6 8 10yx108642-2-4-6-8-1000AA'BCB'C'-10 -8 -6 -4 -2 2 4 6 8 10yx108642-2-4-6-8-1000FF'DED'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. 65

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:When students are working on the coordinate plane when doing transformations, ask:• "What are some advantages to knowing the coordinates of points when you are doing transformations?"• "What changes did we see when reecting points over the x-axis? y-axis?"• "How do you perform a 90 degree clockwise rotation of a point with center (0, 0)?"Time permitting, ask students to apply a few transformations to a point. For example, where does (1, 2) go when• reected over the x-axis? (1, -2)• reected over the y-axis? (-1, 2)• rotated 90 degrees clockwise with center (0, 0)? (2, -1)EXIT TICKETOne of the triangles pictured is a rotation of triangle ABC and one of them is a reection.1. Identify the center of rotation, and label the rotated image PQR.2. Identify the line of reection, and label the reected image XYZ.-8 -6 -4 -2 2 4 6 8yx8642-4-2-6-800BAC© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.66

G8M1 | LESSON 5ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. The center of the rotation taking △ABC to △PQR is (0, 0), and the rotation is 90 degrees in a counterclockwise direction.2. A reection over the x-axis takes △ABC to △XYZ.-8 -6 -4 -2 2 4 6 8yx8642-4-2-6-800BAPQRXYZC© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 6Describing TransformationsLEARNING GOALSCreate a drawing on a coordinate grid of a transformed object using verbal descriptions.Identify what information is needed to transform a polygon. Ask questions to elicit that information.LEARNING GOALS(STUDENT FACING)Let’s transform some polygons in the coordinate plane.LEARNING TARGETS(STUDENT FACING)I can apply transformations to a polygon on a grid if I know the coordinates of its vertices.REQUIRED MATERIALSTemplate for Concept Exploration: Activity 1Geometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesPre-printed slips, cut from copies of the templateREQUIRED PREPARATIONPrint 1 copy of the template for every 2 students. Cut them up ahead of time.From the geometry toolkits, graph paper and tracing paper are especially helpful.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.68

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Prior to this lesson, students have studied and classied dierent types of transformations (translations, rotations, reections). They have practiced applying individual transformations and sequences of transformations to gures both on and o of a coordinate grid. In this lesson, they focus on communicating precisely the information needed to apply a sequence of transformations to a polygon on the coordinate grid. They must think carefully about what information they need and request this information from their partner in a clear, precise way. They also explain why they need each piece of information. The coordinate grid plays a key role in this work, allowing students to communicate precisely about the locations of polygons and how they are transformed.Warm-UpFINDING A CENTER OF ROTATIONInstructional Routine: Think Pair ShareSometimes it is easy to forget to communicate all of the vital information about a transformation. In this case, the center of a rotation is le unspecied. Students do not need to develop a general method for nding the center of rotation, given a polygon and its rotated image. They identify the center in one situation and this can be done via geometric intuition and a little trial and error. LAUNCHArrange students in groups of 2 and provide access to geometry toolkits. Tell students that they have a diagram of a gure and its rotated image and that they need to identify the center of rotation. Give them some quiet work time and an opportunity to share with a partner, followed by a whole-class 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. 69

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASKAndre performs a 90-degree counterclockwise rotation of polygon P and gets polygon P’, but he does not say what the center of the rotation is. Can you nd the center?STUDENT RESPONSEThe rotation takes the horizontal side of P to the vertical side of P’, and the center of rotation is the intersection of the grid lines containing these two sides.DISCUSSION GUIDANCEEmphasize that it is important to communicate clearly. When we perform a transformation, we should provide the information necessary for others to understand what we have done. For a rotation, this means communicating:• The center of the rotation.• The direction of the rotation (clockwise or counterclockwise).• The angle of rotation.The grid provides extra structure that helps to identify these three parts of the rotation. Invite students to share how they identied the center of rotation. Methods may include:• Experimenting with tracing paper.1PP'© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.70

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• Understanding that the rotation does not change the distance between the center of rotation and each vertex, so the center should be the same distance from each vertex and its image.ANTICIPATED MISCONCEPTIONSStudents may have trouble getting started. Suggest that they trace P on to tracing paper and try rotating it 90º. How must they rotate it to get it to land on P’? Where is the center of the “spin”?SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing Time.Provide the image to students who benet from extra processing time to review prior to implementation of this activity.Concept Exploration: Activity 1INFO GAP: TRANSFORMATION INFORMATIONMATERIALS: TEMPLATE FOR CONCEPT EXPLORATION: ACTIVITY 1Instructional Routine: MLR4: Information GapThis info gap activity gives students an opportunity to determine and request the information needed to perform a transformation in the coordinate plane. A sample pair of cards looks as 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. 71

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Students likely need several rounds to determine the information they need.• They need to know which transformations were applied (i.e., translation, rotation, or reection)• They need to determine the order in which the transformations were applied.• They need to remember what information is needed to describe a translation, rotation, or reection.Monitor for students who successfully determine or remember each of these three important pieces of information as well as students who have partially but not completely solved the problem. Students may not realize that the order in which the transformations are applied is important, and this should be addressed in the Synthesis.The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their rst requests do not yield the information they need. It also allows them to rene the language they use and ask increasingly more precise questions until they get the information they need.LAUNCHTell students they will continue to describe transformations using coordinates. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. Provide access to graph paper. In each group, distribute a problem card to one student and a data card to the other student. Aer you review their work on the rst problem, give them the cards for a second problem and instruct them to switch roles.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.72

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 1 You will receive either a problem card or a data card. Do not show or read your card to your partner. Follow the directions in your notes.2If your teacher gives you the data card:1. Silently read the information on your card.2. Ask your partner “What specic information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not gure out anything for your partner!)3. Before telling your partner the information, ask “Why do you need that information?”4. Aer your partner solves the problem, ask them to explain their reasoning and listen to their explanation.If your teacher gives you the problem card:1. Silently read your card and think about what information you need to answer the question.2. Ask your partner for the specic information that you need.3. Explain to your partner how you are using the information to solve the problem.4. Solve the problem and explain your reasoning to your partner.Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your 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. 73

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEDISCUSSION GUIDANCEAer students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Some guiding questions:• “How did using coordinates help in talking about the problem?”• “Was the order in which the transformations were applied important? Why?”• “If this same problem were a picture on a grid without coordinates, how would you talk about the points?”Highlight for students that one advantage of the coordinate plane is that it allows us to communicate information about transformations precisely. Here is what is needed for each type of transformation (consider showing one example of each while going through the dierent transformations):• For a translation, the distance of vertical and horizontal components• For a rotation, the center of rotation, the direction of rotation, and the angle of rotation• For a reection, the line of reectionANTICIPATED MISCONCEPTIONSStudents may struggle to ask their partner for all of the information they need or may ask a question that is not suiciently precise, such as, “What are the transformations?” Ask these students what kinds of transformations they have worked with. What information is needed to 5y4321-1-2-3-4-5x-5 -4 -3 -2 -1 1 2 3 4 5DCBAB'C'D'A'5y4321-1-2-3-4-5x-5 -4 -3 -2 -1 1 2 3 4 5LKNML'K'N'M'© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.74

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:perform a translation? What about a rotation or reection? Encourage them to nd out which transformations they need to perform (Is there a translation? Is there a rotation?) and then nd out the information they need for each transformation.SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversingThis activity uses MLR 4 Information Gap to give students a purpose for discussing information necessary to solve problems involving describing transformations.Design Principle(s): Cultivate ConversationSUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Eliminate Barriers.Demonstrate the steps for the activity or game, in which a group of students and sta play an example round while the rest of the class observes.Digital LessonHere is triangle ABC in the coordinate plane. Draw triangle AʹBʹCʹ, the image of triangle ABC aer each sequence of transformations below.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. On the le plane, draw the image of triangle ABC aer a reection over the x-axis. Label the image triangle A'B'C'. Then, translate triangle A'B'C' 3 units down, and label the image triangle A''B''C''.2. On the right plane, draw the image of triangle ABC aer a translation of 3 units down. Label the image triangle A'B'C'. Then, reect triangle A'B'C' using the x-axis as the line of reection, and label the image triangle A''B''C''.3. Did the order matter for this sequence of transformations? Explain how you know.STUDENT RESPONSES1-2.3. Yes, the order mattered for this sequence of transformations. When I applied the translation rst, the image ended up in a dierent location than when I applied the reection rst.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.76

G8M1 | LESSON 6ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Wrap-UpLESSON SYNTHESISAsk students to choose which of the three transformations they have studied so far (translation, reection, rotation) is their favorite and give students a few minutes to write a few sentences explaining why. Have students rst share their explanations with a partner and then invite students to share their favorite with the class.EXIT TICKET Students describe what information is required to perform a translation and what information is required to perform a reection. They also need to think about the order in which the two transformations are applied as they have just seen that switching the order can impact the outcome.TASKJada applies two transformations to a polygon in the coordinate plane. One of the transformations is a translation and the other is a reection. What information does Jada need to provide to communicate the transformations she has used?STUDENT RESPONSEFor the translation, Jada needs to provide the distance and direction of the vertical displacement and the distance and direction of the horizontal displacement. For the reection, Jada needs to give the line of reection. It is also important for Jada to communicate the order in which the transformations are applied.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

TEMPLATE FOR LESSON 6 CONCEPT EXPLORATION: ACTIVITY 1PAGE 1 OF 2Info Gap: Transformation Information Problem Card 1Polygon A’B’C’D’ is the image of ABCD after some transformations. Find A’B’C’D’ . Info Gap: Transformation InformationData Card 1 Translation: 2 units up and 3 units to the rightRotation: none Reflection: over x -axis Order of Transformations: Translation first and then reflection© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.78

TEMPLATE FOR LESSON 6 CONCEPT EXPLORATION: ACTIVITY 1PAGE 2 OF 2Info Gap: Transformation InformationProblem Card 2 Polygon K’L’M’N’ is the image of KLMN after some transformations. Find K’L’M’N’. Info Gap: Transformation InformationData Card 2 Translation: 1 unit left and 3 units downRotation: 90 degreesDirection of rotation: clockwiseCenter of rotation: (0,0)Reflection: noneOrder of transformations: Rotation first and then translation© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 7No Bending or StretchingLEARNING GOALSComprehend that the phrase “rigid transformation” refers to a transformation where all pairs of “corresponding side lengths” and “corresponding angle measures” in the figure and its image are the same.Draw and label a diagram of the image of a polygon under a rigid transformation and calculate side lengths and angle measures.Identify (orally and in writing) a sequence of rigid transformations using a drawing of a figure and its image.LEARNING GOALS(STUDENT FACING)Let’s compare measurements before and after translations, rotations, and reflections.LEARNING TARGETS(STUDENT FACING)I can describe the effects of a rigid transformation on the lengths and angles in a polygon.REQUIRED MATERIALSGeometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor 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.80

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, students begin to see that translations, rotations, and reections preserve side lengths and angle measures, and for the rst time call them rigid transformations. In earlier lessons, students talked about corresponding points under a transformation. Now they will talk about corresponding sides and corresponding angles of a polygon and its image. As students experiment with measuring corresponding sides and angles in a polygon and its image, they will need to use the structure of the grid as well as appropriate technology, including protractors, rulers, and tracing paper.Warm-UpMEASURING SEGMENTSIn this warm-up, students measure four line segments. They discuss the dierent aspects of making and recording accurate measurements. It is important to highlight the fractional markings and fraction and decimal equivalents used as students explain how they determined the length of the segment.LAUNCHGive students quiet work time followed by whole-class discussion.WARM-UP TASKComplete the problems below. For each question, the unit is represented by the large tick marks with whole numbers.1. Find the length of this segment to the nearest 81 of a unit.2. Find the length of this segment to the nearest 0.1 of a unit.11 2 3 4 5 61 2 3 4 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. 81

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:3. Estimate the length of this segment to the nearest 81 of a unit.4. Estimate the length of the segment in the prior question to the nearest 0.1 of a unit.STUDENT RESPONSE1. 485 units2. 4.7 units3. 343 (or 386 units)4. 3.7 units (or 3.8 units)DISCUSSION GUIDANCEInvite students to share their responses and record them for all to see. Ask the class if they agree or disagree with each response. When there is a disagreement, have students discuss possible reasons for the dierent measurements.Students are likely to have dierent answers for their measure of the third segment. The ruler shown is not as accurate as the question requires as it has not been pre-partitioned into fractional units. Ask 2–3 students with dierent answers to share their strategies for measuring the third segment. There will be opportunities for students to use measuring strategies later in this lesson.ANTICIPATED MISCONCEPTIONSStudents may struggle with the ruler that is not pre-partitioned into fractional units. Encourage these students to use what they know about eighths and tenths to partition the ruler and estimate their answer.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the images to students who benet from extra processing time to review prior to implementation of this activity.1 2 3 4 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.82

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Expressive Language: Eliminate BarriersProvide sentence frames for students to explain their reasoning (i.e., I agree or disagree because .)Concept Exploration: Activity 1SIDES AND ANGLESInstructional Routine: MLR8: Discussion SupportsThe purpose of this activity is for students to see that translations, rotations, and reections preserve side lengths and angle measures. Students can use tracing paper to help them draw the gures and make observations about the preservation of side lengths and angle measures under transformations. While the grid helps measure lengths of horizontal and vertical segments, the students may need more guidance when asked to measure diagonal segments. It is important in the launch to demonstrate for students how to either use the tracing paper or an index card to mark o unit lengths using the grid.Since students are creating their own measuring tool, they can only give an estimate, and some exibility should be allowed in the response. During the discussion, highlight dierent reasonable answers that students nd for the lengths which are not whole numbers.As students work individually, monitor and ask them to explain how they are performing their transformations and nding the side lengths and angle measures. During the discussion, select students who mention corresponding sides and angles, which they learned in grade 7 when making scaled copies, to share. Also select students who estimated the side lengths for Figure C correctly using either the tracing paper or index card.LAUNCHTell students, “In this activity you will be performing transformations. You can use tracing paper to help you draw the images of the gures or to check your work.”Point students to Figure C and tell them, “When you are asked to measure side lengths here, you will need to make a ruler on either tracing paper or on a blank edge of an index card.” This reinforces the strategies and estimates students made in the warm-up.Give students quiet think time. Be sure to save enough time for the 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. 83

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 1Solve the problems below. You can use tracing paper to help you draw the images of the gures or to check your work.21. Translate Polygon A so point P goes to point Q. In the image, write the length of each side, in grid units, next to the side.2. Rotate Triangle B 90 degrees clockwise using R as the center of rotation. In the image, write the measure of each angle in the interior.APQ30°90°BR60°3. Reect Pentagon C across line ℓ.a) In the image, write the length of each side, in grid units, next to the side. You may need to make your own ruler with tracing paper or a blank index card.b) In the image, write the measure of each angle in the interior.ℓC90°90°90°150°120°© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.84

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. The side lengths are measured in units where one unit is the side length of the square in the grid.2. 3. The sides are measured in grid units. The sides that are not whole numbers have been rounded to the nearest tenth.ACTIVITY 1 RECAPAsk selected students to share how they performed the given transformation for each question. Aer each explanation, ask the class if they agree or disagree. Introduce students to the idea of corresponding sides and corresponding angles. Ask students to identify the corresponding angles in the rst question and the corresponding sides in the second since they were not A222P Q11134A2221113430°30°90°90°BR60°60°ℓC90°90°90°2.42.42.43322.421.81.8150°120°C90°90°90°150°120°© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:asked about these attributes the rst time. The point here is not to nd the actual values but to note that the corresponding measurements are equal. Since it is sometimes not possible to measure angles or side lengths exactly, student estimates for these values (both corresponding sides and corresponding angles) may be slightly dierent.Point out that for each of the transformations in this activity, the lengths of the sides of the original gure equal the lengths of the corresponding sides in the image, and the measures of the angles in the original gure equal the measures of the corresponding angles in the image. For this reason, we call these transformations rigid transformations: they behave as if we are moving the shapes around without stretching, bending, or breaking them. An example of a non-rigid transformation is one that compresses a gure vertically, like this:Tell them that a rigid transformation is a transformation where all pairs of corresponding sides and angle measures in the gure and its image are equal. It turns out that translations, reections, and rotations are the building blocks for all rigid transformations, and we will explore that next.ANTICIPATED MISCONCEPTIONSStudents may try to count the grid squares on the diagonal side lengths. Remind students to measure these lengths with their tracing paper or index card. Students may also struggle estimating the diagonal side lengths on their self-marked index card or tracing paper. Remind students of how they estimated the lengths for the questions in the warm-up where the ruler was not marked.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeCheck in with individual students, as needed, to assess for comprehension during each step of the activity.AA'B C 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.86

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion SupportsAs students describe their approaches, press for details in students’ explanations by requesting that students challenge an idea, elaborate on an idea, or give an example of their process. Connect the terms corresponding sides and corresponding angles to students’ explanations multi-modally by using dierent types of sensory inputs, such as demonstrating the transformation or inviting students to do so, using the images and using gestures. This will help students to produce and make sense of the language needed to communicate their own ideas.Design Principle(s): Optimize output (for explanation)Concept Exploration: Activity 2WHICH ONE?Instructional Routines: Anticipate, Monitor, Select, Sequence, Connect; MLR2: Collect and DisplayThe purpose of this activity is to decide if there is a sequence of translations, rotations, and reections that take one gure to another and, if so, to produce one such sequence. Deciding whether or not such a sequence is possible uses the knowledge that translations, rotations, and reections do not change side lengths or angle measures. The triangles ABC and CFG form part of a large pattern of images of triangle ABC that will be examined more closely in future lessons.Monitor for students who use dierent transformations to take triangle ABC to triangle CFG and select them to share during the discussion. (There are two possible sequences in the Student Responses section, but these are not the only two.)LAUNCHProvide access to geometry toolkits. Give students quiet work time and time to discuss with a partner, and then time for a whole-class 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. 87

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 2 TASK 1 Here is a grid showing triangle ABC and two other triangles. You can use a rigid transformation to take triangle ABC to one of the other triangles.1. Which one? Explain how you know.2. Describe a rigid transformation that takes ABC to the triangle you selected.STUDENT RESPONSE1. It’s triangle CFG. Triangle DBE is smaller than △ABC, so no sequence of rigid transformations can take triangle ABC to triangle DBE.2. Answers vary. Here are two possible sequences:• Translate triangle ABC 7 units right so that B matches up with F. Then rotate 90 degrees clockwise around F.• Rotate triangle ABC 90 degrees counterclockwise around point C, and then rotate 180 degrees around the midpoint of M of segment CG.DISCUSSION GUIDANCEAsk a student to explain why triangle ABC cannot be taken to triangle DBE. (We are only using rigid transformations and therefore the corresponding lengths have to be equal and they are not.) If a student brings up that they think triangle DBE is a scale drawing of ABC, bring the discussion back to translations, rotations, and reections, rather than talking about how or why triangle DBE isn’t actually a scale drawing of ABC.33C FGABDE© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.88

