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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) Rotation of 180° about the origin: (x,y) → (-x, -y) Reflection over x-axis: (x, y) → (x, -y) Reflection over y-axis:(x, y) → (-x, y) Reflection over line y = x: (x, y) → (y, x) Translation: (x,y) → (x + a, y + b) or : (x,y) → (x - a, y - b) Vocabulary: Angle of rotation Congruent Line of reflection Reflection Rotation Symmetry Vertical Angles Translation symmetry Basic design element Congruent figures Line of symmetry Reflection symmetry Rotation symmetry Corresponding Angles Supplementary Angles Unit Resources Learning Task: Mathematical reflections Performance Task: Check up(s) / Partner Quiz / Unit Test Project: None Unit Review: Looking Back & Looking Ahead Alternate angles Interior angles Exterior angles Translation Tessellation Transformation Volume Make figures with specified symmetries Identify a basic design element that can be used with a transformation to replicate a given design Perform symmetry transformations of figures, including reflections, translations, and rotations Examine and describe the symmetries of a design made from a figure and its image(s) under a symmetry transformation Give precise mathematical directions for performing reflections, rotations, and translations in terms of the effect of the transformation on points of the original figure 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 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 triangles are congruent Use congruence of triangles to explore congruence of two quadrilaterals Use symmetry and congruence to deduce properties of figures Write coordinate rules for specifying the image of a point under particular transformations Appreciate the power of transformational geometry in the real world Mathematical Practices in Focus: 1. 2. 3. 4. 5. 6. 7. Make sense of problems and preserve and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s 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 1.2 Rotation Symmetry 1.3 Analyzing Symmetries 1.4 Translation Symmetry Math Reflections 1.1: ACE 1-9, 31-33 1.2: ACE 10-19, 34-45, 54 1.3: ACE 20-25, 46-49 1.4: ACE 26-30, 50-53, 55-57 2.1 Line Reflections 2.2 Rotations 2.3 Translations 2.4 Symmetry and Tessellations Math Reflections 2.1: ACE 1-5, 16-18 2.2: ACE 6-7, 19-20, 29-33 2.3: ACE 8-9 ; 21-23 2.4: ACE 10-15, 24-28 3.1 Relating Symmetry and Congruence 3.2 Congruent Triangles 3.3 The Matching Game 3.4 Polystrip Quadrilaterals Math Reflections 3.1: ACE 1-4, 17-18 3.2: ACE 5-6, 19-23 3.3: ACE 7-10, 27-29 3.4: ACE 11-16, 24-26 4.1 Finding Distances Without Measuring 4.2 Using Symmetry to Find Properties of Shape Math Reflections 4.1: ACE 1–10, 16, 17 4.2: ACE 11-15, 18-25 5.1 Coordinate Rules for Reflections 5.2 Coordinate Rules for Translations 5.3 Coordinate Rules for Rotations 5.1: ACE 1-3, 19, 25 5.2: ACE 4, 20-21 5.3: ACE 5–14, 22, 23, 26–28 Investigation 1 Three Types of Symmetry Standards: 8.G.1 (a-c), 8.G.3, 8.G.9 Investigation 2 Symmetry Transformations Standards: 8.G.1 (a-c), 8.G.2, 8.G.9 Investigation 3 Exploring Congruence Standards: 8.G.1 (a-c) Investigation 4 Applications of Congruence and Symmetry Standards: 8.G.1 (a-c), 8.G.3 Investigation 5 Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s 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 Math Reflections Looking Back and Looking Ahead Common Core Investigation 3: (Green Book) 5.4: ACE 15–18, 24 Common Core Investigation 3 Transformations 3.1 Transformations 3.2 Transformations 3.3 Transformations 3.4 Transformations 3.5 Transformations 3.6 Transformations 3.7 Transformations 3.1: ACE 1-3, 13 3.2: ACE 4-8, 13 3.3: ACE 12 3.4: ACE 9-11, 13 3.5: ACE 14, 16-17 3.6: ACE 15, 18 3.7: ACE 19-25 Standards: 8.G.5 Common Core Investigation 3: (Green Book) Common Core Investigation Common Core Investigation 4 Geometry Topics 4.1 Geometry Topics 4.2 Geometry Topics 4.3 Geometry Topics 4.1: ACE 1-8, 17 4.2: ACE 9-10 4.3: ACE 11-16, 18-34 Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4 Common Core Investigation Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s 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 Textbook Correlation Use dynamic geometry software or Excel to o Kaleidoscopes, Hubcaps, and Mirrors (CMP2) explore properties of rotations, reflections, and Investigations 1-5 translations. o Common Core Investigations (CMP2) Investigation 3: Transformations Use a coordinate grid and apply rules such as ( Geogebra x, y) or (x, y+7) to the coordinates of a given Symmetries II figure. Compare the resulting image to the original. Texas Instrument 8.G.1 Lessons CMP2 Resources Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s 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 Verify that a triangle, when rotated, remains a triangle. Problem Task Create a tessellation using rotations, reflections, and translations. CORE CONTENT Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors 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 Textbook Correlation Use geometry software to explore rotations, o Kaleidoscopes, Hubcaps, and Mirrors (CMP2) reflections and translations of two-dimensional Investigations 1-5 figures. o Common Core Investigations (CMP2) Investigations 3: Transformations Use digital photographs with at least two congruent Geogebra shapes and discuss the needed transformations to map one to the other. Symmetries II Texas Instrument 8.G.2 Lessons CMP2 Resources Sample Formative Assessment Tasks Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors Skill-based task Triangle x was transformed to x'. Describe the sequence of transformations that was used to show that Triangle x is congruent to Triangle x'. Triangle x Problem Task Find at least two different ways to describe the transformation(s) that map(s) the first figure onto the second. Triangle x' CORE CONTENT Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors 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 The Transmographer (x, y+7) to the coordinates of a given figure. Compare the resulting image to the original. Geogebra Symmetries II Texas Instrument 8.G.3 Lessons CMP2 Resources Sample Formative Assessment Tasks Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors Skill-based Task Given a triangle with vertices at (5, 2), (-7, 8) and (0, 4), find the new vertices of the triangle after undergoing the transformation described as follows: Problem Task Given an original shape and its image on a coordinate plane, determine the rule or rules that translated the original to the 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 Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors 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 Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s 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 Which of the following transformations will result in a similar figure? Problem Task List the sequence of transformations that verifies the similarity of the two figures. A) (3x , 2y) B) (-x + 2 , y - 2) C) (5x , y + 5) D) (3x , x + y) CORE CONTENT Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard 8.G.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. o Know that the sum of the angles of a triangle equals 180 . 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 Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. 8th Grade Math Unit 7: Kaleidoscopes, Hubcaps, and Mirrors Skill-based Task Identify and name sets of angles of parallel lines cut by a transversal and tell which are congruent. Problem Task The streets 400 E and 900 E run north and south. Euclid Drive cuts 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. Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.