Download Math 8 Unit Guide 8.G.1-8.G.5 (edited 6/28/14) Curriculum Guide

Transformational Geometry and
Angle Properties Unit Guide Grade 8
Big Idea (Cluster): Understand
congruence and similarity using
physical models, transparencies, or
geometry software
.
8.G.1-8.G.5
Edited 6/28/14
Domain: Geometry (8.G)
Big Idea (Cluster): 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.
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.
Standard 8.G.3
Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
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.
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 explain, in terms of transversals why this is so.
Parameters for this cluster of standards:
 Rotations are only multiples of 90 degrees about the origin
 Reflections are only over the x- and y-axes
 Dilations are only with the origin as the center
 Transformations are limited to the coordinate plane.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Relevant Math Practices and Student Actions
MP1
 Make connections between one’s own thinking and other’s
thinking
 Check process, answers and ask “Does this make sense?”
 Use multiple strategies and representations
 Explain why a solution is reasonable
MP 2
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Use representations to make meaning of a problem
Translate a problem from situation to equation
Explain connections between equation and situation
Understand meaning of quantities and units
MP 3
 Justify conclusions and respond to arguments of others
 Communicate and defend mathematical reasoning using objects,
drawings, diagrams, actions, examples and counterexamples
 Use definitions and draw on prior mathematical knowledge when
constructing an argument
MP 4
 Identify quantities necessary to solve a problem and use
representations to map their relationships
 Make assumptions and approximations to make a problem easier
 Analyze mapped relationships mathematically to draw
conclusions
MP 5
 Select and use tools strategically
 Make sound decisions about selecting the appropriate tools to use
 Use technological tools and resources to pose problems, solve
problems and deepen understanding
MP 6
 Communicate accurately mathematical thinking orally and in
writing
 Understand what mathematical symbols and vocabulary mean and
know when to use them appropriately
 Label consistently and accurately
 Calculate accurately and efficiently
MP 7
 Make connections to prior mathematical knowledge to solve new
problems
 Look for, identify, develop and generalize patterns and
relationships
MP 8
 Evaluate reasonableness of solutions and results
 Notice repeated calculations and look for general methods and
shortcuts to solve a problem
 Identify patterns to develop algorithm, formula or calculation
3
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
SBAC Required Evidence (Claim 1)
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The student verifies that rigid transformations preserve distance and angle measures (required evidence #1).
The student describes sequences of rotations, reflections, translations, and dilations that can verify whether two-dimensional figures are similar or
congruent to each other (required evidence #2).
The student constructs a new figure after the original figure is dilated, rotated, reflected, or translated (required evidence #3).
The student describes the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates (required
evidence #4).
A calculator is an allowable tool for this cluster of standards. For more detail and information on the assessment of this cluster of standards, see
SBAC Grade 8 Claim 1 Item Specifications for Target G. This cluster of standards will also be assessed through Claim 2 (Problem Solving) and
Claim 3 (Communicating Reasoning). Based on information from the Claim 1 item specifications, the majority of this cluster will be assessed
through Claim 3, Communicating Reasoning where students will be asked to clearly and precisely construct viable arguments to support their own
reasoning and to critique the reasoning of others.
Vocabulary
Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical
language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter
Balanced assessment items.
Adjacent Angles
Angle
Angle-Angle Criterion
Angle of rotation
Center of dilation
Center of rotation
Clockwise
Complementary Angles
Congruence ( )
Congruent
Congruent Figures
Coordinate plane
Corresponding parts
Counterclockwise
Degree
Dilate
Dilation
Double prime notation
Exterior Angle
Image
Interior Angle
Line of reflection
Line segment
Measure of an angle
Measure of a line segment
Non-rigid transformation
Original
Parallel lines
Prime notation
Ray for dilation
Reflect
Reflection
Rigid Transformation
Rotate
Rotation
Scale factor
Sequence
Similar
Similarity
Supplementary Angles
Transformation
Translate
Translation
Transversal
Vertical Angles
4
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Domain: Geometry (8.G)
Big Idea (Cluster): 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.
See Grade 8 Flip Book pages 33-35 for more explanations and examples.
Learning Objectives
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Understand and describe the impact of a transformation on a figure and its component parts (SBAC ALD Level 3).
Use the properties of reflections, translations, and rotations to perform rigid transformations
Verify mathematically that line segments maintain their original measure following a rotation, reflection, and translation
Verify mathematically that angle measures maintain their original degree measure following a rotation, reflection, and translation
Verify that when parallel lines are rotated, reflected, or translated, each in the same way, they remain parallel lines
Describe transformations from original figure to the image using appropriate geometric labeling and vocabulary
Construct lines, line segments, angles, and parallel lines after the original figures were rotated, reflected, or translated
SBAC Required Evidence (Claim 1)

The student verifies that rigid transformations preserve distance and
angle measures (required evidence #1).