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Oer as many methods for transforming triangle ABC as possible as time permits, selecting previously identied students to share their methods. Include at least two dierent sequences of transformations. Make sure students attend carefully to specifying each transformation with the necessary level of precision. For example, for a rotation, that they specify the center of rotation, the direction, and the angle of rotation.If time allows, consider asking the following questions:“Can triangle ABC be taken to triangle CFG with only a translation?” (No, since CFG is rotated.)“What about with only a reection?” (No, because they have the same orientation.) “What about with a single rotation?” (The answer is yes, but this question does not need to be answered now as students will have an opportunity to investigate this further in a future lesson.)SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversing: MLR 2 Collect and DisplayAs students discuss their work with a partner, listen for and collect the language students use to describe each transformation. Record students’ words and phrases on a visual display (e.g., “Rotate triangle ABC 90 degrees counterclockwise around point C,” “Translate triangle ABC 7 units right,” etc.), and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-group discussions.Design Principle(s): Optimize output (for explanation); Maximize meta-awarenessDigital LessonExplain why the following is not an example of a rigid transformation. TQ© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAnswers vary. Sample response: A rigid transformation is a move that does not change any of the measurements of the gure. Rectangle Q has dierent side lengths than Rectangle T.Wrap-UpLESSON SYNTHESISRemind students that a rigid transformation is a transformation for which all pairs of corresponding side lengths and angle measures in the original gure and its image are equal. Translations, rotations, and reections have this property, so they are rigid transformations. Sequences of these are as well—for example, if you translate a gure then reect the image, the side lengths and angle measures stay the same.Ask students to think of ways they could look at two shapes and tell that one is not the image of the other under a rigid transformation. Give a moment of quiet think time, and then invite students to share their ideas (If two shapes have dierent side lengths or angle measures then there is no rigid transformation taking one shape to the other).When there is a rigid transformation taking one gure to another, there are many ways to do this. Ask students:• “What are some good ways to tell whether one shape can be taken to another with a sequence of rigid transformations?” (Measure all of the side lengths and angle measures and ensure that corresponding measurements are equal. Use tracing paper to see if one shape matches up exactly with the other.)• “What are the three basic types of rigid transformations?” (rotations, translations, and reections)TERMINOLOGYRigid transformation A rigid transformation is a move that does not change any measurements of a gure. Translations, rotations, and reections are rigid transformations, as is any sequence of these.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.90

G8M1 | LESSON 7ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:EXIT TICKETStudents use key dening properties of rigid motions, namely that they preserve side lengths and angle measures, in order to calculate side lengths and angle measures in a polygon and its image under a rigid transformation.LAUNCHProvide access to a geometry toolkit.TASKTrapezoid A′B′C′D′ is the image of trapezoid ABCD under a rigid transformation.Label all vertices on trapezoid A′B′C′D′.On both gures, label all known side lengths and angle measures.STUDENT RESPONSECDAB64130°50°CDAB6644130°50°A'D'C'B'130°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. 91

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 8Rotation PatternsLEARNING GOALSDraw and label rotations of 180 degrees of a line segment from centers of the midpoint, a point on the segment, and a point not on the segment.Generalize (orally and in writing) the outcome when rotating a line segment 180 degrees.Identify (orally and in writing) the rigid transformations that can build a diagram from one starting figure.LEARNING GOALS(STUDENT FACING)Let’s rotate figures in a plane.LEARNING TARGETS(STUDENT FACING)I can describe how to move one part of a figure to another using a rigid transformation.REQUIRED MATERIALSGeometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor and an index card to use as a straightedge or to mark right anglesZEARN MATH TIPThe Independent Digital Lesson is focused on 180 degree rotations, so you may want to focus your instruction on Activity 2: A Pattern of Four 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.92

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, rigid transformations are applied to line segments and triangles. For line segments, students examine the impact of a 180 degree rotation. This is important preparatory work for studying parallel lines and rigid transformations, the topic of the next lesson. For triangles, students look at a variety of transformations where rotations of 90 degrees and 180 degrees are again a focus. This work and the patterns that students build will be important later when they study the Pythagorean Theorem. Throughout the lesson, students use the properties of rigid transformations (they do not change side lengths or angle measures) in order to make conclusions about the objects they are transforming.Warm-UpBUILDING A QUADRILATERALStudents rotate a copy of a right isosceles triangle four times to build a quadrilateral. It turns out that the quadrilateral is a square. Students are not asked or expected to justify this but it can be addressed in the discussion. The fourth question about rotational symmetry of the quadrilateral will help students conclude that it is a square. There are many more opportunities to build gures using rigid transformations in other lessons.LAUNCHProvide access to geometry toolkits, particularly tracing paper.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASK Here is a right isosceles triangle: 1. Rotate triangle ABC 90 degrees clockwise around B.2. Rotate triangle ABC 180 degrees clockwise around B.3. Rotate triangle ABC 270 degrees clockwise around B.4. What would it look like if you rotated the four triangles 90 degrees clockwise around B? 180 degrees clockwise? 270 degrees clockwise?STUDENT RESPONSE1-3. 4. These rotations interchange the 4 triangles.1CBACBA© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.94

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEAsk students what they notice and wonder about the quadrilateral that they have built. Likely responses include:• It looks like a square.• Rotating it 90 degrees clockwise or counterclockwise interchanges the 4 copies of triangle ABC.• Continuing the pattern of rotations, the next one will put ABC back in its original position.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the image to students who benet from extra processing time to review prior to implementation of this activity.Concept Exploration: Activity 1ROTATING A SEGMENTInstructional Routines: Think Pair Share; MLR8: Discussion SupportsThe purpose of this activity is to allow students to explore special cases of rotating a line segment 180°. In general, rotating a segment 180° produces a parallel segment the same length as the original. This activity also treats two special cases:When the center of rotation is the midpoint, the rotated segment is the same segment as the original, except the vertices are switched.When the center of rotation is an endpoint, the segment together with its image form a segment twice as long as the original.Students will make general statements about what happens when a line segment is rotated 180°. They will experiment with a particular line segment but the conclusions that they make, especially in the last problem, are for any line segment.Watch for how students explain that the 180° rotation of segment CD in the second part of the question is parallel to CD. Some students may say that they “look parallel” while others might © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:try to reason using the structure of the grid. Tell them that they will investigate this further in the next lesson.LAUNCHArrange students in groups of 2. Provide access to geometry toolkits. Give students quiet work time, followed by sharing with a partner and a whole-class discussion. ACTIVITY 1 TASK 1 Solve the problems below using the grid. 1. Rotate segment CD 180 degrees around point D. Draw its image and label the image of C as A.2. Rotate segment CD 180 degrees around point E. Draw its image and label the image of C as B and the image of D as F.3. Rotate segment CD 180 degrees around its midpoint, G. What is the image of C?4. What happens when you rotate a segment 180 degrees around a point?2ECGD© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.96

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1-2. 3. The image of the segment lines up with itself, but the endpoints are switched. D is now where C was and C is where D was.4. The new segment may change its location, but it remains the same length. The new segment is parallel to the original segment. When the point of rotation is the midpoint of the segment, then the rotated segment is the same as the original (the endpoints trade places) and when the point of rotation is an end point of the segment, the image connects to the original to form a segment twice as long.DISCUSSION GUIDANCEAsk students why it is not necessary to specify the direction of a 180 degree rotation (because a 180 degree clockwise rotation around point P has the same eect as a 180 degree counterclockwise rotation around P). Invite groups to share their responses. Ask the class if they agree or disagree with each response. When there is a disagreement, have students discuss possible reasons for the dierences.Three important ideas that emerge in the discussion are:Rotating a segment 180° around a point that is not on the original line segment produces a parallel segment the same length as the original.When the center of rotation is the midpoint, the rotated segment is the same segment as the original, except the vertices are switched.When the center of rotation is an endpoint, the segment together with its image form a segment twice as long.FBcGDA© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:If any of the ideas above are not brought up by the students during the class discussion, be sure to make them known.ANTICIPATED MISCONCEPTIONSStudents may be confused when rotating around the midpoint because they think the image cannot be the same segment as the original. Assure students this can occur and highlight that point in the discussion.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion SupportsUse this routine to support whole-class discussion when students discuss whether it is necessary to specify the direction of a 180 degree rotation. Aer a student speaks, call on students to restate and/or revoice what was shared using mathematical language (e.g., rotation, line segment, midpoint, etc.). This will provide more students with an opportunity to produce language as they explore special cases of rotating a line segment 180°.Design Principle(s): Support sense-making; Maximize meta-awarenessSUPPORT FOR ENGLISH LANGUAGE LEARNERSConceptual Processing: Processing TimeBegin with a demonstration of the rst rotation, which will provide access for students who benet from clear and explicit instructions.Concept Exploration: Activity 2A PATTERN OF FOUR TRIANGLESInstructional Routine: MLR8: Discussion SupportsIn this activity, students use rotations to build a pattern of triangles. In the previous lesson, students examined a right triangle and a rigid transformation of the triangle. In this activity, several rigid transformations of the triangle form an interesting pattern.Triangle ABC can be mapped to each of the three other triangles in the pattern with a single rotation. As students work on the rst three questions, watch for any students who see that © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.98

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:a single rotation can take triangle ABC to CDE. The center for the rotation is not drawn in the diagram: it is the intersection of segment AE and segment CG. For students who nish early, guide them to look for a single transformation taking ABC to each of the other triangles.This pattern will play an important role later when students use this shape to understand a proof of the Pythagorean Theorem.Identify students who notice that they have already solved the rst question in an earlier activity. Watch for students who think that CAGE is a square and tell them that this will be addressed in a future lesson. However, encourage them to think about what they conclude about CAGE now. Also watch for students who repeat the same steps to show that ABC can be mapped to each of the other three triangles.LAUNCHArrange students in groups of 2–4. Provide access to geometry toolkits.ACTIVITY 2 TASK 1 You can use rigid transformations of a gure to make patterns. Here is a diagram built with three dierent transformations of triangle ABC.1. Describe a rigid transformation that takes triangle ABC to triangle CDE.2. Describe a rigid transformation that takes triangle ABC to triangle EFG.3. Describe a rigid transformation that takes triangle ABC to triangle GHA.33BCDEFHGA© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:4. Do segments AC, CE, EG, and GA all have the same lengths? Explain your reasoning.STUDENT RESPONSE1. Answers vary. Sample responses:• Translate point B to point D, then rotate 90 degrees clockwise using D as center.• Rotate counterclockwise using C as center until segment CA matches up perfectly with segment CE, then rotate 180 degrees using the midpoint of segment CE as center.2. Answers vary. Sample responses:• Translate B to F and then rotate 180 degrees with center F.• Translate so segment AC matches up with segment GE and then rotate 180 degrees with the midpoint of segment GE as center of rotation.3. Answers vary. Sample responses:• Translate B to H and then rotate 90 degrees counterclockwise with center H.• Rotate with center A so that segment AC matches up with segment AG and then rotate 180 degrees with the midpoint of segment AG as center.4. Yes, because the size and shape of triangle ABC did not change under the rigid transformation. Segment AC can be matched up exactly with segments CE, EG, and GA so the lengths of these segments are all the same.DISCUSSION GUIDANCESelect a student previously identied who noticed how the rst question relates to a previous activity to share their observation. Discuss here how previous work can be helpful in new work, since students may not be actively looking for these connections. The next questions are like the rst, but the triangles have a dierent orientation and dierent transformations are needed. Discuss rigid transformations. Focus especially on the question about lengths. A key concept in this section is the idea that lengths and angle measures are preserved under rigid transformations.Some students may claim CAGE is a square. If this comes up, leave it as an open question for now. This question will be revisited at the end of this mission, once the angle sum in a triangle is known. The last question establishes that CAGE is a rhombus.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.100

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ANTICIPATED MISCONCEPTIONSSome students might not recognize how this work is similar to the previous activity. For these students, ask them to step back and consider only triangles ABC and CDE, perhaps covering the bottom half of the diagram.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion SupportsGive students additional time to make sure that everyone in their group can explain whether the segments AC, CE, EG, and GA all have the same lengths. Then, vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.Design Principle(s): Optimize output (for explanation)SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the image to students who benet from extra processing time to review prior to implementation of this activity.Digital LessonFor the gure shown below, describe where the image would be if you:a) rotated segment AB 180° around point BJBTA© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:b) rotated segment AB 180° around point Jc) rotated segment AB 180° around point TSTUDENT RESPONSEa) The segment is attached at point B and is an extension of segment AB.b) The segment is above point J and is parallel to segment AB.c) The segment is identical to segment AB.Wrap-UpLESSON SYNTHESIS Ask students to describe the possible outcomes when a line segment AB is rotated 180 degrees.• AB is mapped to itself, when the center of rotation is the midpoint of the segment• AB is mapped to another segment collinear with the rst, when the center of rotation is A or B (or any other point on segment AB)• AB is mapped to a parallel segment, when the center of rotation is not on line AB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.102

G8M1 | LESSON 8ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:EXIT TICKET Here are two triangles.Is Triangle B the image of Triangle A aer a rotation?STUDENT RESPONSENo, Triangle B is a reection of Triangle A over line ℓ. A rotation can be used to match two sides of the triangles but will not match one up perfectly with the other. AB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 9Moves in ParallelLEARNING GOALSComprehend that a rotation by 180 degrees about a point of two intersecting lines moves each angle to the angle that is vertical to it.Describe (orally and in writing) observations of lines and parallel lines under rigid transformations, including lines that are taken to lines and parallel lines that are taken to parallel lines.Draw and label rigid transformations of a line and explain the relationship between a line and its image under the transformation.Generalize (orally) that “vertical angles” are congruent using informal arguments about 180 degree rotations of lines.LEARNING GOALS(STUDENT FACING)Let’s transform some lines.LEARNING TARGETS(STUDENT FACING)I can describe the effects of a rigid transformation on a pair of parallel lines.If I have a pair of vertical angles and know the angle measure of one of them, I can find the angle measure of the other.REQUIRED MATERIALSGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, 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.104

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:The previous lesson examines the impact of rotations on line segments and polygons. This lesson focuses on the eects of rigid transformations on lines. In particular, students see that parallel lines are taken to parallel lines and that a 180° rotation about a point on the line takes the line to itself. In grade 7, students found that vertical angles have the same measure, and they justify that here using a 180° rotation.As they investigate how 180° rotations inuence parallel lines and intersecting lines, students are looking at specic examples but their conclusions hold for all pairs of parallel or intersecting lines. No special properties of the two intersecting lines are used so the 180° rotation will show that vertical angles have the same measure for any pair of vertical angles. Warm-UpLINE MOVESIn this warm-up, students continue their work with transformations by shiing from applying rigid transformations to shapes to applying them specically to lines. Each image in this activity has the same starting line and students are asked to name the translation, rotation, or reection that takes this line to the second marked line. Because of their innite and symmetric nature, dierent transformations of lines look the same unless specic points are marked, so 1–2 points on each line are marked.While students have experience transforming a variety of gures, this activity provides the opportunity to use precise language when describing transformations of lines while exploring how sometimes dierent transformations can result in the same nal gures. During the activity, encourage students to look for more than one way to transform the original line.LAUNCHProvide access to tracing paper. Give students quiet work time followed by whole-class discussion.WARM-UP TASKFor each diagram, describe a translation, rotation, or reection that takes line ℓ to line ℓ'. Then plot and label A' and B', the images of A and B.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. 105

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. 2. STUDENT RESPONSE1. Answers vary. Possible responses:• Translation in many possible directions, for example, down 3 units• Reection over a line parallel to ℓ halfway between ℓ and ℓ'• Rotation using a point halfway between ℓ and ℓ' as the center of rotation and an angle of 180°2. Answers vary. Possible responses:• Reection across the vertical line through point A• Reection across the horizontal line through point A• Counterclockwise rotation about point A by the obtuse angle whose vertex is at A• Clockwise rotation about point A by the acute angle whose vertex is at AABℓℓ′ABℓℓ′© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.106

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEInvite students to share the transformations they choose for each problem. Each diagram has more than one possible transformation that would result in the nal gure. If the class only found one, pause for a few minutes and encourage students to see if they can nd another. For the rst diagram, look for a single translation, single rotation, and single reection that work. For the second diagram, look for a single rotation and a single reection.• “Will a translation work for the second diagram? Explain your reasoning.” (A translation will not work. Since translations do not incorporate a turn, translations of a line are parallel to the original line or are the same line.)SUPPORT FOR STUDENTS WITH DISABILITIESExecutive Functioning: Graphic OrganizersProvide a word web for transformations. Students should link important information for future reference (e.g., vocabulary, rules, and procedures).]Concept Exploration: Activity 1PARALLEL LINESInstructional Routine: MLR1: Stronger and Clearer Each TimeIn this activity, students will investigate the question, “What happens to parallel lines under rigid transformations?” by performing three dierent transformations on a set of parallel lines. Aer applying each transformation, they will jot down what they notice by answering the questions for each listed transformation.As students work through these problems they may remember essential features of parallel lines (they do not meet, they remain the same distance apart). Rigid transformations do not change either of these features which means that the image of a set of parallel lines aer a rigid transformation is another set of parallel lines.Identify the students who saw that the orientation of the lines changes but the lines remain parallel to each other regardless and select them to share during the discussion.Arrange students in groups of 3. Provide access to tracing paper. Each student in the group does one of the problems and then the group discusses their ndings. © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 LAUNCHBefore beginning, review with students what happens when we perform a rigid transformation. Demonstrate by moving the tracing paper on top of the image to replicate an example transformation (for example, rotation of the lines clockwise 90° around the center K). Tell students that the purpose of this activity is to investigate, “What happens to parallel lines when we perform rigid transformations on them?”ACTIVITY 1 TASK 1Use a piece of tracing paper to trace lines a and b and point K. Then use that tracing paper to draw the images of the lines under the three dierent transformations listed in your notes.As you perform each transformation, think about the question:What is the image of two parallel lines under a rigid transformation?1. Translate lines a and b 3 units up and 2 units to the right.a) What do you notice about the changes that occur to lines a and b aer the translation?b) What is the same in the original and the image?2. Rotate lines a and b counterclockwise 180 degrees using K as the center of rotation.Kabaˈbˈ2Khab© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.108

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:a) What do you notice about the changes that occur to lines a and b aer the rotation?b) What is the same in the original and the image?3. Reect lines a and b across line h.a) What do you notice about the changes that occur to lines a and b aer the reection?b) What is the same in the original and the image?STUDENT RESPONSE1. Translation: 3 grid square units up and 2 grid square units to the rightAnswers vary. Sample Responses:a) All 4 lines, a, b, a', and b' are parallel. The lines a' and b' look like a and b but shied upward.b) The pair of lines remain parallel. The distance between the lines did not change.2. Rotation around K.Answers vary. Sample Responses:a) The new pair of lines a' and b' are parallel to the original lines a and b.aba'b'Kab'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. 109

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:b) The lines a' and b' are still parallel and they are the same distance apart as a and b.3. Reection over line h.Answers vary. Sample responses:a) Line a is above line b whereas line b' is above line a'.b) Lines a' and b' are still parallel and are the same distance apart as lines a and b. All four lines are parallel to one another. DISCUSSION GUIDANCEAsk previously selected students who saw that the images of the parallel lines were parallel to the original in all three cases to share how they would answer the main question “What is the image of two parallel lines under a rigid transformation?” Make sure students understand that in general if ℓ and m are parallel lines and ℓ' and m' are their images under a rigid transformation then:• ℓ' and m' are parallel.• ℓ and m are not necessarily parallel to ℓ' and m' (refer to the 90 degree rotation shown during the launch).In addition to the fact that the parallel lines remain parallel to each other when rigid transformations are performed, the distance between the lines stay the same. What can change is the position of the lines in the plane, in relative terms (i.e., which line is ‘on top’) or in absolute terms (i.e., does a line contain a particular point in the plane).Give students quiet time to write a response to the main question, “What is the image of two parallel lines under a rigid transformation?”ANTICIPATED MISCONCEPTIONSStudents may not perform the transformations on top of the original image. Ask these students to place the traced lines over the original and perform each transformation from there.habb'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.110

