8Th Grade Math Unit 6: Kaleidoscopes, Hubcaps, and Mirrors

8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors

Enduring understanding (Big Idea): Students will be able to identify figures with different kinds of symmetry, describe types of symmetry in terms of the images made by applying reflections, rotations, tessellations, and translations to points of the original figure, and use transformations to construct figures with different kinds of symmetry. Essential Questions: How can I use symmetry to describe the shapes and properties of figures in a design or a problem? Which figures in a pattern are congruent? What parts of a figure will be matched by a congruence transformation?

BY THE END OF THIS UNIT: Students will know… Students will be able to… Rotation of 90° about the origin: (x,y) → (-y, x)  Make figures with specified symmetries Rotation of 180° about the origin: (x,y) → (-x, -y)  Identify a basic design element that can be used with a transformation to replicate a given Reflection over x-axis: (x, y) → (x, -y) design  Perform symmetry transformations of figures, including reflections, translations, and rotations Reflection over y-axis:(x, y) → (-x, y)  Examine and describe the symmetries of a design made from a figure and its image(s) under a Reflection over line y = x: (x, y) → (y, x) symmetry transformation Translation: (x,y) → (x + a, y + b) or : (x,y) → (x - a, y - b)  Give precise mathematical directions for performing reflections, rotations, and translations in Vocabulary: terms of the effect of the transformation on points of the original figure Angle of rotation Basic design element Alternate angles  Draw conclusions about a figure in terms of the effect of the transformation on points of the original figure based on what symmetry or symmetries the figure has  Understand that figures with the same shape and size are congruent Congruent Congruent figures Interior angles  Use symmetry transformations to explore whether two figures are congruent  Give examples of minimum sets of measures of angles and sides that will guarantee that two Line of reflection Line of symmetry Exterior angles triangles are congruent  Use congruence of triangles to explore congruence of two quadrilaterals Reflection Reflection symmetry Translation  Use symmetry and congruence to deduce properties of figures  Write coordinate rules for specifying the image of a point under particular transformations Rotation Rotation symmetry Tessellation  Appreciate the power of transformational geometry in the real world Unit Resources Mathematical Practices in Focus: Learning Task: 1. Make sense of problems and preserve and persevere in solving them.  Mathematical reflections 2. Reason abstractly and quantitatively. Performance Task: 3. Construct viable arguments and critique the reasoning of others.  Check up(s) / Partner Quiz / Unit Test 4. Model with mathematics. Project: 5. Use appropriate tools strategically.  None 6. Attend to precision. Unit Review: 7. Look for and make use of structure.  Looking Back & Looking Ahead 8. Look for and express regularity in repeated reasoning. Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors

Unit Plans Investigation Suggested ACE Questions Standards : 8.G.1(a-c), 8.G.9 1.1 Reflection Symmetry 1.1: ACE 1-9, 31-33 1.2 Rotation Symmetry 1.2: ACE 10-19, 34-45, 54 Investigation 1 1.3 Analyzing Symmetries 1.3: ACE 20-25, 46-49 Three Types of Symmetry 1.4 Translation Symmetry 1.4: ACE 26-30, 50-53, 55-57 Math Reflections

Standards : 8.G.1 (a-c), 8.G.3, 8.G.9 2.1 Line Reflections 2.1: ACE 1-5, 16-18 2.2 Rotations 2.2: ACE 6-7, 19-20, 29-33 Investigation 2 2.3 Translations 2.3: ACE 8-9 ; 21-23 Symmetry Transformations 2.4 Symmetry and Tessellations 2.4: ACE 10-15, 24-28 Math Reflections

Standards : 8.G.1 (a-c), 8.G.2, 8.G.9 3.1 Relating Symmetry and Congruence 3.1: ACE 1-4, 17-18 3.2 Congruent Triangles 3.2: ACE 5-6, 19-23 Investigation 3 3.3 The Matching Game 3.3: ACE 7-10, 27-29 Exploring Congruence 3.4 Polystrip Quadrilaterals 3.4: ACE 11-16, 24-26 Math Reflections

Standards : 8.G.1 (a-c) 4.1 Finding Distances Without Measuring 4.1: ACE 1–10, 16, 17 4.2 Using Symmetry to Find Properties of 4.2: ACE 11-15, 18-25 Investigation 4 Shape Applications of Congruence and Math Reflections Symmetry

