Download 8th Grade Math Unit 6: Kaleidoscopes, Hubcaps

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
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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:
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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.
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
1.2 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
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
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
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
 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
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
Verify that a triangle, when rotated, remains a triangle.
Problem Task
Create a tessellation using rotations, reflections, and translations.
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.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
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
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'
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.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
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
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:
(x,y)  (x + 6, y - 3)
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.
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'.
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.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
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)
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.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
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
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.
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.