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESFine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allowing students who struggle with ne motor skills to dictate tracing transformations as needed.SUPPORT FOR ENGLISH LANGUAGE LEARNERSWriting, Speaking: MLR 1 Stronger and Clearer Each TimeUse this routine to help students rene a written explanation about what happens to two parallel lines under a rigid transformation. Give students time to meet with 2–3 partners, to share and get feedback on their writing. Encourage listeners to press for details and clarity as appropriate based on what each speaker produces. Display prompts for feedback that will help individuals strengthen their ideas and clarify their language, such as: “Can you give or draw an example?”, “When are the lines parallel or do they intersect?”, and “What changes in a rigid transformation?” Give students time to revise their writing based on the feedback they receive. Design Principle(s): Optimize output (for explanation)Concept Exploration: Activity 2LET’S DO SOME 180’SInstructional Routines: MLR7: Compare and ConnectIn this activity, students apply their understanding of the properties of rigid transformations to 180° rotations of a line about a point on the line in order to establish the vertical angle theorem. Students have likely already used this theorem in grade 7, but this lesson informally demonstrates why the theorem is true. The demonstration of the vertical angle theorem exploits the structure of parallel lines and properties of both 180 degree rotations (studied in the previous lesson) and rigid transformations. Students begin the activity by rotating a line with marked points 180° about a point on the line. Unlike the example discussed in the launch, this line contains marked points other than the center of rotation. Then students rotate an angle 180° about a point on the line to draw conclusions about lengths and angles. Finally, students are asked to consider the intersection of two lines, the angles formed, and how the measurements of those angles can be deduced using a 180° rotation about the intersection of the lines, which is the vertical angle theorem.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:While students are working, encourage the use of tracing paper to show the transformations directly over the original image in order to help students keep track of what is happening with lines in each 180° rotation.LAUNCHRotations require the students to think about rotating an entire gure. It would be good to remind students about this before the start of this activity. This might help students see what is happening in the rst question better.Before students read the activity, draw a line ℓ with a marked point D for all to see. Ask students to picture what the gure rotated 180° around point D looks like. Aer some quiet think time, invite students share what they think the transformed gure would look like. Make sure all students agree that ℓ' looks “the same” as the original. If no students bring it up in their explanations, ask for suggestions of features that would make it possible to quickly tell the dierence between the ℓ' and ℓ, such as another point or if the line were dierent colors on each side of point D.Provide access to tracing paper.ACTIVITY 2 TASK 1Answer the questions in your notes. 1. The diagram shows a line with points labeled A, B, C, and D. a) On the diagram, draw the image of the line and points A, C, and B aer the line has been rotated 180 degrees around point D.b) Label the images of the points A', B', and C'.c) What is the order of all seven points? Explain or show your reasoning.2. The diagram shows a line with points A and C on the line and a segment AD where D is not on the line.a) Rotate the gure 180 degrees about point C. Label the image of A as A' and the image of D as D'.33A C D 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.112

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:b) What do you know about the relationship between angle CAD and angle CA'D'? Explain or show your reasoning.3. The diagram shows two lines ℓ and m that intersect at a point O with point A on ℓ and point D on m.a) Rotate the gure 180 degrees around O. Label the image of A as A' and the image of D as D'.b) What do you know about the relationship between the angles in the gure? Explain or show your reasoning.STUDENT RESPONSE1. A, C, B', D, B, C', A'2. Lengths of segment CA and segment CA' are the same, lengths of segment AD and segment A'D' are the same, and angles CAD and CA'D' have the same measure because both distances and angle measures are preserved under rigid transformations.3. Possible responses: Angles AOD and A'OD' have the same measure. Angles DOA' and D'OA have the same measure.ACDAODℓmA A′C C′D BB′AA′CDD′AA′ODD′ℓm© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEThe focus of the discussion should start with the relationships students nd between the lengths of segments and angle measures and then move to the nal problem, which establishes the vertical angle theorem as understood through rigid transformations. Questions to connect the discussion include:• "What relationships between lengths did we nd aer performing transformations?" (They are the same.)• "What relationships between angle measures did we nd aer performing transformations?" (They are the same.)• "What does this transformation informally prove?" (Vertical angles are congruent.)If time permits, consider discussing how the vertical angle theorem was approached in grade 7, namely by looking for pairs of supplementary angles. Pairs of vertical angles have the same measure because they are both supplementary to the same angle. The argument using 180 degree rotations is dierent because no reference needs to be made to the supplementary angle. The 180 degree rotation shows that both pairs of vertical angles have the same measure directly by mapping them to each other!ANTICIPATED MISCONCEPTIONS In the second question, students may not understand that rotating the gure includes both segment CA and segment AD since they have been working with rotating one segment at a time. Explain to these students that the gure refers to both of the segments. Encourage them to use tracing paper to help them visualize the rotation.SUPPORT FOR STUDENTS WITH DISABILITIESFine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allowing students who struggle with ne motor skills to dictate transformations 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.114

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSRepresenting, Speaking: MLR 7 Compare and ConnectUse this routine when students share what they noticed about the relationships between the angle measures. Ask students to consider what changes and what stays the same when rigid transformations are applied to lines and segments. Draw students’ attention to the associations between the rigid transformation, lengths of segments, and angle measures. These exchanges strengthen students’ mathematical language use and reasoning based on rigid transformations of lines and will lead to the informal argument of the vertical angle theorem.Design Principle(s): Maximize meta-awarenessDigital LessonThe diagram shows two lines that intersect at point B. Use the diagram to nd the measures of each angle. Explain your reasoning.a) Find the measure of angle ABCb) Find the measure of angle CBDc) Find the measure of angle DBESTUDENT RESPONSEAnswers vary. Sample response: a) Angle EBA and angle ABC form a straight angle. To nd the measure of angle ABC, I can subtract 85 degrees from 180 degrees. Therefore, angle ABC 85ºAB CDE© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:measures 95 degrees. b) Angle EBA and angle CBD are vertical angles, so they have the same measure. Therefore, angle CBD measures 85 degrees. c) Angle DBE and angle ABC have the same measure. Therefore, angle DBE measures 95 degrees. Wrap-UpLESSON SYNTHESISIn this lesson, students apply dierent rigid transformations to lines with a focus on parallel lines. They should be able to articulate what happens to parallel lines when a rigid transformation is performed on them. In addition, students gain a better understanding of why the vertical angle theorem they learned in grade 7 is true.To highlight how transformations aect parallel lines, ask students:• "When we perform rigid transformations on parallel lines, what do we know about their image?"• "Does the distance between the lines change?"To help students make a connection to how rotations aect lines in the second activity, ask:• "When we rotate a line 180° around a point on the line where does the line land?"• "How does the rotation aect the angle measurements for a pair of intersecting lines?"• "How does this help us prove the vertical angle theorem?"Students should see that a rotation of two intersecting lines about the point of intersection by 180° moves each angle to the angle that is vertical to it. Since rotation is a rigid transformation, the vertical angles must have the same measure.In general, rigid transformations help us see that when we transform lines it might change the orientation but the lines retain their original properties.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.116

G8M1 | LESSON 9ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TERMINOLOGYVertical angles Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.For example, angles AEC and DEB are vertical angles. If angle AEC measure 120°, then angle DEB must also measure 120°.Angles AED and BEC are another pair of vertical angles.EXIT TICKETPoints A', B', and C' are the images of 180-degree rotations of A, B, and C, respectively, around point O.Answer each question and explain your reasoning without measuring segments or angles.1. Name a segment whose length is the same as segment AO.2. What is the measure of angle A'OB'?STUDENT RESPONSE1. Segment A'O, because A' is the image of A aer a 180 degree rotation with center at O. This rotation preserves distances and takes segment AO to segment A'O.2. 79 degrees, the same measure as AOB, because the 180 degree rotation with center at O takes AOB to A'OB'. The rotation preserves angle measures.EACD BBOCAC′A′B′79°35°© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 10Composing FiguresLEARNING GOALSDraw and label images of triangles under rigid transformations and then describe (orally and in writing) properties of the composite figure created by the images.Generalize that side lengths and angle measures are preserved under any rigid transformation.Identify sides and angles that have equivalent measurements in composite shapes and explain (orally and in writing) why they are equivalent.LEARNING GOALS(STUDENT FACING)Let’s use reasoning about rigid transformations to find measurements without measuring.LEARNING TARGETS(STUDENT FACING)I can find missing side lengths or angle measures using properties of rigid transformations.REQUIRED MATERIALSGeometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor and an index card to use as a straightedge or to mark right anglesTEACHER INSTRUCTION ONLYZEARN MATH TIPThere is no Independent Digital Lesson for this lesson. It is a further exploration of the work done so far in this mission. If you feel your students still need work with these concepts, we recommend you teach it with your whole group or teach it during your concept exploration and give students digital time to ensure they are on pace with your instruction heading into the next topic.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.118

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, students create composite shapes using translations, rotations, and reections of polygons and continue to observe that the side lengths and angle measures do not change. They use this understanding to draw conclusions about the composite shapes. Later, they will use these skills to construct informal arguments. For example, they will construct informal arguments about the sum of the angles in a triangle.When students rotate around a vertex or reect across the side of a gure, it is easy to lose track of the center of rotation or line of reection since they are already part of the gure. It can also be challenging to name corresponding points, segments, and angles when a gure and its transformation share a side. Students attend to these details carefully in this lesson.Consider using the optional activity if you need to reinforce students’ belief that rigid transformations preserve side lengths and angle measures aer the main activities.Warm-UpANGLES OF AN ISOSCELES TRIANGLEIsosceles triangles are triangles with (at least) one pair of congruent sides. Isosceles triangles also have (at least) one pair of congruent angles. In this warm-up, students show why this is the case by exploiting the fact that rigid motions of the plane do not change angle measures.LAUNCHGive students quiet work time followed by whole-class discussion.WARM-UP TASKHere is triangle ABC. Solve the problems below. 1ABC323© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. Reect triangle ABC over line AB. Label the image of C as C′.2. Rotate triangle ABC′ around A so that C′ matches up with B.3. What can you say about the measures of angles B and C?STUDENT RESPONSE1. 2. Rotating ABC′ as described takes ABC′ back to the original triangle.3. The measures of angles B and C are the same. Neither the rotation nor the reection changed the angle measures, and since these transformations take the angle at C to the angle at B, they must have the same measure.DISCUSSION GUIDANCEMake sure that students are precise in their response to the third question, indicating both the center of the rotation and the angle of rotation. Time permitting, mention that it is also true that when a triangle has two angles with the same measure then the sides opposite those angles have the same length (i.e., the triangle is isosceles). This can also be shown with rigid transformations; reect the triangle rst and then line up the sides containing the pairs of angles with the same measure.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the image to students who benet from extra processing time to review prior to implementation of this activity.ABC′ C323© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.120

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Concept Exploration: Activity 1TRIANGLE PLUS ONEInstructional Routines: Think Pair Share; MLR1: Stronger and Clearer Each TimeThe purpose of this task is to use rigid transformations to describe an important picture that students saw in grade 6 when they developed the formula for the area of a triangle. They rst found the area of a parallelogram to be base ∙ height and then, to nd 21 base ∙ height for the area of a triangle, they “composed” two copies of a triangle to make a parallelogram. The language “compose” is a grade 6 appropriate way of talking about a 180° rotation. The focus of this activity is on developing this precise language to describe a familiar geometric situation.Students need to remember and use an important property of 180 degree rotations, namely that the image of a line aer a 180 degree rotation is parallel to that line. This is what allows them to conclude that the shape they have built is a parallelogram.LAUNCHArrange students in groups of 2. Provide access to geometry toolkits. Give students quiet work time, followed by sharing with a partner and a whole-class discussion.ACTIVITY 1 TASK 1Here is triangle ABC. Solve the problems below. 1. Draw midpoint M of side AC.2. Rotate triangle ABC 180 degrees using center M to form triangle CDA. Draw and label this triangle.3. What kind of quadrilateral is ABCD? Explain how you know.2AB 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. 121

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1-2. 3. A parallelogram. The 180 degree rotation around M takes line AB to line CD and so these are parallel. It also takes line BC to line AD so these lines are also parallel. That means that ABCD is a parallelogram.DISCUSSION GUIDANCEBegin the discussion by asking, “What happens to points A and C under the rotation?” (They end up at C and A, respectively.) This type of rotation and analysis will happen several times in upcoming lessons.Next ask, “How do you know that the lines containing opposite sides of ABCD are parallel?” (They are taken to one another by a 180 degree rotation.) As seen in the previous lesson, the image of a 180° rotation of a line ℓ is parallel to ℓ. Students also saw that when 180° rotations were applied to a pair of parallel lines it resulted in a (sometimes) new pair of parallel lines which were also parallel to the original lines. The logic here is the same, except that only one line is being rotated 180° rather than a pair of lines. This does not need to be mentioned unless it is brought up by students.Finally, ask students “How is the area of parallelogram ABCD related to the area of triangle ABC?” (The area of the parallelogram ABCD is twice the area of triangle ABC because it is made up of ABC and CDA which has the same area as ABC.) Later in this mission, areas of shapes and their images under rigid transformations will be studied further.ANTICIPATED MISCONCEPTIONSStudents may struggle to see the 180° rotation using center M. This may be because they do not understand that M is the center of rotation or because they struggle with visualizing a 180° rotation. Oer these students patty paper, a transparency, or the rotation overlay from earlier in this mission to help them see the rotated triangle.AMDB 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.122

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSWriting, Speaking: Math Language Routine 1 Stronger and Clearer Each TimeThis is the rst time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and are asked to create a rst dra response. Students meet with 2–3 partners to share and rene their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second dra of their response reecting ideas from partners and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and rene their ideas through verbal and written means.Design Principle(s): Optimize output (for justication)How It Happens:1. Use this routine to provide students a structured opportunity to rene their justication for the question asking “What kind of quadrilateral is ABCD? Explain how you know.” Give students time to individually create rst dra responses in writing.2. Invite students to meet with 2–3 other partners for feedback. Instruct the speaker to begin by sharing their ideas without looking at their written dra, if possible. Listeners should press for details and clarity. Provide students with these prompts for feedback that will help individuals strengthen their ideas and clarify their language: “What do you mean when you say...?”, “Can you describe that another way?”, “How do you know the lines are parallel?”, and “What happens to lines under rotations?” Be sure to have the partners switch roles. Allow time to discuss.3. Signal for students to move on to their next partner and repeat this structured meeting.4. Close the partner conversations and invite students to revise and rene their writing in a second dra. Students can borrow ideas and language from each partner to strengthen the nal product. Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Quadrilateral ABCD is a because.…” and “Another way to verify this is.…”. Here is an example of a second dra: “ABCD is a parallelogram, I know this because a 180-degree rotation creates new lines that are parallel to the original lines. In this gure, the 180-degree rotation takes line AB to line CD and line BC to line AD. I checked this by copying the triangle onto patty paper and rotating it 180 degrees. This means the new shape will have two pairs of parallel sides. Quadrilaterals that have two pairs of parallel sides are called parallelograms.”5. If time allows, have students compare their rst and second dras. If not, have the students move on by discussing other aspects of the 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. 123

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeCheck in with individual students, as needed, to assess for comprehension during each step of the activity.Concept Exploration: Activity 2TRIANGLE PLUS TWOInstructional Routine: MLR3: Clarify, Critique, CorrectThis activity continues the previous one, building a more complex shape this time by adding an additional copy of the original triangle. The three triangle picture in the task will be important later in this mission when students show that the sum of the three angles in a triangle is 180°. To this end, encourage students to notice that the points E, A, and D all lie on a line.As with many of the lessons applying transformations to build shapes, students are constantly using their structural properties to make conclusions about their shapes. Specically, that rigid transformations preserve angle measures and side lengths.LAUNCHKeep students in the same groups. Provide access to geometry toolkits. Allow for some quiet work time, followed by sharing with a partner and a whole-class discussion.ACTIVITY 2 TASK 1The picture shows 3 triangles. Triangle 2 and Triangle 3 are images of Triangle 1 under rigid transformations.33ACBDE3 21© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.124

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. Describe a rigid transformation that takes Triangle 1 to Triangle 2. What points in Triangle 2 correspond to points A, B, and C in the original triangle?2. Describe a rigid transformation that takes Triangle 1 to Triangle 3. What points in Triangle 3 correspond to points A, B, and C in the original triangle?3. Find two pairs of line segments in the diagram that are the same length, and explain how you know they are the same length.4. Find two pairs of angles in the diagram that have the same measure, and explain how you know they have the same measure.STUDENT RESPONSE1. Answers vary. Sample response: a 180-degree rotation using the midpoint of side AC as center. In triangle 2, point C corresponds to A in the original, D corresponds to B in the original, and A corresponds to C in the original.2. Answers vary. Sample responses: a 180-degree rotation using the midpoint of side AB as center or a 180 degree rotation using the midpoint of segment AC as center followed by a translation taking A to E. In triangle 3, point A corresponds to B in the original, B corresponds to A in the original, and E corresponds to C in the original.3. Answers vary. Sample response: segment AE and segment BC are the same length and segments AB and CD are also the same length. This is true because a rigid transformation doesn’t change a gure’s side lengths.4. Answers vary. Sample response: angle D and angle ABC have the same measure and so do angles E and ACB. This is true because a rigid transformation doesn’t change a gure’s angle measures.DISCUSSION GUIDANCEAsk students to list as many dierent pairs of matching line segments as they can nd. Then, do the same for angles. Record these for all to see. Students may wonder why there are fewer pairs of line segments: this is because of shared sides AB and AC. If they don’t ask, there’s no reason to bring it up.If you create a visual display of these pairs, hang on to the information about angles that have the same measure. The same diagram appears later in this mission and is used for a proof about the sum of the angle measures in a triangle.Aer this activity, ask students to summarize their understanding about side lengths and angle measures under rigid transformations. If students don’t say it outright, you should: “Under any rigid transformation, side lengths and angle measures are preserved.”© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSWriting: Math Language Routine 3 Clarify, Critique, CorrectThis is the rst 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, reect 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.Design Principle(s): Maximize meta-awarenessHow It Happens:1. In the class discussion for this activity, present this incomplete description of a rigid transformation that takes Triangle 1 to Triangle 2: “CB and AD are the same because you turn ABC.” Prompt students to identify the ambiguity of this response. Ask students, “What do you think this person is trying to say? What is unclear?”2. Give students individual time to respond to the questions in writing, and then time to discuss with a partner. As pairs discuss, provide these sentence frames for scaolding: “I think that what the author meant by ‘turn ABC’ was….”, “The part that is the most unclear to me is … because.…”, and “I think this person is trying to say.…”. 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. Listen for students using appropriate geometry terms such as “transformation” and “rotation” in explaining why the two sides are equivalent.3. Then, ask students to write a more precise version and explain their reasoning in writing with their partner. Improved responses should include for each step an explanation, order/time transition words (rst, next, then, etc.), and/or reasons for decisions made during steps. Here is a sample improved response: “First, I used tracing paper to create a copy of triangle ABC because I wanted to transform it onto the new triangle. Next, I labeled the vertices on my tracing paper. Then, using the midpoint of side AC as my center, I rotated the tracing paper 180 degrees, because then it matched up on top of Triangle 2. So, sides CB and AD are equivalent.”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 class, and invite the class to make comparisons among the responses shared and their own responses. Listen for responses that identify the correct pair of equivalent sides and explain how they know. In this conversation, also allow students the opportunity to name other equivalent sides and angles.5. Close the conversation with the generalization that side lengths and angle measures are preserved under any rigid transformation, and then move on to the next lesson 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.126

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer TutorsPair students with their previously identied peer tutors.Wrap-UpLESSON SYNTHESISBriey review the properties of rigid transformations (they preserve side lengths and angle measures). Go over some examples of corresponding sides and angles in gures that share points. For example, talk about which sides and angles correspond if this image is found by reecting ABCD across line AD.Point out to students that sides on a reection line do not move, so they are their own image when we reect across a side. Also, the center of rotation does not move, so it is its own image when we rotate around it. All points move with a translation.EXIT TICKETStudents apply the fact that rigid motions preserve side lengths and angle measures. They are presented with a gure and two transformed images, and they use what they know to nd the side lengths and angle measures of the transformed gures.ABCD© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 10ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TASKHere is a diagram showing triangle ABC and some transformations of triangle ABC.On the le side of the diagram, triangle ABC has been reected across line AC to form quadrilateral ABCD. On the right side of the diagram, triangle ABC has been rotated 180 degrees using midpoint M as a center to form quadrilateral ABCE.Using what you know about rigid transformations, side lengths and angle measures, label as many side lengths and angle measures as you can in quadrilaterals ABCD and ABCE.STUDENT RESPONSEADCB2.73.2AECB2.73.2M64.3° 64.3°ADCB2.72.73.23.264.3°64.3°AECB2.72.73.23.264.3°M64.3°© 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.128