Standards : 8.G.1 (a-c), 8.G.3 5.1 Coordinate Rules for Reflections 5.1: ACE 1-3, 19, 25 5.2 Coordinate Rules for Translations 5.2: ACE 4, 20-21 Investigation 5 5.3 Coordinate Rules for Rotations 5.3: ACE 5–14, 22, 23, 26–28 Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors

Transforming Coordinates 5.4 Combinations of Transformations 5.4: ACE 15–18, 24 Math Reflections Looking Back and Looking Ahead Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4 Common Core Investigation 3: (Green Book) Common Core Investigation

Common Core Investigation 3 3.1 Transformations 3.1: ACE 1-3, 13 Transformations 3.2 Transformations 3.2: ACE 4-8, 13 3.3 Transformations 3.3: ACE 12 3.4 Transformations 3.4: ACE 9-11, 13 3.5 Transformations 3.5: ACE 14, 16-17 3.6 Transformations 3.6: ACE 15, 18 3.7 Transformations 3.7: ACE 19-25

Standards: 8.G.5 Common Core Investigation 3: (Green Book) Common Core Investigation

Common Core Investigation 4 4.1 Geometry Topics 4.1: ACE 1-8, 17 Geometry Topics 4.2 Geometry Topics 4.2: ACE 9-10 4.3 Geometry Topics 4.3: ACE 11-16, 18-34

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors

CORE CONTENT Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard 8.G.1: Verify experimentally the properties of rotations, reflections, and translations: a) Lines are taken to lines, and line segments to line segments of the same length. b) Angles are taken to angles of the same measure. c) Parallel lines are taken to parallel lines. Concepts and Skills to Master  Verify that congruence of line segments and angles is maintained through rotation, reflection, and translation.  Verify that congruence of line segments and angles is maintained through rotation, reflection, and translation.  Verify that when parallel lines are rotated, reflected, or translated, each in the same way, they remain parallel lines. SUPPORTS FOR TEACHERS Critical Background Knowledge  Know definitions and properties of angles, segments, lines, and parallel lines.  Measure angles and line segments. Academic Vocabulary Line, angle, segment, parallel line, rigid motion, congruent, center of rotation, line of reflection, rotation, reflection, translation, transformation Suggested Instructional Strategies Resources  Use dynamic geometry software or Excel to explore  Textbook Correlation properties of rotations, reflections, and translations. o Kaleidoscopes, Hubcaps, and Mirrors (CMP2) . Investigations 1-5  Use a coordinate grid and apply rules such as (-x, o Common Core Investigations (CMP2) y) or (x, y+7) to the coordinates of a given figure. . Investigation 3: Transformations Compare the resulting image to the original.  Geogebra  Symmetries II  Texas Instrument 8.G.1 Lessons Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors

 CMP2 Resources

Sample Formative Assessment Tasks Skill-based task Problem Task Verify that a triangle, when rotated, remains a Create a tessellation using rotations, reflections, and triangle. translations.

CORE CONTENT Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard 8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Concepts and Skills to Master  Understand that the congruency of two dimensional figures is maintained while undergoing rigid transformations.  Describe the transformation of a figure as a rotation, reflection, translation or a combination of transformations. SUPPORTS FOR TEACHERS Critical Background Knowledge  Identify rotations, reflections, and translations with lines, segments and angles. Academic Vocabulary rotation, reflection, translation, congruent, center of rotation, line of reflection, angle of rotation Suggested Instructional Strategies Resources  Use geometry software to explore rotations,  Textbook Correlation reflections and translations of two-dimensional o Kaleidoscopes, Hubcaps, and Mirrors (CMP2) figures. . Investigations 1-5 o Common Core Investigations (CMP2)  Use digital photographs with at least two congruent . Investigations 3: Transformations shapes and discuss the needed transformations to  Geogebra map one to the other.  Symmetries II

 Texas Instrument 8.G.2 Lessons

 CMP2 Resources

Sample Formative Assessment Tasks Skill-based task Problem Task Triangle x was transformed to x'. Describe the sequence of Find at least two different ways to describe the transformation(s) transformations that was used to show that Triangle x is that map(s) the first figure onto the second. congruent to Triangle x'.