A calculator is an allowable tool for this cluster of standards. For more detail and
information on the assessment of this cluster of standards, see SBAC Grade 8 Claim 1
Item Specifications for Target G. This cluster of standards will also be assessed
through Claim 2 (Problem Solving) and Claim 3 (Communicating Reasoning).
Based on information from the Claim 1 item specifications, the majority of this
cluster will be assessed through Claim 3, Communicating Reasoning where students
will be asked to clearly and precisely construct viable arguments to support their own
reasoning and to critique the reasoning of others.
Connections to Prior Learning
Standard 7.G.2
Draw (freehand, with ruler and protractor, and with technology) geometric
shapes with given conditions. Focus on constructing triangles from three
measures of angles or sides, noticing when the conditions determine a
unique triangle, more than one triangle, or no triangle.
Standard 7.G.5
Use facts about supplementary, complementary, vertical, and adjacent
angles in a multi-step problem to write and solve simple equations for an
unknown angle in a figure.
5
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Connections to Curricular Materials
Additional Resources/Technology Resources
Kaleidoscopes, Hubcaps and Mirrors Investigation 2 will cover
verifying properties of reflections, rotations and translations by
describing the properties of each of these transformations by analyzing
lines and angles between the image and pre-image. The concern with
this unit is that the investigation is off grid and students will only
be presented with transformations in the coordinate plane during
an assessment. The focus of this CMP2 unit is on symmetry while the
Common Core State Standards are using transformations to verify
congruence between image and pre-image by verifying corresponding
angles congruent and corresponding sides congruent. Modifications
should be made at the building level to provide more practice with
transformations in the coordinate plane given the parameters on
page 2 which are also found in the SBAC item specifications for
Claim 1 Target G.
8.G.1 – 8.G.5 Transformational Geometry PowerPoint from NJCTL
Playing with Transformations (NVLM)
Playing with Reflections (NVLM)
Playing with Rotations (NVLM)
Transformation Golf
Transformation Games (OnlineMathLearning.com)
Field Guide to Geometric Similarity
Explanations and Examples
In a translation, every point of the original figure is moved the same distance and in the same direction to form the image.
reflection is the “flipping” of an object over a line, known as the line of reflection. A rotation is a transformation that is
performed by “spinning” the figure around a fixed point known as the center of rotation. The figure may be rotated
clockwise or counterclockwise.
A
Students need multiple opportunities to explore the transformation of figures so that they can appreciate that points stay the same distance apart,
corresponding sides lengths between pre-image and image are the same length and corresponding angles have the same measure after they have been
rotated, reflected, and/or translated.
6
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Domain: Geometry (8.G)
Big Idea (Cluster): 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.
See Grade 8 Flip Book page 36 for more examples and explanations.
Learning Objectives
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Use or describe a sequence of transformations to determine or exhibit the congruence of two figures (SBAC ALD Level 3)
Describe the transformation of a figure as a rotation, reflection, translation or a combination of transformations
Develop an understanding of two-dimensional congruence
Understand that the congruency of two dimensional figures is maintained while undergoing rigid transformations
Explain how transformations can be used to prove that two figures are congruent
Verify mathematically that congruence of line segments and angles is maintained through rotation, reflection, and translation
Construct new figures after the original figures were rotated, reflected, or translated
Describe sequences of rotations, reflections, and translations that show whether two-dimensional figures are congruent to each other
Describe a sequence of rigid transformations that takes an original figure to its image
SBAC Required Evidence (Claim 1)

The student describes sequences of rotations, reflections,
translations, and dilations that can verify whether two-dimensional
figures are similar or congruent to each other (required evidence
#2).
A calculator is an allowable tool for this cluster of standards. For more detail and
information on the assessment of this cluster of standards, see SBAC Grade 8 Claim 1
Item Specifications for Target G. This cluster of standards will also be assessed
through Claim 2 (Problem Solving) and Claim 3 (Communicating Reasoning).
Based on information from the Claim 1 item specifications, the majority of this
cluster will be assessed through Claim 3, Communicating Reasoning where students
will be asked to clearly and precisely construct viable arguments to support their own
reasoning and to critique the reasoning of others.