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 11What Is the Same?LEARNING GOALSCompare and contrast (orally and in writing) side lengths, angle measures, and areas using rigid transformations to explain why a shape is or is not congruent to another.Comprehend that congruent figures have equal corresponding side lengths, angle measures, and areas.Describe (orally and in writing) two figures that can be moved to one another using a sequence of rigid transformations as “congruent.”LEARNING GOALS(STUDENT FACING)Let’s decide whether shapes are the same.LEARNING TARGETS(STUDENT FACING)I can decide visually whether or not two figures are congruent.REQUIRED MATERIALSGeometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor and an index card to use as a straightedge or to mark right anglesREQUIRED PREPARATIONColored pencils are supposed to be a usual part of the geometry toolkit, but they are called out here because one activity asks students to shade rectangles using different colors.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, students explore what it means for shapes to be “the same” and learn that the term congruent is a mathematical way to talk about gures being the same that has a precise meaning. Specically, they learn that two gures are congruent if there is a sequence of translations, rotations, and reections that moves one to the other. They learn that gures that are congruent can have dierent orientations, but corresponding lengths and angle measures are equal. Agreeing upon and formulating the denition of congruence requires careful use of precise language and builds upon all of the student experiences thus far in this mission, moving shapes and trying to make them match up.As they work to decide whether or not pairs of shapes are congruent, students will look for and make use of structure. For shapes that are not congruent, what property can be identied in one that is not shared by the other? This could be an angle measure, a side length, or the size of the shape. For shapes that are congruent, is there any way to tell other than experimenting with tracing paper? In some cases, like the rectangles, students discover that looking at the length and width is enough to decide if they are congruent.In elementary grades, deciding if two shapes are the “same” usually involves making sure that they are the same general shape (for example, triangles or circles) and that the size is the same. As shapes become more complex and as we develop new ways to measure them (angles for example), something more precise is needed. The denition of congruence here states that two shapes are congruent if there is a sequence of translations, rotations, and reections that matches one shape up exactly with the other. This denition has many advantages:• It does not require measuring all side lengths or angles.• It applies equally well to all shapes, not just polygons.• It is precise and unambiguous: certain moves are allowed and two shapes are congruent when one can be moved to align exactly with the other.The material treated here will be taken up again in high school from a more abstract point of view. In grade 8, it is essential for students to gain experience executing rigid motions with a variety of tools (tracing paper, coordinates, technology) to develop the intuition that they will need when they study these moves (or transformations) in greater depth later.Warm-UpFIND THE RIGHT HANDSInstructional Routine: Think Pair ShareIn this activity, students get their rst formal introduction to the idea of mirror orientation, sometimes called “handedness” because le and right hands are reections of each other. The easiest way to decide which are the right hands is to hold one’s hands up and rotate them until © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.130

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:they match a particular gure (or don’t). This prepares them for a discussion about whether gures with dierent mirror orientation are the same or not.LAUNCHArrange students in groups of 2, and provide access to geometry toolkits. Give students some quiet work time, followed by time for sharing with a partner and a whole-class discussion. Show students this image or hold up both hands and point out that our hands are mirror images of each other. These are hands shown from the back. If needed, clarify for students that all of the hands in the task are shown from the back.WARM-UP TASKA person’s hands are mirror images of each other. In the diagram, a le hand is labeled. Shade all of the right hands.Le hand Right hand1Le hand© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEDISCUSSION GUIDANCEAsk students to think about the ways in which the le and right hands are the same, and the ways in which they are dierent.Some ways that they are the same include:• The side lengths and angles on the le and right hands match up with one another.• If a le hand is ipped, it can match it up perfectly with a right hand (and vice versa).Some ways that they are dierent include:• They can not be lined up with one another without ipping one of the hands over.• It is not possible to make a physical le and right hand line up with one another, except as “mirror images.”SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the image to students who benet from extra processing time to review prior to implementation of this activity.Le hand© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Concept Exploration: Activity 1ARE THEY THE SAME?Instructional Routines: MLR7: Compare and Connect; MLR2: Collect and Display; Anticipate, Monitor, Select, Sequence, ConnectIn previous work, students learned to identify translations, rotations, and reections. They started to study what happens to dierent shapes when these transformations are applied. They used sequences of translations, rotations, and reections to build new shapes and to study complex congurations in order to compare, for example, vertical angles made by a pair of intersecting lines. Starting in this lesson, rigid transformations are used to formalize what it means for two shapes to be the same, a notion which students have studied and applied since the early grades of elementary school.In this activity, students express what it means for two shapes to be the same by considering carefully chosen examples. Students work to decide whether or not the dierent pairs of shapes are the same. Then the class discusses their ndings and comes to a consensus for what it means for two shapes to be the same: the word “same” is replaced by “congruent” moving forward.There may be discussion where a reection is required to match one shape with the other. Students may disagree about whether or not these should be considered the same and discussion should be encouraged. This activity encourages students to construct viable arguments and critique the reasoning of others as students need to explain why they believe that a pair of gures is the same or is not the same.Monitor for students who use these methods to decide whether or not the shapes are the same and invite them to share during the discussion:• Observation (this is oen suicient to decide that they are not the same): Encourage students to articulate what feature(s) of the shapes help them to decide that they are not the same.• Measuring side lengths using a ruler or angles using a protractor: Then use dierences among these measurements to argue that two shapes are not the same.• Cutting out one shape and trying to move it on top of the other: A variant of this would be to separate the two images and then try to put one on top of the other or use tracing paper to trace one of the shapes. This is a version of applying transformations studied extensively prior to this lesson.LAUNCHGive students quiet work time followed by a discussion. Provide access to geometry toolkits.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 1For each pair of shapes, decide whether or not they are the same. 1. 2. 3. 4. 5. STUDENT RESPONSE1. The two shapes are the same. Rotating the shape on the le (by 180 degrees) around the top point and moving it down and to the right it matches up perfectly with the shape on the right.2. They are not the same. Possible strategy: The side lengths of the shapes are the same but the angles are not. The shape on the right is more squished down (and has less area) so they are not the same.3. They are the same. Possible strategy: The shapes are both curly arrows and look like they are the same size. Reecting over a vertical line halfway between the two shapes, they appear to match up perfectly with one another. Or: They are not the same. Possible strategy: The curly arrow on the le moves in a clockwise direction while the curly arrow on the right moves in a counterclockwise direction.4. They are not the same. Possible strategy: The general shapes are the same and the angles match up but the side lengths are dierent. The shape on the le is bigger than the shape on the right.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.134

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:5. They are not the same. Possible strategy: The part that sticks out of the right side is higher on the rst piece and lower on the second piece. Building a puzzle, both shapes would not t in the same spot.DISCUSSION GUIDANCEFor each pair of shapes, poll the class. Count how many students decided each pair was the same or not the same. Then for each pair of shapes, select at least one student to defend their reasoning. (If there is unanimous agreement over any of the pairs of shapes, these can be dealt with quickly, but allow the class to hear at least one argument for each pair of shapes.) Sequence these explanations in the order suggested in the Activity Narrative: general observations, taking measurements, and applying rigid transformations with the aid of tracing paper.The most general and precise of these criteria is the third which is the foundation for the mathematical denition of congruence: The other two are consequences. The moves allowed by rigid transformations do not change the shape, size, side lengths, or angle measures.There may be disagreement about whether or not to include reections when deciding if two shapes are the same. Here are some reasons to include reections:• A shape and its reected image can be matched up perfectly (using a reection).• Corresponding angles and side lengths of a shape and its reected image are the same.And here are some reasons against including reections:• A le foot and a right foot (for example) do not work exactly the same way. If we literally had two le feet it would be diicult to function normally!• Translations and rotations can be enacted, for example, by putting one sheet of tracing paper on top of another and physically translating or rotating it. For a reection the typical way to do this is to li one of the sheets and ip it over.If this disagreement doesn’t come up, ask students to think about why someone might conclude that the pair of gures in C were not the same. Explain to students that people in the world can mean many things when they say two things are “the same.” In mathematics there is oen a need to be more precise, and one kind of “the same” is congruent. (Two gures are congruent if one is a reection of the other, but one could, if one wanted, dene a dierent term, a dierent kind of “the same,” where ipping was not allowed!)Explain that Figure A is congruent to Figure B if there is a sequence of translations, rotations, and reections which make Figure A match up exactly with Figure B.Combining this with the earlier discussion a few general observations about congruent gures include:© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• Corresponding sides of congruent gures are congruent.• Corresponding angles of congruent gures are congruent.• The area of congruent gures are equal.What can be “dierent” about two congruent gures? The location (they don’t have to be on top of each other) and the orientation (requiring a reection to move one to the other) can be dierent.ANTICIPATED MISCONCEPTIONSStudents may think all of the shapes are the same because they are the same general shape at rst glance. Ask these students to look for any dierences they can nd among the pairs of shapes.SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversing, Representing: MLR 2 Collect and DisplayAs students work on comparing shapes, circulate and listen to students talk. Record common or important phrases (e.g., side length, rotated, reected, etc.), together with helpful sketches or diagrams on a display. Pay particular attention to how students are using transformational language while determining whether the shapes are the same. Scribe students’ words and sketches on a visual display to refer back to during whole-class discussions throughout this lesson and the rest of the mission. This will help students use mathematical language during their group and whole-class discussions.Design Principle(s): Support sense-makingSpeaking: MLR 7 Compare and ConnectTo support students when explaining each pair of shapes, ask students to consider why a shape is the same as or dierent from another. Draw students’ attention to the association between general observations, taking measurements and applying rigid transformations to use for comparing segments, angles, reections and rotations ( e.g., ask, “For shapes that are not congruent, what property can be identied in one that is not shared by the other?” Or, “For shapes that are congruent, is there any way to tell other than experimenting with tracing paper?”). These exchanges will strengthen students’ mathematical language use and reasoning based on rigid transformations.Design Principle(s): Support sense-making© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.136

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer TutorsPair students with their previously identied peer tutors.Concept Exploration: Activity 2AREA, PERIMETER, AND CONGRUENCEInstructional Routines: Think Pair Share, MLR1: Stronger and Clearer Each TimeSometimes people characterize congruence as “same size, same shape.” The problem with this is that it isn’t clear what we mean by “same shape.” All of the gures in this activity have the same shape because they are all rectangles, but they are not all congruent. Students examine a set of rectangles and classify them according to their area and perimeter. Then they identify which ones are congruent. Because congruent shapes have the same side lengths, congruent rectangles have the same perimeter. But rectangles with the same perimeter are not always congruent. Congruent shapes, including rectangles, also have the same area. But rectangles with the same area are not always congruent. Highlighting important features, like perimeter and area, which can be used to quickly establish that two shapes are not congruent, develops students’ ability to look for and make use of structure as they identify fundamental properties shared by any pair of congruent shapes.LAUNCHTell students that they will investigate further how nding the area and perimeter of a shape can help show that two gures are not congruent. It may have been a while since students have thought about the terms area and perimeter. If necessary, to remind students what these words mean and how they can be computed, display a rectangle like this one for all to see. Ask students to explain what perimeter means and how they can nd the perimeter and area of this rectangle.Arrange students in groups of 2. Provide access to geometry toolkits (colored pencils are specically called for). Give students quiet work time followed by time for sharing with a partner and 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. 137

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 2 TASK 1Answer the questions in your notes. 1. Which of these rectangles have the same area as Rectangle R but dierent perimeter?2. Which rectangles have the same perimeter as Rectangle R but dierent area?3. Which have the same area as Rectangle R and the same perimeter?4. Use materials from the geometry tool kit to decide which rectangles are congruent. Shade congruent rectangles with the same color.STUDENT RESPONSEThe perimeter of Rectangle R is 10 units since 3 + 2 + 3 + 2 = 10 while its area is 6 square units since 2∙ 3 = 6 . All of the rectangles in the picture share at least one of these properties (either the perimeter or the area) but only the 2 unit by 3 unit rectangles share both:1. Rectangles B and C have the same area (6 square units) but dierent perimeter (14 units) 2. Rectangles D and F have the same perimeter (10 units) but dierent area (4 square units) 3. Rectangles A and E have the same area and perimeter: only their position and orientation on the page is dierent.4. The 2 by 3 rectangles are congruent to Rectangle R. In each case, Rectangle R can be translated and rotated so that it matches up perfectly with the 2-by-3 rectangle. The same argument shows that Rectangles B and C are congruent as are Rectangles D and F.332ABDEF11443322CR33226611441111663 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.138

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEInvite students who used the language of transformations to answer the nal question to describe how they determined that a pair of rectangles are congruent.Perimeter and area are two dierent ways to measure the size of a shape. Ask the students:• “Do congruent rectangles have the same perimeter? Explain your reasoning.” (Yes. Rigid motions do not change distances, and so congruent rectangles have the same perimeter.)• “Do congruent rectangles have the same area? Explain your reasoning.” (Yes. Rigid motions do not change area or rigid motions do not change distances and so do not change the length times the width in a rectangle.)• “Are rectangles with the same perimeter always congruent?” (No. Rectangles D and F have the same perimeter but they are not congruent.)• “Are rectangles with the same area always congruent?” (No. Rectangles B and C have the same area but are not congruent.)One important take away from this lesson is that measuring perimeter and area is a good method to show that two shapes are not congruent if these measurements dier. When the measurements are the same, more work is needed to decide whether or not two shapes are congruent.A risk of using rectangles is that students may reach the erroneous conclusion that if two gures have both the same area and the same perimeter, then they are congruent. If this comes up, challenge students to think of two shapes that have the same area and the same perimeter, but are not congruent. Here is an example:21144332233BCAD226611441111663 32A 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. 139

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ANTICIPATED MISCONCEPTIONSWatch for students who think about the nal question in terms of “same shape and size.” Remind them of the denition of congruence introduced in the last activity.SUPPORT FOR STUDENTS WITH DISABILITIESExecutive Functioning: Graphic OrganizersProvide a Venn diagram with which to compare the similarities and dierences between the rectangles.SUPPORT FOR ENGLISH LANGUAGE LEARNERSWriting, Speaking: MLR 1 Stronger and Clearer Each TimeUse this routine with successive pair shares to give students a structured opportunity to revise their written strategies for deciding which rectangles are congruent. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language (e.g., “How was a sequence of transformations used to…?”, “What properties do the shapes share?”, “What was dierent and what was the same about each pair?”, etc.). Students can borrow ideas and language from each partner to strengthen their nal product.Design Principle(s): Optimize output (for explanation)Digital Lesson1. If two rectangles have the same perimeter, do they have to be congruent? Explain how you know.2. If two rectangles are congruent, do they have to have the same perimeter? Explain how you know.STUDENT RESPONSEAnswers vary. Sample response:© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.140

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. If two rectangles have the same perimeter, they are not necessarily congruent. They could be congruent if they both have the same side lengths. For example, if both rectangles have a length of 8 units and a width of 6 units. They could also not be congruent. For example, if one rectangle had a length of 9 units and a width of 6 units, and the other had a length 8 units and a width of 7 units. In both cases the rectangles have a perimeter of 30 units, but they are not congruent, since there isn’t a sequence of rigid transformations that would make them match up.2. If two rectangles are congruent, then they have the same perimeter. That’s because when 2 shapes are congruent, their angle measures and side lengths are equal. So if the two rectangles have the same side lengths, then they also have the same perimeter.Wrap-UpLESSON SYNTHESISAsk students to state their best denition of congruent. (Two shapes are congruent when there is a sequence of translations, rotations, and reections that take one shape to the other.) Some important concepts to discuss:• “How can you check if two shapes are congruent?” (For rectangles, the side lengths are enough to tell. For more complex shapes, experimenting with transformations is needed.)• “Are a shape and its mirror image congruent?” (Yes, because a reection takes a shape to its mirror image.)• “What are some ways to know that two shapes are not congruent?” (Two shapes are not congruent if they have dierent areas, side lengths, or angles.)• “What are some properties that are shared by congruent shapes?” (They have the same number of sides, same length sides, same angles, same 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. 141

G8M1 | LESSON 11ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TERMINOLOGYCongruent One gure is congruent to another if it can be moved with translations, rotations, and reections to t exactly over the other.In the gure, Triangle A is congruent to Triangles B, C, and D. A translation takes Triangle A to Triangle B, a rotation takes Triangle B to Triangle C, and a reection takes Triangle C to Triangle D.EXIT TICKETThroughout this mission, students have been using translations, rotations, and reections to move gures in the plane. In this lesson, students have learned that Figure A is congruent to Figure B when there is a sequence of translations, rotations, and reections that take Figure A to Figure B. Here they apply this to two non-polygonal gures, one of which is a reection of the other.TASKFigure B is the image of Figure A when reected across line l. Are Figure A and Figure B congruent? Explain your reasoning.STUDENT RESPONSEYes, they are congruent. There is a rigid transformation that takes one gure to the other, so they are congruent.ABCDFigure A Figure 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.142

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 12Congruent PolygonsLEARNING GOALSComprehend that figures with the same area and perimeter may or may not be congruent.Critique arguments (orally) that two figures with congruent corresponding sides may be non-congruent figures.Justify (orally and in writing) that two polygons on a grid are congruent using the definition of congruence in terms of transformations.LEARNING GOALS(STUDENT FACING)Let’s decide if two figures are congruent.LEARNING TARGETS(STUDENT FACING)I can decide using rigid transformations whether or not two figures are congruent.REQUIRED MATERIALSGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor, and an index card to use as a straightedge or to mark right anglesToothpicks, pencils, straws, or other objects© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, students nd rigid transformations that show two gures are congruent and make arguments for why two gures are not congruent. They learn that, for many shapes, simply having corresponding side lengths that are equal will not guarantee the gures are congruent.In the previous lesson, students dened what it means for two shapes to be congruent and started to apply the denition to determine if a pair of shapes is congruent. In the rst part of this lesson, students continue to determine whether or not pairs of shapes are congruent, but here they have the extra structure of a grid. With this extra structure, students attend to precision when describing translations, reections, and rotations. For example:• Instead of “translate down and to the le,” students can say, “translate 3 units down and 2 units to the le”• Instead of “reect the shape,” students can say, “reect the shape over this vertical line.”In addition, students have to be careful how they name congruent polygons, making sure that corresponding vertices are listed in the proper order. Warm-UpTRANSLATED IMAGESThis task helps students think strategically about what kinds of transformations they might use to show two gures are congruent. Being able to recognize when two gures have either a mirror orientation or rotational orientation is useful for planning out a sequence of transformations.LAUNCHProvide access to geometry toolkits. Allow quiet time followed by a whole-class 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.144