Triangle x Triangle x'

CORE CONTENT Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard 8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Concepts and Skills to Master  Understand how to dilate, translate, rotate, and reflect two-dimensional figures on the coordinate plane.  Describe the effects of dilations, translations, rotations, and reflections using coordinate notation.  Given an image and its transformed image, use coordinate notation to describe the transformation. SUPPORTS FOR TEACHERS Critical Background Knowledge • Plot or identify points on the coordinate plane. Academic Vocabulary rotation, reflection, translation, congruent, center of rotation, line of reflection, angle of rotation Suggested Instructional Strategies Resources • Have students take pictures of transformations they see  Textbook Correlation in the world around them. Then overlay the picture with o Kaleidoscopes, Hubcaps, and Mirrors (CMP2) the coordinate plane and describe the transformation . Investigations 2 and 5 using coordinate notation. o Common Core Investigations (CMP2) . Investigations 3: Transformations • Use a coordinate grid and apply rules such as (-x, y) or (x, y+7) to the coordinates of a given figure. Compare the  The Transmographer resulting image to the original.  Geogebra  Symmetries II  Texas Instrument 8.G.3 Lessons  CMP2 Resources

Sample Formative Assessment Tasks Skill-based Task Problem Task Given a triangle with vertices at (5, 2), (-7, 8) and (0, 4), Given an original shape and its image on a coordinate plane, find the new vertices of the triangle after undergoing the determine the rule or rules that translated the original to the transformation described as follows: resulting image.

(x,y)  (x + 6, y - 3) The vertices of Triangle A are (1,0), (1,1), (0,0) and Triangle A' are (2,1), (2,2), (3,1). Describe the series of transformations performed on Triangle A that result in Triangle A'.

CORE CONTENT Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard 8.G.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Concepts and Skills to Master  Understand that any combination of transformations will result in similar figures.  Describe the sequence of transformations needed to show how one figure is similar to another.  Make dilations of figures by a given scale factor. SUPPORTS FOR TEACHERS Critical Background Knowledge  Perform rotations, translations, reflections, and dilations.  Understand proportions. Academic Vocabulary similar, similarity, dilation, rotation, reflection, translation, transformation Suggested Instructional Strategies Resources • Use Patty Paper transformations.  Textbook Correlation o Kaleidoscopes, Hubcaps, and Mirrors (CMP2) • Use dynamic geometry software to make . Investigations 2 transformations and compare transformations. o Common Core Investigations (CMP2) . Investigations 3: Transformations

 The Transmographer  Geogebra  Symmetries II  Texas Instrument 8.G.4 Lessons  CMP2 Resources Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors

Sample Formative Assessment Tasks Skill-based Task Problem Task Which of the following transformations will result in a List the sequence of transformations that verifies the similarity of similar figure? the two figures.

A) (3x , 2y) B) (-x + 2 , y - 2) C) (5x , y + 5) D) (3x , x + y)

CORE CONTENT Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard T.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument, in terms of transversals, why this is so. Concepts and Skills to Master  Understand that the measure of an exterior angle of a triangle is equal to the sum of the measures of the non-adjacent angles.  Know that the sum of the angles of a triangle equals 180o.  Determine the relationship between corresponding angles, alternate interior angles, alternate exterior angles, vertical pairs, and supplementary pairs when parallel lines are cut by a transversal.  Recognize that if two triangles have two congruent angles, then they are similar triangles (angle-angle). SUPPORTS FOR TEACHERS Critical Background Knowledge • Understand the definition of similar figures. • Know how to measure angles. Academic Vocabulary corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles, supplementary pairs, vertical pairs, transversal, adjacent, non-adjacent, exterior angle of a triangle, remote interior angles of a triangle Suggested Instructional Strategies Resources • Use a series of transformations of triangles to produce  Textbook Correlation parallel lines and examine the properties of resultant o Common Core Investigations (CMP2) angles and triangles. . Investigations 4: Geometric Topics

 Texas Instrument 8.G.5 Lessons

 CMP2 Resources

Sample Formative Assessment Tasks Skill-based Task Problem Task Identify and name sets of angles of parallel lines cut by a The streets 400 E and 900 E run north and south. Euclid Drive cuts transversal and tell which are congruent. both of these streets at an angle from SE to NW. Pythagoras Way passes through all three streets SW to NE. Are all possible triangles created by the intersection of the streets similar? Justify.

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.