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASK All of these triangles are congruent. Sometimes we can take one gure to another with a translation. Shade the triangles that are images of triangle ABC under a translation.STUDENT RESPONSE1ABCABC© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEPoint out to students that if we just translate a gure, the image will end up pointed in the same direction. (More formally, the gure and its image have the same mirror and rotational orientation.) Rotations and reections usually (but not always) change the orientation of a gure.For a couple of the triangles that are not translations of the given gure, ask what sequence of transformations would show that they are congruent, and demonstrate any rotations or reections required.ANTICIPATED MISCONCEPTIONSIf any students assert that a triangle is a translation when it isn’t really, ask them to use tracing paper to demonstrate how to translate the original triangle to land on it. Inevitably, they need to rotate or ip the paper. Remind them that a translation consists only of sliding the tracing paper around without turning it or ipping it.Concept Exploration: Activity 1CONGRUENT PAIRS (PART 1)MATERIALS: GEOMETRY TOOLKITSInstructional Routines: MLR7: Compare and Connect; Anticipate, Monitor, Select, Sequence, ConnectIn the previous lesson, students formulated a precise mathematical denition for congruence and began to apply this to determine whether or not pairs of gures are congruent. This activity is a direct continuation of that work with the extra structure of a square grid. The square grid can be a helpful structure for describing the dierent transformations in a precise way. For example, with translations we can talk about translating up or down or to the le or right by a specied number of units. Similarly, we can readily reect over horizontal and vertical lines and perform some simple rotations. Students may also wish to use tracing paper to help execute these transformations. Choosing an appropriate method to show that two gures are congruent encourages using appropriate tools strategically.Students are given several pairs of shapes on grids and asked to determine if the shapes are congruent. The congruent shapes are deliberately chosen so that more than one transformation will likely be required to show the congruence. In these cases, students will likely nd dierent ways to show the congruence. Monitor for dierent sequences of transformations that show congruence. For example, for the rst pair of quadrilaterals, some dierent ways are:© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.146

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• Translate EFGH 1 unit to the right, and then rotate its image 180º about (0,0).• Reect ABCD over the x-axis, then reect its image over the y-axis, and then translate this image 1 unit to the le.For the pairs of shapes that are not congruent, students need to identify a feature of one shape not shared by the other in order to argue that it is not possible to move one shape on top of another with rigid motions. At this early stage, arguments can be informal. Monitor for these situations:• The side lengths are dierent so it is not possible to make them match up.• The angles are dierent so the two shapes can not be made to match up.• The areas of the shapes are dierent.LAUNCHProvide access to geometry toolkits. Allow quiet work time followed by a discussion.ACTIVITY 1 TASK 1 For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.1. 2EFGHDCBAyx© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:2. 3. 4. yxABCDEFGHIJyxDEFABCI JKLMNOPA BCDEFGHyx© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.148

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. These are congruent. Sample response: Rotate quadrilateral ABCD around D by 180 degrees, and then translate le 3 units and down 2 units. It matches up perfectly with HGFE.2. These are not congruent. Sample response 1: They are both pentagons, but ABCDE has a pair of opposite parallel sides while FGHIJ does not. Sample response 2: Angle D in ABCDE measures more than 180 degrees, while all angles in FGHIJ measure less than 180 degrees. Sample response 3: Side DE measures one unit in length, while all sides of FGHIJ measure more than 1 unit in length.3. These are congruent. Sample response: Rotate triangle ABC around (0, 0) counterclockwise by 90 degrees, and then translate it down 2 units and le 3 units. It matches up with triangle DEF perfectly.4. These are not congruent. Sample response: Both are regular octagons, but ABCDEFGH is larger than IJKLMNOP. This can be seen by comparing the images or by looking at sides AB and IJ. Side AB is 2 units in length, while side IJ is less than 2 units in length.DISCUSSION GUIDANCEPoll the class to identify which shapes are congruent (A and C) and which ones are not (B and D). For the congruent shapes, ask which motions (translations, rotations, or reections) students used, and select previously identied students to show dierent methods. Sequence the methods from most steps to fewest steps when possible.For the shapes that are not congruent, invite students to identify features that they used to show this and ask students if they tried to move one shape on top of the other. If so, what happened? It is important for students to connect the dierences between identifying congruent vs non-congruent gures.The purpose of the discussion is to understand that when two shapes are congruent, there is a rigid transformation that matches one shape up perfectly with the other. Choosing the right sequence takes practice. Students should be encouraged to experiment, using technology and tracing paper when available. When two shapes are not congruent, there is no rigid transformation that matches one shape up perfectly with the other. It is not possible to perform every possible sequence of transformations in practice, so to show that one shape is not congruent to another, we identify a property of one shape that is not shared by the other. For the shapes in this problem set, students can focus on side lengths: for each pair of non congruent shapes, one shape has a side length not shared by the other. Since transformations do not change side lengths, this is enough to conclude that the two shapes are not congruent.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ANTICIPATED MISCONCEPTIONSStudents may want to visually determine congruence each time or explain congruence by saying, “They look the same.” Encourage those students to explain congruence in terms of translations, rotations, reections, and side lengths. For students who focus on features of the shapes such as side lengths and angles, ask them how they could show the side lengths or angle measures are the same or dierent using the grid or tracing paper.SUPPORT FOR ENGLISH LANGUAGE LEARNERSRepresenting, Conversing, Listening: MLR 7 Compare and Connect.As students prepare their work for discussion, look for approaches that focus on visually determining congruence and on approaches that focus on features of the shapes such as side lengths and angles. Encourage students to explain congruence in terms of translations, rotations, reections, and side lengths and to show physical representations of congruence of side lengths and angle measures using grid or tracing paper. Emphasize transformational language used to make sense of strategies to identify congruent and non-congruent gures.Design Principle(s): Maximize meta-awareness; Support sense-makingSUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer Tutors.Pair students with their previously identied peer tutors.Concept Exploration: Activity 2CONGRUENT PAIRS (PART 2)Instructional Routines: Take Turns; MLR8: Discussion SupportsStudents take turns with a partner claiming that two given polygons are or are not congruent and explaining their reasoning. The partner’s job is to listen for understanding and challenge their partner if their reasoning is incorrect or incomplete. This activity presents an opportunity for students to justify their reasoning and critique the reasoning of others.This activity continues to investigate congruence of polygons on a grid. Unlike in the previous activity, the non-congruent pairs of polygons share the same side lengths.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.150

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHArrange students in groups of 2, and provide access to geometry toolkits. Tell students that they will take turns on each question. For the rst question, Student A should claim whether the shapes are congruent or not. If Student A claims they are congruent, they should describe a sequence of transformations to show congruence, while Student B checks the claim by performing the transformations. If Student A claims the shapes are not congruent, they should support this claim with an explanation to convince Student B that they are not congruent. For each question, students exchange roles.Ask for a student volunteer to help you demonstrate this process using the pair of shapes here.Then, students work through this same process with their own partners on the questions in the activity.ACTIVITY 2 LAUNCHAre the two shapes congruent? 33A BCDEFGHI JK L© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 2 TASK 1For each pair of shapes, decide whether or not shape A is congruent to shape B. Explain how you know.1. 2. 3. 34yxGH IJ KLAPQRSTUByxBP UT SRQG HIJKLAyxGHI JAQRSPB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.152

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:4. 5. STUDENT RESPONSE1. Yes, they are congruent. Sample response: Rotate hexagon A 90 degrees clockwise with center (0, 0), and then translate it 7 units to the right. It matches up perfectly with hexagon B.2. No, they are not congruent. Sample response: Hexagon A has greater area (8 square units) than hexagon B (6 square units). They are not congruent because translations, rotations, and reections do not change the area of a gure.3. Yes, they are congruent. Sample response: Reect quadrilateral A over the y-axis, and then translate one unit to the right and one unit down. It matches up perfectly with quadrilateral B.4. No, they are not congruent. Sample response: Both shapes are quadrilaterals, and the side lengths all appear to be 5 units in length. But the angles are not the same. Quadrilateral B is a square with 4 right angles. Quadrilateral A is a rhombus. Angles G and I are acute while angles H and J are obtuse. Since translations, rotations, and yxAGHJIP QRSByxG HIJPQRSAB© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:reections do not change angle measures, there is no way to match up any of the angles of these quadrilaterals.5. No, they are not congruent. Sample response 1: Rotate quadrilateral B about by 45 degrees counterclockwise and then translate to the le by 7 units. Angle PSR matches up with angle GHI, but the sides of quadrilateral B are a little shorter than those of quadrilateral A, so the two shapes are not congruent. Sample response 2: The area of square quadrilateral A is 9 square units. The area of quadrilateral B (which is also a square) is 8 square units because it contains 4 whole unit squares and then 8 half unit squares that make 4 more unit squares. Congruent shapes have the same area so these two shapes are not congruent.DISCUSSION GUIDANCETo highlight student reasoning and language use, invite groups to respond to the following questions:• “For which shapes was it easiest to give directions to your partner? Were some transformations harder to describe than others?”• “For the pairs of shapes that were not congruent, how did you convince your partner? Did you use transformations or did you focus on some distinguishing features of the shapes?”• “Did you use any measurements (length, area, angle measures) to help decide whether or not the pairs of shapes are congruent?”For more practice articulating why two gures are or are not congruent, select students with dierent methods to share how they showed congruence (or not). If the previous activity provided enough of an opportunity, this may not be necessary.ANTICIPATED MISCONCEPTIONSFor Problem 5, students may be correct in saying the shapes are not congruent but for the wrong reason. They may say one is a 3-by-3 square and the other is a 2-by-2 square, counting the diagonal side lengths as one unit. If so, have them compare lengths by marking them on the edge of a card, or measuring them with a ruler.In discussing congruence for problem 3, students may say that quadrilateral GHIJ is congruent to quadrilateral PQRS, but this is not correct. Aer a set of transformations is applied to quadrilateral GHIJ, it corresponds to quadrilateral QRSP. The vertices must be listed in this order to accurately communicate the correspondence between the two congruent quadrilaterals.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.154

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSListening, Representing: MLR 8 Discussion Supports.During the nal discussion, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.Design Principle(s): Maximize meta-awareness.SUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer Tutors.Pair students with their previously identied peer tutors.Digital LessonIf two shapes are congruent, do they have the same side lengths? If two shapes have the same side lengths, are they denitely congruent? Explain your thinking. Consider drawing shapes to support your answer.STUDENT RESPONSETwo shapes are congruent if there is a sequence of rigid transformations that you can perform to get the shapes to match up. As a result, congruent shapes must have the same side lengths. However, if 2 shapes have the same side lengths, they are not necessarily congruent. Although all the side lengths might be the same, if their angles have dierent measures, then the shapes are not congruent.Not congruent:(4 cm)(4 cm)(4 cm) (4 cm)(4 cm)(4 cm)(4 cm) (4 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. 155

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Wrap-UpLESSON SYNTHESIS The main points to highlight at the conclusion of the lesson are:• Two gures are congruent when there is a sequence of translations, rotations, and reections that match one gure up perfectly with the other (this is from the previous lesson but it is vital to thinking in this lesson as well.).• When showing that two gures are congruent on a grid, we use the structure of the grid to describe each rigid motion. For example, translations can be described by indicating how many grid units to move le or right and how many grid units to move up or down. Reections can be described by indicating the line of reection (an axis or a particular grid line are readily available). • Two gures are not congruent if they have dierent side lengths, dierent angles, or dierent areas.• Even if two gures have the same side lengths, they may not be congruent. With four sides of the same length, for example, we can make many dierent rhombuses that are not congruent to one another because the angles are dierent.EXIT TICKETStudents describe explicit transformations that take one polygon to another. Several solutions are possible. Though students may use tracing paper to help visualize the dierent transformations, at this point they should be able to articulate the move abstractly using the language of translations, rotations, and reection. © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.156

G8M1 | LESSON 12ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:EXIT TICKET Describe a sequence of reections, rotations, and translations that shows that quadrilateral ABCD is congruent to quadrilateral EFGH.STUDENT RESPONSEAnswers vary. Sample response: Translate ABCD down 1 and 5 to the right. Then reect over line GH.A DB CEHFG© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 13CongruenceLEARNING GOALSDetermine whether shapes are congruent by measuring corresponding points.Draw and label corresponding points on congruent figures.Justify (orally and in writing) that congruent figures have equal corresponding distances between pairs of points.LEARNING GOALS(STUDENT FACING)Let’s find ways to test congruence of interesting figures.LEARNING TARGETS(STUDENT FACING)I can use distances between points to decide if two figures are congruent.REQUIRED MATERIALSGeometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor and an index card to use as a straightedge or to mark right anglesTEACHER INSTRUCTION ONLYZEARN MATH TIPThere is no Independent Digital Lesson for this lesson. It is a further exploration of the work done so far in this mission. If you feel your students still need work with these concepts, we recommend you teach it with your whole group or teach it during your concept exploration and give kids digital time to ensure they are on pace with your instruction heading into the next topic.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.158

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:So far, we have mainly looked at congruence for polygons. Polygons are special because they are determined by line segments. These line segments give polygons easily dened distances and angles to measure and compare. For a more complex shape with curved sides, the situation is a little dierent (unless the shape has special properties such as being a circle). The focus here is on the fact that the distance between any pair of corresponding points of congruent gures must be the same. Because there are too many pairs of points to consider, this is mainly a criterion for showing that two gures are not congruent: that is, if there is a pair of points on one gure that are a dierent distance apart than the corresponding points on another gure, then those gures are not congruent.One of the mathematical practices that takes center stage in this lesson is attending to precision. For congruent gures built out of several dierent parts (for example, a collection of circles) the distances between all pairs of points must be the same. It is not enough that the constituent parts (circles for example) be congruent: they must also be in the same conguration, the same distance apart. This follows from the denition of congruence: rigid transformations do not change distances between points, so if gure 1 is congruent to gure 2 then the distance between any pair of points in gure 1 is equal to the distance between the corresponding pair of points in gure 2.Warm-UpNOT JUST THE VERTICESPolygons are special shapes because once we know the vertices, listed in order, we can join them by line segments to produce the polygon. This is important when performing rigid transformations. Because rigid transformations take line segments to line segments, once we track where the vertices of a polygon go, we can join them in the correct order with segments to nd the image of the polygon.In this warm-up, students begin to explore this structure, nding corresponding points in congruent polygons which are not vertices. LAUNCHGive quiet think time followed by a whole-class 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. 159

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASKTrapezoids ABCD and A′B′C′D′ are congruent. • Draw and label the points on A′B′C′D′ that correspond to E and F.• Draw and label the points on ABCD that correspond to G′ and H′.• Draw and label at least three more pairs of corresponding points.STUDENT RESPONSEAnswers vary. Here are some possibilities:DISCUSSION GUIDANCERemind students that when two gures are congruent, every point on one gure has a corresponding point on the other gure.Ask students what methods they used to nd their corresponding points. Possible answers include:• Using the grid and the corresponding vertices to keep track of distances and place the corresponding point in the right place• Using tracing paper and rigid transformations taking one polygon to the other1CBDEAFC'D'G'H'B'A'CB IDEJ G KAFLHMC'D'E' J' G' K'F'L'H'B' I'A'M'© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.160

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ANTICIPATED MISCONCEPTIONSStudents may struggle to nd corresponding points that are not vertices. Suggest that they use tracing paper or the structure of the grid to help identify these corresponding points.Concept Exploration: Activity 1CONGRUENT OVALSMATERIALS: GEOMETRY TOOLKITInstructional Routines: Think Pair Share, MLR5: Co-Cra Questions and ProblemsThis activity begins a sequence that looks at gures that are not polygons. From the point of view of congruence, polygons are special shapes because they are completely determined by the set of vertices. For curved shapes, we usually cannot check that they are congruent by examining a few privileged points, like the vertices of polygons. We can ascertain that they are not congruent by identifying a feature of one shape not shared by the other (for example, this oval is 3 units wide while this one is only 221 units wide). But to show that two curved shapes are congruent, we need to apply the denition of congruence and try to move one shape so that it matches up exactly with the other aer some translations, rotations, and reections.In this activity, students begin to explore the subtleties of congruence for curved shapes. Make sure that students provide a solid mathematical argument for the shapes which are congruent, beyond saying that they look the same. Providing a viable argument requires careful thinking about the meaning of congruence and the structure of the shapes. Monitor for groups who use precise language of transformations as they attempt to move one traced oval to match up perfectly with another. Invite them to share their reasoning during the discussion. Also monitor for arguments based on measurement for why neither of the upper ovals can be congruent to either of the lower ones.LAUNCHArrange students in groups of 2. Provide access to geometry toolkits. Give students quiet work time, then invite them to share their reasoning with a partner, followed by a whole-class 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. 161

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 1Are any of the ovals congruent to one another? Explain how you know.STUDENT RESPONSESample response: All four shapes are ovals. For the top two shapes, they can be surrounded by rectangles that measure 3 units by 2 units. For the bottom two shapes, they measure about 221 (possibly a little less) units by 2 units. This means that the top shapes are not congruent to the bottom shapes.The top two shapes and the bottom two shapes have the same width and height (though the orientation is dierent). More is needed, however, to determine whether or not the two upper (and two lower) ovals are congruent. For this, we can trace one of the upper two ovals on tracing paper and check that it can be placed on top of the other and similarly for the lower pair of ovals. This can be done with, for example, a 90 degree clockwise rotation with center at the point shown here:DISCUSSION GUIDANCEInvite groups to explain how they determined that the upper ovals are not congruent to the lower ones, with at least one explanation focusing on diering measurable attributes 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.162

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:(for example length and width). Also invite previously selected groups to show how they demonstrated that the two upper (and two lower) ovals are congruent, focusing on the precise language of transformations.Emphasize that showing that two oval shapes are congruent requires using the denition of congruence: is it possible to move one shape so that it matches up perfectly with the other using only rigid transformations? Experimentation with transformations is essential when showing that two of the ovals match up because, unlike polygons, these shapes are not determined by a nite list of vertices and side lengths.Students have seen that rectangles that have the same side lengths are congruent and will later nd criteria for determining whether two triangles are congruent. For more complex curved shapes, the denition of congruence is required.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: Math Language Routine 5 Co-Cra QuestionsThis is the rst time Math Language Routine 5 is suggested as a support in this course. In this routine, students are given a context or situation, oen in the form of a problem stem with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?” The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.Design Principle(s): Cultivating Conversations; Maximize meta-awarenessHow It Happens:1. Display the four images of the ovals without the directions. Ask students, “What mathematical questions could you ask about this situation?”2. Give students individual time to jot some notes, and then time to share ideas with a partner. As pairs discuss, support students in using conversation skills to generate and rene their questions collaboratively by seeking clarity, referring to students’ written notes, and revoicing oral responses, as necessary. Listen for how students use language about transformations and/or refer to measurements of the gures when talking about the curved polygons.3. Ask each pair of students to contribute one written question to a poster, the whiteboard, or digital projection. Call on 2–3 pairs of students to present their question to the whole class, and invite the class to make comparisons among the questions shared and their own questions. Listen for questions comparing dierent features of the ovals to determine congruence, especially those that use distances between corresponding points. Revoice student ideas with an emphasis on measurements wherever it serves to clarify a question.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:4. Reveal the question, “Are any of the ovals congruent to one another?” and give students time to compare it to their own question and those of their classmates. Identify similarities and dierences. Consider providing these prompts: “Which of your questions is most similar to/dierent from the one provided? Why?”, “Is there a main mathematical concept that is present in both your questions and the one provided? If so, describe it.”, and “How do your questions relate to one of the lesson goals of measuring corresponding points to determine congruence?”5. Invite students to choose one question to answer (from the class or from the curriculum), and then have students move on to the discussion.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the image to students who benet from extra processing time to review prior to implementation of this activity.Fine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allow students who struggle with ne motor skills to dictate shape overlay as needed.Concept Exploration: Activity 2CORRESPONDING POINTS IN CONGRUENT FIGURESMATERIALS: GEOMETRY TOOLKITInstructional Routines: MLR7: Compare and Connect, MLR8: Discussion SupportsCorresponding sides of congruent polygons have the same length. For shapes like ovals, examined in the previous activity, there are no “sides.” However, if points A and B on one gure correspond to points A′ and B′ on a congruent gure, then the length of segment AB is equal to the length of segment A′B′ because translations, rotations, and reections do not change distances between points. Students have seen and worked with this idea in the context of polygons and their sides. This remains true for other shapes as well.Because rigid transformations do not change distances between points, corresponding points on congruent gures (even oddly shaped gures!) are the same distance apart. This is one more good example of looking for and making use of structure, as the fundamental mathematical property of rigid transformations is that they do not change distances between corresponding points. This idea holds for any points on any congruent gures.There are two likely strategies for identifying corresponding points on the two corresponding gures: © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.164

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• Looking for corresponding parts of the gures such as the line segments• Performing rigid transformations with tracing paper to match the gures upBoth are important. Watch for students using each technique and invite them to share during the discussion.LAUNCHKeep students in the same groups. Provide quiet work time followed by sharing with a partner and a whole-class discussion. Provide access to geometry toolkits (rulers are needed for this activity).ACTIVITY 2 TASK 1Here are two congruent shapes with some corresponding points labeled.1. Draw the points corresponding to B, D, and E, and label them B′, D′, and E′.2. Draw line segments AD and A′D′ and measure them. Do the same for segments BC and B′C′ and for segments AE and A′E′. What do you notice?3. Do you think there could be a pair of corresponding segments with dierent lengths? Explain.33A'C'CBAED© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. 2. The lengths are the same. The rigid transformations of the plane used to show the congruence of these shapes do not change distances between points. So, the distance between A and D, for example, is the same as the distance between A′ and D′. The same is true for the other pairs of corresponding points. The segments connecting these points are all shown here:3. Sample response: No, rigid transformations do not change distances between points. Corresponding segments in the two congruent gures must have the same length.A'C'D'E'B'CBAEDA'C'D'E'B'CBAED© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.166

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEAsk selected students to show how they determined the points corresponding to B, D, and E, highlighting dierent strategies (identifying key features of the shapes and performing rigid transformations). Ask students if these strategies would work for nding C′ if it had not been marked. Performing rigid transformations matches the shapes up perfectly, and so this method allows us to nd the corresponding point for any point on the gure. Identifying key features only works for points such as A, B, D, and E, which are essentially like vertices and can be identied by which parts of the gures are “joined” at that point.While it is challenging to test “by eye” whether or not complex shapes like these are congruent, the mathematical meaning of the word “congruent” is the same as with polygons: two shapes are congruent when there is a sequence of translations, reections, and rotations that match up one shape exactly with the other. Because translations, reections, and rotations do not change distances between points, any pair of corresponding segments in congruent gures will have the same length.If time allows, have students use tracing paper to make a new gure that is either congruent to the shape in the activity or slightly dierent. Display several for all to see and poll the class to see if students think the gure is congruent or not. Check to see how the class did by lining up the new gure with one of the originals. Work with these complex shapes is important because we tend to rely heavily on visual intuition to check whether or not two polygons are congruent. This intuition is usually reliable unless the polygons are complex or have very subtle dierences that cannot be easily seen. The meaning of congruence in terms of rigid transformations and our visual intuition of congruence can eectively reinforce one another:• If shapes look congruent, then we can use this intuition to nd the right motions of the plane to demonstrate that they are congruent.• Through experimenting with rigid transformations, we increase our visual intuition about which shapes are congruent.SUPPORT FOR ENGLISH LANGUAGE LEARNERSRepresenting, Conversing, Listening: MLR 7 Compare and ConnectAs students work, look for students who perform rigid transformations with tracing paper to test congruence of two gures. Call students’ attention to the dierent ways they match up gures to identify corresponding points, and to the dierent ways these operations are made visible in each representation (e.g., lengths of segments that are equal, translations, rotations and reections do not change distances between points). Emphasize and amplify the mathematical language students use when determining if two gures are congruent.Design Principle(s): Maximize meta-awareness; Support sense-making© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESReceptive/Expressive Language: Processing TimeStudents who benet from extra processing time would also be aided by MLR 8 (Discussion Supports). Use “think aloud modeling” to make visual intuition more concrete.Wrap-UpLESSON SYNTHESISThis lesson wraps up work on congruence. Important points to highlight include:• Two gures are congruent when there is a sequence of translations, rotations, and reections matching up one gure with the other.• To show that two gures are not congruent it is enough to nd corresponding points on the gures which are not the same distance apart, or corresponding angles that have dierent measures.• The distance between pairs of corresponding points in congruent gures is the same (this says that corresponding sides on polygons have the same length but it also applies to curved gures or to any pair of points, not necessarily vertices, on polygons).• Some gures are made up of several parts. For example, these two designs are each made up of three circles:All six of the circles are congruent (as we could check using tracing paper). But in the le design, each circle touches both of the other two, but this is not true in the design on the right. The distances between any two circle centers in one design will be dierent than the distances between any two circle centers in the other design.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.168

G8M1 | LESSON 13ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:EXIT TICKETStudents decide whether or not two ovals are congruent. These particular ovals are visually quite distinct so expect students to use one of these methods:• Identify a distance on one oval that is dierent than the corresponding distance on the other oval.• Trace one of the ovals and observe that it does not match up with the other one.LAUNCHMake tracing paper available.TASKAre Figures A and B congruent? Explain your reasoning.STUDENT RESPONSEAnswers vary. Sample response: These gures are not congruent because if they were, the longest horizontal distances between two points would be the same. However, for A it is less than 4 units, and for B it is about 4 units.Figure A Figure 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. 169

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 14Alternate Interior AnglesLEARNING GOALSCalculate angle measures using alternate interior, adjacent, vertical, and supplementary angles to solve problems.Justify (orally and in writing) that “alternate interior angles” made by a “transversal” connecting two parallel lines are congruent using properties of rigid motions.LEARNING GOALS(STUDENT FACING)Let’s explore why some angles are always equal.LEARNING TARGETS(STUDENT FACING)If I have two parallel lines cut by a transversal, I can identify alternate interior angles and use that to find missing angle measurements.REQUIRED MATERIALSGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesREQUIRED PREPARATION Students need rulers and tracing paper from the geometry toolkits.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.170

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, students justify that alternate interior angles are congruent, and use this and the vertical angle theorem, previously justied, to solve problems.Thus far in this mission, students have studied dierent types of rigid motions, using them to examine and build dierent gures. This work continues here, with an emphasis on examining angles. In a previous lesson, 180 degree rotations were employed to show that vertical angles, made by intersecting lines, are congruent. The warm-up recalls previous facts about angles made by intersecting lines, including both vertical and adjacent angles. Next a third line is added, parallel to one of the two intersecting lines. There are now 8 angles, 4 each at the two intersection points of the lines. At each vertex, vertical and adjacent angles can be used to calculate all angle measures once one angle is known. But how do the angle measures compare at the two vertices? It turns out that each angle at one vertex is congruent to the corresponding angle (via translation) at the other vertex and this can be seen using rigid motions: translations and 180 degree rotations are particularly eective at revealing the relationships between the angle measures.Students will notice as they calculate angles that they are repeatedly using vertical and adjacent angles and that oen they have a choice which method to apply. They will also notice that the angles made by parallel lines cut by a transversal are the same and this observation is the key structure in this lesson.Warm-UpANGLE PAIRSThis task is designed to prompt students to recall their prior work with supplementary angles. While they have seen this material in grade 7, this is the rst time it has come up explicitly in grade 8. As students work on the task, listen to their conversations specically for the use of vocabulary such as supplementary and vertical angles. If no students use this language, make those terms explicit in the discussion.Some students may wish to use protractors, either to double check work or to investigate the dierent angle measures. This is an appropriate use of technology, but ask these students what other methods they could use instead.LAUNCHProvide access to geometry toolkits. Before students start working, make sure they are familiar with the convention for naming an angle using three points, where the middle letter denotes the angle’s vertex.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASKUse the angle below to complete the questions. 1. Find the measure of angle JGH. Explain or show your reasoning.2. Find and label a second 30° degree angle in the diagram. Find and label an angle congruent to angle JGH.STUDENT RESPONSE1. 150°. Sample response: In the diagram, the given 30° angle and angle JGH are supplementary, so they add up to 180°.2. Angles are labeled as shown using reasoning about vertical or supplementary angles.DISCUSSION GUIDANCEDisplay the image for all to see. Invite students to share their responses, adding onto the image as needed to help make clear student thinking. If not mentioned by students, make sure to highlight the term supplementary angles to describe, for example, angles FGJ and JGH, and vertical angles to describe, for example, angles JGF and HGI.1JFHGI30°JFHGI30°30°150°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.172

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Processing TimeProvide the image to students who benet from extra processing time to review prior to implementation of this activity.Fine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allowing students who struggle with ne motor skills to dictate the use of protractors as needed.Concept Exploration: Activity 1CUTTING PARALLEL LINES WITH A TRANSVERSALInstructional Routines: Think Pair Share; MLR2: Collect and DisplayIn this task, students explore the relationship between angles formed when two parallel lines are cut by a transversal line. Students investigate whether knowing the measure of one angle is suicient to gure out all the angle measures in this situation. They also consider whether the relationships they found hold true for any two lines cut by a transversal. The last two questions in this activity are optional, to be completed if time allows. Make sure to leave enough time for the next activity, “Alternate Interior Angles are Congruent.”As students work with their partners, they begin to ll in supplementary angles and vertical angles. To nd the measures of corresponding and alternate interior, students may use tracing paper and some of the strategies found earlier in the mission. For example, they may use tracing paper to translate vertex B to vertex E. They might try to translate B′ to E′ in the third picture and observe that the angles at those two vertices are not congruent. Similarly, to nd measures of vertical angles students may use a 180° rotation like they did earlier in this mission when showing that vertical angles are congruent. Monitor for students who use these dierent strategies and select them to share during the discussion.For students who nish early, ask them to think of dierent methods they could use to determine the angles: For example, all of the congruent angles can be shown to be congruent with transformations.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHA transversal (or transversal line) for a pair of parallel lines is a line that meets each of the parallel lines at exactly one point. Introduce this idea and provide a picture such as this picture where line k is a transversal for parallel lines ℓ and m:Arrange students in groups of 2. Provide access to geometry toolkits. Give students quiet think time to plan on how to nd the angle measures in the picture then time to share their plan with their partner. Give partners time for the rest of the task, followed by a discussion. Instruct students to stop aer the third question if you’ve decided to skip the last two questions.ACTIVITY 1 TASK 1Lines AC and DF are parallel. They are cut by transversal HJ. Use this diagram to answer the questions.1. With your partner, nd the seven unknown angle measures in the diagram. Explain your reasoning.2. What do you notice about the angles with vertex B and the angles with vertex E?mk2AJCBFEHD63°???????© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.174

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:3. Using what you noticed, nd the measures of the four angles at point B in the second diagram. Lines AC and DF are parallel.4. The next diagram resembles the rst one, but the lines form slightly dierent angles. Work with your partner to nd the six unknown angles with vertices at points B and E.5. What do you notice about the angles in this diagram as compared to the earlier diagram? How are the two diagrams dierent? How are they the same?34°????GFEDAHBC63°108°??????ABCJFHED© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license. 175

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. Explanations vary. Sample strategy 1: Tracing paper helped nd the three 117 degree angles. Each of the other four angles is supplementary to a 117 degree angle, so they are all 63 degree angles. Sample strategy 2: Using pairs of vertical angles shows that angle CBJ is a 63 degree angle. The other angles at vertex B can be found using supplementary angles. The angles at vertex E can be found the same way aer using tracing paper to nd one of them.2. Answers vary. Sample response: The angles in the same place relative to the transversal have the same measure.3. Answers vary. Sample response: Angle ABH is a 34 degree angle because it forms a vertical pair with the marked 34 degree angle aer translating E to B. Angle HBC is a 146 degree angle because it is supplementary to the 34 degree angle found by translating E to B.4. 5. Answers vary. Sample response: In both pictures, the two pair of vertical angles at each vertex are congruent. Also adjacent angles at each vertex are supplementary. In the rst picture, the angle measures at the two vertices are the same while in the second picture they are dierent.AJCBFEHD63°63°63°63°117°117°117°117°63°72°72°63°117°117°108°108°ABCJFHED© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.176

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 RECAPThe purpose of this discussion is to make sure students noticed relationships between the angles formed when two parallel lines are cut by a transversal and to introduce the term alternate interior angles to students. Display the images from the Task Statement for all to see one at a time and invite groups to share their responses. Encourage students to use precise vocabulary, such as supplementary and vertical angles, when describing how they gured out the dierent angle measurements. Aer students point out the matching angles at the two vertices, dene the term alternate interior angles and ask a few students to identify some pairs of angles from the activity.Consider asking some of the following questions:• “What were some tools you used to nd or conrm angle measures?” (Tracing paper, protractor, transformations)• “What were some angle relationships you used to nd missing measures?” (Vertical angles, supplementary angles)• “What do you notice about the angles at vertex B compared to the angles at vertex E?” (They have the same angle measures for angles in the same position relative to the transversal.)• “Which angle relationships were true for all three pictures? Which were true for only one or two of the pictures?” (Congruent vertical and supplementary angles around a vertex were always true. Congruent angles in corresponding positions at the two vertices were only true in the rst two pictures, which had parallel lines.)ANTICIPATED MISCONCEPTIONSIn the rst image, students may ll in congruent angle measurements based on the argument that they look the same size. Ask students how they can be certain that the angles don’t dier in measure by 1 degree. Encourage them a way to explain how we can know for sure that the angles are exactly the same measure.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversing, Representing: MLR 2 Collect and DisplayAs students work, listen for and collect the language students use to describe the relationships they notice between the angles formed when two parallel lines are cut by a transversal. Record students’ words and phrases onto a visual display and update it throughout the lesson. It may also be helpful include sketches or diagrams. Pay attention to how students use mathematical language when they work together to determine angle congruence. This will help students use mathematical language during discussions.Design Principle: Support sense-makingConcept Exploration: Activity 2ALTERNATE INTERIOR ANGLES ARE CONGRUENTInstructional Routines: MLR8: Discussion SupportsThe goal of this task is to experiment with rigid motions to help visualize why alternate interior angles (made by a transversal connecting two parallel lines) are congruent. This result will be used in a future lesson to establish that the sum of the angles in a triangle is 180 degrees. The second question is optional if time allows. This provides a deeper understanding of the relationship between the angles made by a pair of (not necessarily parallel) lines cut by a transversal.Expect informal arguments as students are only beginning to develop a formal understanding of parallel lines and rigid motions. They have, however, studied the idea of 180 degree rotations in a previous lesson where they used this technique to show that a pair of vertical angles made by intersecting lines are congruent. Consider recalling this technique especially to students who get stuck and suggesting the use of tracing paper.Given the diagram, the tracing paper, and what they have learned in this mission, students should be looking for ways to demonstrate that alternate interior angles are congruent using transformations. Make note of the dierent strategies (including dierent transformations) students use to show that the angles are congruent and invite them to share their strategies during the discussion. Approaches might include:• A 180 degree rotation about M• First translating P to Q and then applying a 180-degree rotation with center Q© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.178

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHProvide access to geometry toolkits. Tell students that in this activity, we will try to gure out why we saw all the matching angles we did in the last activity.ACTIVITY 2 TASK 1Lines l and k are parallel and t is a transversal. Point M is the midpoint of segment PQ. Find a rigid transformation showing that angles MPA and MQB are congruent.STUDENT RESPONSERotate the picture 180° with center MSince 180 degrees is half of a circle this takes each point on the circle to its “opposite.” Point A maps to point B and B maps to A. So the 180 degree rotation will interchange P and Q. The rotation interchanges lines ℓ and m and also angles MQB and MPA so the angles are congruent.33PQMBAtkPQMBAtk© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 2 TASK 2In this picture, lines l and k are no longer parallel. M is the still the midpoint of segment PQ. Does your argument in the earlier problem apply in this situation? Explain.STUDENT RESPONSEIf ℓ and m are not parallel, a 180 degree rotation around M still takes P to Q and Q to P. The problem is that it does not take ℓ to m, and it does not take m to ℓ because m is not parallel to ℓ. So this rotation does not take angle MQB to angle MPA and vice versa. The argument from 1 does not apply unless ℓ and m are parallel.DISCUSSION GUIDANCESelect students to share their explanations. Pay close attention to which transformations students use in the rst question and make sure to highlight dierent possibilities if they arise. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent.Highlight the fact that students are using many of the transformations from earlier sections of this mission. The argument here is especially close to the one used to show that vertical angles made by intersecting lines are congruent. In both cases a 180 degree rotation exchanges pairs of angles. For vertical angles, the center of rotation is the common point of intersecting lines. For alternate interior angles, the center of rotation is the midpoint of the transverse between the two parallel lines. But the structure of these arguments is identical.34PQMBAtk© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.180

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR STUDENTS WITH DISABILITIESSpeaking: MLR 8 Discussion SupportsUse this routine to amplify students’ mathematical uses of language when describing and demonstrating transformations used for showing alternate interior angles are congruent. Aer a student shares their response, invite another student to repeat the reasoning using mathematical language (e.g., “vertical angles”, “180 degree rotation”, “center of rotation”, “intersecting lines”, “midpoint”, “parallel lines”, etc.). Invite all students to chorally repeat the phrases that include these words in context.Design Principles: Support sense-making, Optimize output (for explanation)Digital LessonLines m and n are parallel, and the measure of angle b is 27 degrees.a) Explain why the measure of angle f is 27 degrees.b) What is the measure of angle h? Explain or show your thinking.STUDENT RESPONSEa) Angle b and angle c are vertical angles, so they are congruent. This means that the measure of angle c is 27 degrees. Angle c and angle f are alternate interior angles, so they are congruent. This means that the measure of angle f is 27 degrees.b) The measure of angle h is 153 degrees. Angle f and angle h are supplementary angles, so their measures add up to 180 degrees.bamncefgdh© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Wrap-UpLESSON SYNTHESISDisplay the image of two parallel lines cut by a transversal. Tell students that in cases like this, translations and rotations can be particularly useful in guring out angle measurements since they move angles to new positions, but the angle measure does not change.Select students to point out examples of alternate interior, vertical, and supplementary angles in the picture. They should also be able to articulate which angles are congruent to one another and give an example of a rigid transformation that explains why.In particular, make sure students can articulate:• c = 60 because it is the measure of an angle forming an alternate interior angle with the given 60 degree angle.• e = d = 120 because they are also alternate interior angles, each supplementary to a 60 degree angle.• The rest of the angle measures can be found using vertical or supplementary angles.TERMINOLOGYAlternate interior angles Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.This diagram shows two pairs of alternate interior angles. Angles a and d are one pair and angles b and c are another pair.b°a°c°e°60°f°g°d°abcdtransversal© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.182

G8M1 | LESSON 14ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TransversalA transversal to two parallel lines is a line that cuts across them, intersecting each one.This diagram shows a transversal line k intersecting parallel lines m and ℓ.EXIT TICKETThe diagram shows two parallel lines cut by a transversal. One angle measure is shown.Find the values of a, b, c, d, e, f, and g.STUDENT RESPONSEa: 126, b: 54, c: 126, d: 54, e: 126, f: 54, g: 126mke°d°a°b°c°g°f°54°© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 15Adding the Angles in a TriangleLEARNING GOALSComprehend that a straight angle can be decomposed into 3 angles to construct a triangle.Justify (orally and in writing) that the sum of angles in a triangle is 180 degrees using properties of rigid motions.LEARNING GOALS(STUDENT FACING)Let’s explore angles in triangles.LEARNING TARGETS(STUDENT FACING)If I know two of the angle measures in a triangle, I can find the third angle measure.REQUIRED MATERIALSTemplate for Concept Exploration: Activity 1Template for Concept Exploration: Activity 2Copies of templateGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right anglesPre-printed slips, cut from copies of the templateREQUIRED PREPARATIONPrepare 1 copy of the Activity 1 template for every 15 students. Cut these up ahead of time.Print copies of the Activity 2 template. Prepare 1 copy for every group of 4 students. From the geometry toolkit, students will need scissors.ZEARN MATH TIPThe digital lesson focuses on Activity 2 in this lesson, so we recommend focusing the class time on Activity 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.184

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this lesson, the focus is on the interior angles of a triangle. What can we say about the three interior angles of a triangle? Do they have special properties?The lesson opens with an activity looking at dierent types of triangles with a particular focus on the angle combinations of specic acute, right, and obtuse triangles. Aer being given a triangle, students form groups of 3 by identifying two other students with a triangle congruent to their own. Aer collecting some class data on all the triangles and their angles, they nd that the sum of the angle measures in all the triangles turns out to be 180 degrees.In the next activity, students observe that if a straight angle is decomposed into three angles, it appears that the three angles can be used to create a triangle. Together the activities provide evidence of a close connection between three positive numbers adding up to 180 and having a triangle with those three numbers as angle measures.A new argument is needed to justify this relationship between three angles making a line and three angles being the angles of a triangle. This is the topic of the following lesson.Warm-UpCAN YOU DRAW IT?Students try to draw triangles satisfying dierent properties. They complete the table and then check with a partner whether or not they agree that the pictures are correct or that no such triangle can be drawn. The goals of this warm-up are:• Reviewing dierent properties and types of triangles.• Focusing on individual angle measures in triangles in preparation for studying their sum.Note that we use the inclusive denition of isosceles triangle having at least two congruent sides. It is possible that in students’ earlier experiences, they learned that an isosceles triangle has exactly two congruent sides. This issue may not even come up, but be aware that students may be working under a dierent denition of isosceles than what is written in the task statement.LAUNCHArrange students in groups of 2. Give quiet work time to complete the table followed by partner and whole-class 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. 185

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:WARM-UP TASKComplete the questions below. 1. Complete the table by drawing a triangle in each cell that has the properties listed for its column and row. If you think you cannot draw a triangle with those properties, write “impossible” in the cell.2. Share your drawings with a partner. Discuss your thinking. If you disagree, work to reach an agreement.Acute (all angles acute)Right (has a right angle)Obtuse (has an obtuse angle)Scalene (side lengths all dierent)Isosceles (at least two side lengths are equal)Equilateral (three side lengths equal)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.186

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAcute (all angles acute)Right (has a right angle)Obtuse (has an obtuse angle)Scalene (side lengths all dierent)Isosceles (at least two side lengths are equal)Equilateral (three side lengths equal)Impossible ImpossibleWARM-UP RECAPInvite students to share a few triangles they were able to draw such as:• Right and isosceles• Equilateral and acuteAsk students to share which triangles they were unable to draw and why. For example, there is no right equilateral or obtuse equilateral triangle because the side opposite the right (or obtuse) angle is longer than either of the other two sides. SUPPORT FOR STUDENTS WITH DISABILITIESExecutive Functioning: Visual AidsCreate an anchor chart (i.e., the triangle table) publicly displaying important denitions, rules, formulas or concepts for future reference.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Concept Exploration: Activity 1FIND ALL THREEMATERIALS: TEMPLATE FOR CONCEPT EXPLORATION: ACTIVITY 1, GEOMETRY TOOLKITSInstructional Routines: MLR8: Discussion SupportsThis is a matching activity where each student receives a card showing a triangle and works to form a group of three. Each card has a triangle with the measure of only one of its angles given. Students use what they know about transformations and estimates of angle measures to nd partners with triangles congruent to theirs. Each unique triangle’s three interior angles are then displayed for all to see. Students notice that the sum of the measures of the angles in each triangle is 180 degrees.During this activity, students can think about applying rigid motions to their triangle to see if it might match up with another student’s triangle. Or they may identify that their triangle is acute, right, or obtuse and use this as a criterion when they search for a congruent copy.You will need the Activity 1 template for this activity.LAUNCHProvide access to geometry toolkits. Distribute one card to each student, making sure that all three cards have been distributed for each triangle. (If the number of students in your class is not a multiple of three, it’s okay for one or two students to take ownership over two cards showing congruent triangles.) Explain that there are two other students who have a triangle congruent to theirs that has been re-oriented in the coordinate plane through combinations of translations, rotations, and reections. Instruct students to look at the triangle on their card and estimate the measures of the other two angles. With these estimates and their triangle in mind, students look for the two triangles congruent to theirs with one of the missing angles labeled.Prepare and display a table for all to see with columns angle 1, angle 2, angle 3 and one row for each group of three students. It should look something like this:Student Groups Angle 1 Angle 2 Angle 3Once the three partners are together, they complete one row in the posted table for their triangle’s angle measures. Discussion to follow.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.188

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Students might ask if they can use tracing paper to nd congruent triangles. Ask how they would use it and listen for understanding of transformations to check for congruence. Respond that this seems to be a good idea.ACTIVITY 1 TASK 1Your teacher will give you a card with a picture of a triangle. 1. The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.2. Find two other students with triangles congruent to yours but with a dierent angle labeled. Conrm that the triangles are congruent, that each card has a dierent angle labeled, and that the angle measures make sense.3. Enter the three angle measures for your triangle on the table your teacher has posted.STUDENT RESPONSEThe angle combinations are: 40, 50, 90; 40, 60, 80; 50, 50, 80; 20, 20, 140; 20, 40, 120DISCUSSION GUIDANCEBegin the discussion by asking students:• “What were your thoughts as you set about to nd your partners?”• “How did you know that you found a correct partner?”Expect students to discuss estimates for angle measures and their experience of how dierent transformations inuence the position and appearance of a polygon.Next look at the table of triangle angles and ask students:• “Is there anything you notice about the combinations of the three angle measures?”• “Is there something in common for each row?”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. 189

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Guide students to notice that the sum in each row is the same, 180 degrees. Ask whether they think this is always be true for any triangle. Share with students that in the next activity, they will work toward considering whether this result is true for all triangles.SUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: Eliminate BarriersDemonstrate the steps for the activity or game, in which a group of students and sta play an example round while the rest of the class observes.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion SupportsUse this routine to support group discussion. Display sentence frames for students to use when they discuss the relationships between the three angles. For example, “ We know that our triangles are congruent because . . .” and amplify mathematical language use in the students’ explanations (e.g., transformations, types of triangles, estimates of angle measures, etc.). This will help students to explain their reasoning about angle combinations of acute, right and obtuse triangles.Design Principle(s): Support sense-making; Optimize output (for explanation)Concept Exploration: Activity 2TEAR IT UPMATERIALS: TEMPLATE FOR CONCEPT EXPLORATION: ACTIVITY 2, GEOMETRY TOOLKITSInstructional Routines: MLR7: Compare and Connect, Group PresentationsIn Activity 1, students found that the sum of the angles of all the triangles on the cards was 180 degrees and questioned if all triangles have the same angle sum. In this activity, students experiment with the converse: If we know the measures of three angles sum to 180 degrees, can these three angles be the interior angles in a triangle?Students cut out three angles that form a line, and then try to use these three angles to make a triangle. Students also get to create their own three angles from a line and check whether they can construct a triangle with their angles. Watch for students who successfully make triangles out of each set of angles and select them to share (both the nished product and how they worked to arrange the angles) during the © 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.190

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:discussion. Watch also for how students divide the blank line into angles. It is helpful if the rays all have about the same length as in the pre-made examples.LAUNCHArrange students in groups of 4. Provide access to geometry toolkits, especially scissors. Distribute 1 copy of the template to each group.Instruct students to cut the four individual pictures out of the template. Each student will work with one of these. Instruct the student with the blank copy to use a straightedge to divide the line into three angles (dierent from the three angles that the other students in the group have). Demonstrate how to do this if needed. If needed, you may wish to demonstrate “making a triangle” part of the activity so students understand the intent. With three cut-out 60 degree angles, for example, you can build an equilateral triangle. Here is a picture showing three 60 degree angles arranged so that they can be joined to form the three angles of an equilateral triangle. The students will need to arrange the angles carefully, and they may need to use a straightedge in order to add the dotted lines to join the angles and create a triangle.ACTIVITY 2 TASK 1Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?STUDENT RESPONSEAnswers vary. Sample response: We were all able to build triangles with the given sets of angles. One is a right triangle, one acute, and one obtuse. The three angles we chose also made a triangle.60º60º 60º33© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEIf time allows, have students do a “gallery walk” at the start of the discussion. Ask students to compare the triangle they made to the other triangles made from the same angles and be prepared to share what they noticed. (For example, students might notice that all the other triangles made with their angles looked pretty much the same, but were dierent sizes.) If students do not bring it up, direct students to notice that all of the “create three of your own angles” students were able to make a triangle, not just students with the ready-made angles.Ask previously selected students to share their triangles and explain how they made the triangles. To make the triangles, some trial and error is needed. A basic method is to line up the line segments from two angles (to get one side of the triangle) and then try to place the third angle so that it lines up with the rays coming from the two angles already in place. Depending on the length of the rays, they may overlap, or the angles may need to be moved further apart. Ask questions to make sure that students see the important connection:• “How do you know the three angles you were given sum to 180 degrees?” (They were adjacent to each other along a line.)• “How do you know these can be the three angles of a triangle?” (We were able to make a triangle using these three angles.)• “What do you know about the three angles of the triangle you made and why?” (Their measures sum to 180 because they were the same three angles that made a line.)Ask students if they think they can make a triangle with any set of three angles that form a line and poll the class for a positive or negative response. Tell them that they will investigate this in the next lesson and emphasize that while experiments may lead us to believe this statement is true, the methods used are not very accurate and were only applied to a few sets of angles. If time permits, perform a demonstration of the converse: Start with a triangle, tear o its three corners, and show that these three angles when placed adjacent each other sum to a line.SUPPORT FOR STUDENTS WITH DISABILITIESFine Motor Skills: Peer TutorsPair students with their previously identied peer tutors and allowing students who struggle with ne motor skills to dictate cutting triangles 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.192

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSRepresenting, Conversing, Listening: MLR 7 Compare and ConnectAs students prepare their work for discussion, look for those who successfully construct triangles out of each set of angles and for those who successfully create their own three angles from a line and create triangles. Encourage students to explain how they worked to arrange the angles. Emphasize language used to make sense of strategies used to nd that the sum of the angles in a triangle is 180 degrees.Design Principle(s): Maximize Meta-awareness, Support sense-makingDigital LessonExplain what you observed about the angle measurements of a triangle made from a straight line.STUDENT RESPONSEThe angle measurements of a triangle made from a straight line always sum to 180. The angle measures of a triangle were the same three angles that made a straight line, and the angle formed by a straight line is always 180 degrees. In this example, the angles that are labeled with the same shapes are the same measurements, so we can see that the three angles of the triangle are the same measures as the three angles that form a straight line.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Wrap-UpLESSON SYNTHESISSome guiding questions for the discussion include:• “What did we observe about the sum of the angles inside a triangle?” (The sum of the angles inside a triangle seem to always add up to 180 degrees.)• “When you know two angles inside a triangle, how can you nd the third angle?” (If all three angles add up to 180 degrees, then subtracting two of the angles from 180 will give the measure of the third angle.)• “Are there pairs of angles that cannot be used to make a triangle?” (Yes. If the two angles are both bigger than or equal to 90 degrees, then you cannot make a triangle.)Emphasize that we were able to see for multiple triangles that the sum of their angles is 180° and that using several sets of three angles adding to 180° we were able to build triangles with those angles. In the next lesson we will investigate and explain this interesting relationship.TERMINOLOGYStraight angle A straight angle is an angle that forms a straight line. It measures 180 degrees.EXIT TICKET Students have experimented to see that the sum of the angles in a triangle is 180 degrees. While they will prove this in the next lesson, here they apply the concept in order to give examples of dierent kinds of triangles with one given angle measure.LAUNCHIf needed, tell students to use their conjecture that the sum of the angles in a triangle is always 180 degrees as they work on these problems.straight 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.194

G8M1 | LESSON 15ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:TASKIn triangle ABC, the measure of angle B is 50 degrees. 1. Give possible values for the measures of angles A and C if ABC is an acute triangle.2. Give possible values for the measures of angles A and C if ABC is an obtuse triangle.3. Give possible values for the measures of angles A and C if ABC is a right triangle.STUDENT RESPONSEAnswers vary. Sample responses:1. To make an acute triangle, the other two angles must measure less than 90 degrees (for example: 60, 70).2. To make an obtuse triangle, one of the two angles must be greater than 90 degrees (for example: 100, 30).3. There is only one way to make a right triangle (90, 40).© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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 15 CONCEPT EXPLORATION: ACTIVITY 1PAGE 1 OF 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.196

TEMPLATE FOR LESSON 15 CONCEPT EXPLORATION: ACTIVITY 2PAGE 1 OF 2Use a straight edge to create three of your own 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. 197

TEMPLATE FOR LESSON 15 CONCEPT EXPLORATION: ACTIVITY 2PAGE 2 OF 2Use a straightedge to create your own 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.198

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 16Parallel Lines and the Angles in a TriangleLEARNING GOALSCreate diagrams using 180-degree rotations of triangles to justify (orally and in writing) that the measure of angles in a triangle sum up to 180 degrees.Generalize the Triangle Sum Theorem using rigid transformations or the congruence of alternate interior angles of parallel lines cut by a transversal.Apply known angle relationships to solve missing angle problems involving exterior angles of a triangle.LEARNING GOALS(STUDENT FACING)Let’s see why the angles in a triangle add to 180 degrees and use that fact to solve missing angle problems.LEARNING TARGETS(STUDENT FACING)I can explain using pictures why the sum of the angles in any triangle is 180 degrees.I can use the fact that the sum of the interior angles of any triangle is 180 degrees to solve missing angle problems.REQUIRED MATERIALSGeometry toolkits: tracing paper, graph paper, colored pencils, scissors, ruler, protractor, 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. 199

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Earlier in this mission, students learned that when a 180º rotation is applied to a line l, the resulting line is parallel to l. Here is a picture students worked with earlier in the mission:The picture was created by applying 180º rotations to △ABC with centers at the midpoints of segments AC and AB. Notice that E, A, and D all lie on the same grid line that is parallel to line BC. In this case, we have the structure of the grid to help see why this is true. This argument uses the structure of the grid to help explain why the three angles in this triangle add to 180 degrees.In this lesson, students begin by examining the argument using grid lines described above. Then they examine a triangle o of a grid, PQR. Here an auxiliary line plays the role of the grid lines: the line parallel to line PQ through the opposite vertex R.The three angles sharing vertex R make a line and so they add to 180 degrees. Using what they have learned earlier in this mission (either congruent alternate interior angles for parallel lines cut by a transverse or applying rigid transformations explicitly), students argue that the sum of the angles in triangle PQR is the same as the sum of the angles meeting at vertex R. This shows that the sum of the angles in any triangle is 180 degrees. The idea of using an auxiliary line in a construction to solve a problem is a critical thinking skill.Aer establishing this fact, students apply their understanding of angle relationships within a triangle to explore other angle relationships and solve missing angle problems. The nal activity in this lesson allows students to consider relationships involving the exterior angles of a triangle.AEB3 21CDP QR© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.200

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Warm-UpTRUE OR FALSE: COMPUTATIONAL RELATIONSHIPSInstructional Routine(s): True or FalseThis warm-up encourages students to reason algebraically about various computational relationships and patterns. While students may evaluate each side of the equation to determine if it is true or false, encourage students to think about the properties of arithmetic operations in their reasoning.Seeing the structure that makes the equations true or false helps students learn how to look for and make use of structure.LAUNCHDisplay one problem at a time. Tell students to give a signal when they have an answer and a strategy. Aer each problem, give students quiet think time and follow with a whole-class discussion.WARM-UP TASK Is this equation true or false? 1. 62 – 28 = 60 – 302. 3 ∙ -8 = (2 ∙ -8) – 83. -216 + -224 = -240STUDENT RESPONSE1. False. Explanations vary. Possible response: Think about a number line. The dierence between numbers is how far apart they are. 62 and 28 are further apart than 60 and 30.2. True. Explanations vary. Possible response: Rewrite (3 ∙ -8) as (2 ∙ -8) + (1 ∙ -8).3. True. Explanations vary. Possible response: Since 16 + 24 = 40 both sides of the equation are equal to -240.1© 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. 201

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:DISCUSSION GUIDANCEAsk students to share their strategies for each problem. Record and display their explanations for all to see. Ask students how they decided upon a strategy. To involve more students in the conversation, consider asking:• Do you agree or disagree? Why?• Who can restate ___’s reasoning in a dierent way?• Does anyone want to add on to _____’s reasoning?Aer each true equation, ask students if they could rely on the reasoning used on the given problem to think about or solve other problems that are similar in type. Aer each false equation, ask students how we could make the equation true.ANTICIPATED MISCONCEPTIONSIn the rst question, students may think you can round or adjust numbers in a subtraction problem in the same way as in addition problems. For example, when adding 62 + 28, taking 2 from the 62 and adding it to the 28 does not change the sum. However, using that same strategy when subtracting, the distance between the numbers on the number line changes and the dierence does not remain the same.SUPPORT FOR STUDENTS WITH DISABILITIESMemory: Processing Time.Provide sticky notes or mini whiteboards to aid students with working memory challenges.Concept Exploration: Activity 1ANGLE PLUS TWOInstructional Routines: MLR8: Discussion SupportsIn the previous lesson, students conjectured that the measures of the interior angles of a triangle add up to 180 degrees. The purpose of this activity is to explain this structure in some cases. Students apply 180º rotations to a triangle in order to calculate the sum of its three angles. They have applied these transformations earlier in the context of building 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.202

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:using rigid transformations. Here they exploit the structure of the coordinate grid to see in a particular case that the sum of the three angles in a triangle is 180 degrees. The next activity removes the grid lines and gives an argument that applies to all triangles.LAUNCHArrange students in groups of 2–3. Provide access to geometry toolkits. Give some work time building the diagram and measuring angles. Then allow for a short check-in about angle measurement error, and then provide time for students to complete the task.ACTIVITY 1 TASK 1 Here is triangle ABC. Answer the questions below. 1. Rotate triangle ABC 180º around the midpoint of side AC. Label the new vertex D.2. Rotate triangle ABC 180º around the midpoint of side AB. Label the new vertex E.3. Look at angles EAB, BAC, and CAD. Without measuring, write what you think is the sum of the measures of these angles. Explain or show your reasoning.4. Is the measure of angle EAB equal to the measure of any angle in triangle ABC? If so, which one? If not, how do you know?AB C2AB 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. 203

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:5. Is the measure of angle CAD equal to the measure of any angle in triangle ABC? If so, which one? If not, how do you know?6. What is the sum of the measures of angles ABC, BAC, and ACB?STUDENT RESPONSE1. When I rotate by 180º around the midpoint of segment AC, C and A trade places and B goes to the new point labeled D in the picture.2. Rotating 180º around the midpoint of segment AB, points A and B trade places and C goes to the new point labeled E in the picture.3. They look like they will add to 180º, because they appear to form a straight angle and there are 180º in a straight angle.4. Yes, angle ABC. When triangle ABC is rotated 180 degrees with center the midpoint of segment AB, ∠ABC goes to ∠EAB.5. Yes, angle ACB. When triangle ABC is rotated 180 degrees with center the midpoint of segment AC, ∠ACB goes to ∠CAD.6. The total measure of these angles should be 180º, because it is the same as the total measure of angles EAB, BAC, and CAD and these angles add up to 180º.ACTIVITY 1 RECAPAsk students how the grid lines helped to show that the sum of the angles in this triangle is 180 degrees. Important ideas to bring out include:• A 180º rotation of a line BC (with center the midpoint of AB or the midpoint of AC) is parallel to line BC and lays upon the horizontal grid line that point A is on.• The grid lines are parallel so the rotated angles lie on the (same) grid line.AEBCD© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.204

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Consider asking students, “Is it always true that the sum of the angles in a triangle is 180º?” (Make sure students understand that the argument here does not apply to most triangles, since it relies heavily on the fact that BC lies on a grid line which means we know the images of BC, AE and DA, also lie on grid line that point A is on.It turns out that the key to showing the more general result lies in studying the rotations that were used to generate the three triangle picture. This investigation is the subject of the next activity.ANTICIPATED MISCONCEPTIONSSome students may have trouble with the rotations. If they struggle, remind them of similar work they did in a previous lesson. Help them with the rst rotation, and allow them to do the second rotation on their own.SUPPORT FOR ENGLISH LANGUAGE LEARNERSSpeaking: MLR 8 Discussion Supports.Use this routine to support discussion. For each observation that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the group. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language about transformations.Design Principle(s): Support sense-makingConcept Exploration: Activity 2EVERY TRIANGLE IN THE WORLDInstructional Routines: MLR5: Co-Cra Questions and ProblemsThe previous activity recalls the 180º rotations used to create an important diagram. This diagram shows that the sum of the angles in a triangle is 180º if the triangle happens to lie on a grid with a horizontal side. The purpose of this activity is to provide a complete argument, not depending on the grid, of why the sum of the three angles in a triangle is 180º.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this activity, rather than building a complex shape from a triangle and its rotations, students begin with a triangle and a line parallel to the base through the opposite vertex:In this image, lines AC and DE are parallel. The advantage to this situation is that we know that points D, B, and E all lie on a line. In order to calculate angles DBA and EBC, students can use either the rotation idea of the previous activity or congruence of alternate interior angles of parallel lines cut by a transversal. In either case, they need to analyze the given constraints and decide on a path to pursue to show the congruence of angles. Students then conclude from the fact that D, B, and E lie on the same line that a + b + c = 180.There is a subtle distinction in the logic between this lesson and the previous. The previous lesson suggests that the sum of the angles in a triangle is 180º using direct measurements of a triangle on a grid. This activity shows that this is the case by using a generic triangle and reasoning about parallel lines cut by a transversal. But, in order to do so, we needed to know to draw the line parallel to line AC through B and that this idea came through experimenting with rotating triangles.LAUNCHKeep students in the same groups. Tell students they’ll be working on this activity without the geometry toolkit.In case students have not seen this notation before, explain that m∠DBA is shorthand for “the measure of angle DBA.”Begin with some quiet work time. Give groups time to compare their arguments, then have a discussion.ACTIVITY 2 TASK 1Here is triangle ABC. Line DE is parallel to line AC. B ECDAaºbºcº33B ECDAaºbºcº© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.206

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:1. What is m∠DBA + b + m∠CBE? Explain how you know.2. Use your answer to explain why a + b + c = 180.3. Explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is 180º.STUDENT RESPONSE1. Angles DBA, ABC, and CBE make a line. So the sum of their angle measures is 180º.2. Angles DBA and CAB are congruent because these are alternate interior angles for the parallel lines AC and DE with transversal AB. Angles EBC and BCA are congruent because these are alternate interior angles for the parallel lines AC and DE with transversal BC. Angles DBA, ABC, and CBE make a line, and so their angle measures add up to 180º. Then a + b + c = 180.3. For any triangle, draw a line parallel to one side, containing the opposite vertex: With this picture, use the same argument to show that the sum of the three angles of the triangle is 180º. This works for every triangle.ANTICIPATED MISCONCEPTIONSSome students may say that a, b, and c are the three angles in a triangle, so they add up to 180. Make sure that these students understand that the goal of this activity is to explain why this must be true. Encourage them to use their answer to the rst question and think about what they know about dierent angles in the diagram.For the last question students may not understand why their work in the previous question only shows a + b + c = 180 for one particular triangle. Consider drawing a dierent triangle (without the parallel line to one of the bases), labeling the three angle measures a, b, c, and asking the student why a + b + c = 180 for this triangle.DISCUSSION GUIDANCEAsk students how this activity diers from the previous one, where △ABC had a horizontal side lying on a grid line. Emphasize that:• This argument applies to any triangle ABC.AB 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. 207

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• The prior argument relies on having grid lines and having the base of the triangle lie on a grid line.• This argument relies heavily on having the parallel line to AC through B drawn, something we can always add to a triangle.The key inspiration in this activity is putting in the line DE through B parallel to AC. Once this line is drawn, previous results about parallel lines cut by a transversal allow us to see why the sum of the angles in a triangle is 180º. Tell students that the line DE is oen called an ‘auxiliary construction’ because we are trying to show something about △ABC and this line happens to be very helpful in achieving that goal. It oen takes experience and creativity to hit upon the right auxiliary construction when trying to prove things in mathematics.SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversing, Representing, Writing: MLR 5 Co-Cra Questions.Display only the image and the rst line of this task (i.e., “Here is triangle ABC. Line DE is parallel…”). Ask pairs of students to write possible mathematical questions about the representation. Then, invite pairs to share their responses with the group. Listen for students who talk about relationships between the sum of angles in a triangle and in a straight angle. This will help students produce the language of mathematical questions prior to being asked to analyze another’s reasoning for the task.Design Principle(s): Cultivate conversation; Support sense-makingSUPPORT FOR STUDENTS WITH DISABILITIESSocial-Emotional Functioning: Peer Tutors.Pair students with their previously identied peer tutors.Concept Exploration: Activity 3APPLYING ANGLE RELATIONSHIPSNow that students have suiciently proven that the sum of the interior angles of a triangle equals 180°, this activity allows them to apply that fact to solve missing angle problems involving triangles. Additionally, each problem in this activity involves nding the measure of an exterior angle of a 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.208

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:LAUNCHKeep students in the same groups. Tell students they will continue working without the geometry toolkit to nd the measure of the unknown angles in each problem.Begin with some quiet work time. Give groups time to compare their answers and strategies, then have a discussionACTIVITY 3 TASK 1For each gure below, determine the missing angle measures. Note that the gures may not be drawn to scale. 1. Line l is parallel to line k.2. 3. 34lk115°40°70°60°50°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. 209

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSE1. 2. 3. DISCUSSION GUIDANCESelect students to share their explanations for how they found the missing angle measures in each problem. Encourage students to use precise vocabulary, such as straight angle and alternate interior angles, when describing how they gured out the dierent missing angle measures.Tell students that in each problem they found the missing angle measure for at least one exterior angle, which is an angle formed when one of the sides of a triangle is extended beyond the vertex. Ask them to identify the exterior angles in each problem as well as consider the relationship between the exterior angle and the interior angles. The purpose of this discussion is to explore the relationship between the exterior and interior angles of a triangle. It’s not necessary for students to have a formal understanding of any specic relationships, so allow them to name any relationship they observe. That said, a key relationship that you might point out for students is that the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. lk115°40°140°25°130°70°60°50°30°130°150°100°80°© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.210

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Digital Lessona) Find the measure of the missing angle in triangle GHJ, shown below.b) Explain how you can nd the measure of a missing angle in any triangle if you know the measures of the other two angles.STUDENT RESPONSEa) Angle GHJ is 76º.b) You can nd the measure of a missing angle in any triangle if you know the measures of the other two angles because the sum of the three angles in a triangle always equals 180. So, you can write an equation with the two known angles plus x to represent the unknown angle equals 180. Then, you can solve for the value of x to nd the unknown angle measurement.Wrap-UpLESSON SYNTHESIS Revisit the basic steps in the proof that shows that the sum of the angles in a triangle is 180º. Consider asking a student to make a triangle that you can display for all to see and add onto it showing each step.• We had a triangle, and a line through one vertex parallel to the opposite side.• We knew that the three angles with their vertices on the line summed to 180º.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 16ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:• We knew that two of these angles were congruent to corresponding angles in the triangle, and the third one was inside the triangle.• Therefore, the three angles in the triangle must also sum to 180º.Tell students that this is one of the most useful results in geometry and they will get to use it again and again in the future.EXIT TICKETStudents sketch dierent triangles and list angle possibilities. Knowing that the sum of the angles in a triangle is 180 degrees establishes that each angle in an equilateral triangle measures 60 degrees.TASK1. In an equilateral triangle, all side lengths are equal and all angle measures are equal. Sketch an equilateral triangle. What are the measures of its angles?2. In an isosceles triangle, which is not equilateral, two side lengths are equal and two angle measures are equal. Sketch three dierent isosceles triangles.3. List two dierent possibilities for the angle measures of an isosceles triangle.STUDENT RESPONSE1. The three angle measures must add to 180º but must all be equal. Therefore each measure is 60º, since 60 = 31 ∙ 180.2. Answers vary. Triangles should each have two sides that are the same length but not three.3. Answers vary. Sample responses: 70º, 70º, 40º and 30º, 30º, 120º.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.212

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:GRADE 8 / MISSION 1 / LESSON 17Rotate and TessellateLEARNING GOALSCreate tessellations and designs with rotational symmetry using rigid transformations.Explain (orally and in writing) the rigid transformations needed to move a tessellation or design with rotational symmetry onto itself.LEARNING GOALS(STUDENT FACING)Let’s make complex patterns using transformations.LEARNING TARGETS(STUDENT FACING)I can repeatedly use rigid transformations to make interesting repeating patterns of figures.I can use properties of angle sums to reason about how figures will fit together.REQUIRED MATERIALSTemplate for Warm-Up, copies of template Blank paper, graph paper, Isometric graph paper Geometry toolkits: Tracing paper, graph paper, colored pencils, scissors, centimeter ruler, protractor and an index card to use as a straightedge or to mark right anglesREQUIRED PREPARATIONPrint the Deducing Angle Measures template. Prepare 1 copy for every 2 students. Cut the copies in half, so that there are enough copies for each student to receive a half-sheet. If possible, make these copies on cardstock so that students will have an easier time tracing shapes after they cut them out. If available, pattern blocks also work well for this.Students may benefit from using graph paper and isometric graph paper, but these materials are optional.ZEARN MATH TIPThere is no Independent Digital Lesson for this lesson. It is a further exploration of the work done so far in this mission. If you want to continue the exploration, we recommend you engage in one or two of the activities with your whole group or teach it during your concept exploration and give kids digital time to ensure they are on pace with your instruction heading into the next mission.TEACHER INSTRUCTION ONLY© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:In this mission, students have learned how to name dierent types of rigid transformations of the plane and have studied how to move dierent gures (lines, line segments, polygons, and more complex shapes). They have also used rigid transformations to dene what it means for gures to be congruent and have used rigid transformations to investigate the sum of the angles in a triangle. In this lesson, students use the language of transformations to produce, describe, and investigate patterns in the plane. This is a direct extension of earlier work with triangles.• three triangles were arranged in the plane to show that the sum of the angles in a triangle is 180 degrees• four copies of a triangle were arranged in a large square, cutting out a smaller square in the middleHere the focus is more creative. Students will examine and create dierent patterns of shapes, including tessellations (patterns that ll the entire plane), and complex designs that exhibit rotational symmetry (that is, the design is congruent to itself by several rotations). Depending on the time available, students might work on both activities or choose one of the two.As with many activities in this lesson, looking for and making use of structure is central as students use the structure of a given set of polygons to produce a tessellation. The side lengths and angles of the polygons are constraints and through experimenting and abstract reasoning, students discover a repeating pattern.Warm-UpDEDUCING ANGLE MEASURESMATERIALS: TEMPLATE FOR WARM-UPInstructional Routine: MLR8: Discussion SupportsThroughout this lesson, students build dierent patterns with copies of some polygons. In this activity, they make copies of each polygon and arrange them in a circle. They calculate some of the angles of the polygons while also gaining an intuition for how the polygons t together. Here are the gures included 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.214

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Students might use a protractor to measure angles, but the measures of all angles can also be deduced. In the rst question in the task, students are instructed to t copies of an equilateral triangle around a single vertex. Six copies t, leading them to deduce that each angle measures 60º because 360 ÷ 6 = 60. For the other shapes, they can reason about angles that sum to 360º, angles that sum to a line, and angles that sum to a known angle.LAUNCHProvide access to geometry toolkits. Distribute one half-sheet (that contains 7 shapes) to each student. It may be desirable to demonstrate how to use tracing paper to position and trace copies of the triangle around a single vertex, as described in the rst question.WARM-UP TASK Your teacher will give you some shapes. Use them to answer the questions below.1. How many copies of the equilateral triangle can you t together around a single vertex, so that the triangles’ edges have no gaps or overlaps? What is the measure of each angle in these triangles?2. What are the measures of the angles in the:a) square?b) hexagon?c) parallelogram?d) right triangle?e) octagon?f) pentagon?STUDENT RESPONSE1. 6, 60º2. Measures of angles: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. 215

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:a) Square: 90ºb) Hexagon: 120ºc) Parallelogram: 120º and 60ºd) Right triangle: 45º and 90ºe) Octagon: 135ºf) Pentagon: 90º, 120º, and 150ºDISCUSSION GUIDANCEWhen deducing angle measures, it is important to know that angles “all the way around” a vertex sum to 360º. It is also important to know that angles that make a line when adjacent sum to 180º. Monitor for students who need to be reminded of these facts.ANTICIPATED MISCONCEPTIONSFor the remainder of the lesson, it is not so important that the degree measures of the angles are known, so don’t dwell on the answers. Select a few students who deduced angle measures by tting pieces together to present their work. Make sure students see lots of examples of shapes tting together like puzzle pieces.Recall from the previous lesson that the 3 congruent angles in an equilateral triangle make a line or 180-degree angle, so it makes sense that 6 copies of this angle make a full circle.SUPPORT FOR STUDENTS WITH DISABILITIESReceptive/Expressive Language: Processing TimeStudents who benet from extra processing time would also be aided by MLR 8 (Discussion Supports).© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.216

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:Concept Exploration: Activity 1TESSELLATE THISInstructional Routines: Group Presentations, MLR1: Stronger and Clearer Each TimeEach activity in this lesson (this one, creating a tessellation, and the next one, creating a design with rotational symmetry) could easily take an entire class period or more. Consider letting students choose to pursue one of the two activities.A tessellation of the plane is a regular repeating pattern of one or more shapes that covers the entire plane. Some of the most familiar examples of tessellations are seen in bathroom and kitchen tiles. Tiles (for ooring, ceiling, bathrooms, kitchens) are oen composed of copies of the same shapes because they need to t together and extend in a regular pattern to cover a large surface.ACTIVITY 1 LAUNCHShare with students a denition of tessellation like, “A tessellation of the plane is a regular repeating pattern of one or more shapes that covers the entire plane.” Consider showing several examples of tessellations. A true tessellation covers the entire plane: While this is impossible to show, we can identify a pattern that keeps going forever in all directions. This is important when we think about tessellations and symmetry. One denition of symmetry is, “You can pick it up and put it down a dierent way and it looks exactly the same.” In a tessellation, you can perform a translation and the image looks exactly the same. In the example of this tiling, the translation that takes point Q to point R results in a gure that looks exactly the same as the one you started with. So does the translation that takes S to Q. Describing one of these translations shows that this gure has translational symmetry.Provide access to geometry toolkits. Suggest to students that if they cut out a shape, it is easy to make many copies of the shape by tracing it. Encourage students to use the shapes from the previous activity (or pattern blocks if available) and experiment putting them together. They do not need to use all of the shapes, so if students are struggling, suggest that they try using copies of a couple of the simpler shapes.QRS© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 1 TASK 1 Solve the problems below. 1. Design your own tessellation. You will need to decide which shapes you want to use and make copies. Remember that a tessellation is a repeating pattern that goes on forever to ll up the entire plane.2. Find a partner and trade pictures. Describe a transformation of your partner’s picture that takes the pattern to itself. How many dierent transformations can you nd that take the pattern to itself? Consider translations, reections, and rotations.3. If there’s time, color and decorate your tessellation.STUDENT RESPONSE1. Answers vary.2. Answers vary. For example, in the tessellation given previously, we could reect across the dashed line, or rotate 90 degrees clockwise around the point marked T.DISCUSSION GUIDANCEInvite students to share their designs and also describe a transformation that takes the design to itself. Consider decorating your room with their nished products.ANTICIPATED MISCONCEPTIONSWatch out for students who choose shapes that almost-but-don’t-quite t together. Reiterate that the pattern has to keep going forever—oen small gaps or overlaps become more obvious when you try to continue the pattern.2T© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.218

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:SUPPORT FOR ENGLISH LANGUAGE LEARNERSWriting, Speaking, Listening: MLR 1 Stronger and Clearer Each TimeUse this routine with successive pair shares to give students a structured opportunity to revise their initial explanation for creating their tessellations. Aer writing an initial explanation, ask students to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “Can you explain how you created your tessellation?”, “How many rigid transformations will take the pattern to itself?”, etc.). Students can borrow ideas and language from each partner to strengthen the nal explanation. They can then return to their seats and write down their revised tessellation description. Design Principle(s): Optimize output (for generalization)Concept Exploration: Activity 2ROTATE THATInstructional Routines: Group Presentations, MLR2: Collect and DisplayEach activity in this lesson (the previous one, creating a tessellation, and this one, creating a design with rotational symmetry) could easily take an entire class period or more. Consider letting students choose to pursue one of the two activities.In this activity, using their geometry toolkits, students can make their own design that has rotational symmetry. They then share designs and nd the dierent rotations (and possibly reections) that make the shape match up with itself.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:ACTIVITY 2 LAUNCHAsk students what transformation they could perform on the gure so that it matches up with its original position. There are a number of rotations using A as the center that would work: 72º or any multiple of 72º. Make sure students understand that the 5 triangles in this pattern are congruent and that 5 ∙ 72 = 360: This is why multiples of 72º with center A match this gure up with itself. They need to be careful in selecting angles for the shapes in their pattern. If they struggle, consider asking them to use pattern tiles or copies of the shapes from the previous activity to help build a pattern.If possible, show students several examples of gures that have rotational symmetry. Provide access to geometry toolkits. If possible, provide access to square graph paper or isometric graph paper.ACTIVITY 2 TASK 1Solve the problems below. 1. Make a design with rotational symmetry.2. Find a partner who has also made a design. Exchange designs and nd a transformation of your partner’s design that takes it to itself. Consider rotations, reections, and translations.3. If there’s time, color and decorate your design.A33© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up Resources’ 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.220

G8M1 | LESSON 17ZEARN MATH TEACHERLESSON MATERIALSYOUR NOTES:STUDENT RESPONSEAnswers vary. An example shape is below.DISCUSSION GUIDANCEInvite students to share their designs and also describe a transformation that takes the design to itself. Consider decorating your room with their nished products.SUPPORT FOR ENGLISH LANGUAGE LEARNERSConversing, Representing: MLR 2 Collect and DisplayCirculate and listen to students talk about their tessellations during pair work or group work, and jot notes about common or important words and phrases (e.g., reect, rotate, center, degrees, multiples), together with helpful sketches or diagrams. Scribe students’ words and sketches on a visual display to refer back to during whole-class discussions throughout the lesson. This will help students use mathematical and transformational language when describing their rotational symmetry designs.Design Principle(s): Support sense-making; Maximize meta-awarenessSUPPORT FOR STUDENTS WITH DISABILITIESConceptual Processing: ManipulativesProvide manipulatives (i.e., pattern blocks or shape cut-outs) to aid students who benet from hands-on activities.© 2023 Zearn. Licensed to you pursuant to Zearn’s Terms of Use.This work is a derivative of Open Up 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

TEMPLATE FOR LESSON 17 WARM-UPPAGE 1 OF 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.222

GRADE8TEACHER EDITIONMission 1Math1 2 3GRADE 84 75 86 9Mission 1 Rigid Transformations and CongruenceMission 2 Dilations, Similarity, and Introducing SlopeMission 3 Linear RelationshipsMission 4Linear Equations and Linear SystemsMission 5Functions and VolumeMission 6Associations in DataMission 7Exponents and Scientific NotationMission 8Pythagorean Theorem and Irrational NumbersMission 9Putting It All TogetherTEACHER EDITIONMathGRADE 8TEACHER EDITIONGrade 8 | Mission 1