Connections to Prior Learning
Standard 7.G.2 (see above), Standard 7.G.5 (see above) and Standard
8.G.1
In Grade 4, students will have drawn and measured points, lines, line
segments, angles and parallel lines. In Grade 5, students are introduced to
the first quadrant of the coordinate plane and drawing two-dimensional
figures on a coordinate plane. In Grade 6, students will draw polygons in a
coordinate plane given the vertices within all four quadrants. In Grade 7,
students will have learned and used the properties of angle relationships in
order to identify unknown angles of a figure.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Connections to Curricular Materials
Kaleidoscopes, Hubcaps and Mirrors Investigation 3 will cover
exploring congruence through analyzing properties of reflections,
rotations and translations between the image and pre-image. The
concern with this unit is that the investigation is off grid and
students will only be presented with transformations in the
coordinate plane during an assessment. The focus of this CMP2 unit
is on symmetry while the Common Core State Standards are using
transformations to verify congruence between image and pre-image by
verifying corresponding angles congruent and corresponding sides
congruent. Modifications should be made at the building level to
provide more practice with transformations in the coordinate plane
given the parameters on page 2 which are also found in the SBAC
item specifications for Claim 1 Target G.
Additional Resources/Technology Resources
8.G.1 – 8.G.5 Transformational Geometry PowerPoint from NJCTL
Core Math Tools (interactive software tool from NCTM)
Graphing translations video
Graphing reflection of a shape video
Graphing rotations video
Representing and Combining Transformations task (MAP)
http://www.Geogebra.org
Composing Reflections
Composing Transformations
Identifying Unknown Transformations
Visualizing Transformations
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Explanations and Examples
This standard is the students’ introduction to congruency. Congruent figures have the same shape and size. Translations, reflections and rotations are
examples of rigid transformations. A rigid transformation is one in which the original figure and its image both have exactly the same size and shape
since the measures of the corresponding angles and corresponding line segments remain equal (are congruent).
Students examine two figures to determine congruency by identifying the rigid transformation(s) that produced the figures. Students recognize the
symbol for congruency ( ) and write statements of congruency. Within the SBAC item specifications, the students will have at most three
transformations in a sequence of transformations.
Translations
Translations move the figure so that every point of the figure moves in the same direction as well as the same distance. In a
translation, the translated figure (image) is congruent to its original figure.
Reflections
A reflection is the “flipping” of an object over a line, known as the “line of reflection”. In the 8th grade, the line of
reflection will be the x-axis and the y-axis.
Rotations
A rotation is a transformation performed by “spinning” the figure around a fixed point known as the center of rotation. The
figure may be rotated clockwise or counterclockwise up to 360º. In the 8th grade, rotations will be about the origin and
multiples of 90º. In a rotation, the rotated figure (image) is congruent to its original figure.
Example: Is Figure A congruent to Figure A’? Explain how you know.
Solution: These figures are congruent since A’ was produced by translating each vertex of Figure A three to the right and one
down from the original position. Each corresponding line segment between original and image has the same length.
Example: Describe the sequence of transformations that result in the transformation of Figure A to Figure A’.
Solution: Figure A’ was produced by a 90 clockwise rotation around the origin.
9
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Domain: Geometry (8.G)
Big Idea (Cluster): Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.3 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
See Grade 8 Flip Book pages 37-38 for more explanations and examples.
Learning Objectives
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Understand and describe the impact of a transformation on a figure and its component parts with coordinates (SBAC ALD Level 3).
Describe properties of dilations using scale factor
Understand how to dilate, translate, rotate, and reflect two-dimensional figures on the coordinate plane
Use coordinate notation to describe the transformation of a figure within a coordinate plane
Describe the changes to the x- and y- coordinates of a figure after a translation, reflection, rotation, and/or dilation both verbally and in writing
Determine the coordinates of the image of an original figure after one or more rigid transformations
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates
Determine the coordinates of original figure when given a sequence of transformations and an image
Describe how the coordinates of the image relate to the coordinates of an original figure under a given transformation
10
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
SBAC Required Evidence (Claim 1)
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The student describes sequences of rotations, reflections,
translations, and dilations that can verify whether two-dimensional
figures are similar or congruent to each other (required evidence
#2).
The student constructs a new figure after the original figure is
dilated, rotated, reflected, or translated (required evidence #3).
The student describes the effect of dilations, translations, rotations,
and reflections on two-dimensional figures using coordinates
(required evidence #4).
Connections to Prior Learning
Standard 7.G.2 (see above), Standard 7.G.5 (see above), Standard
8.G.1 and Standard 8.G.2 (see above)
Standard 6.G.3
Draw polygons in the coordinate plane given coordinates for the vertices;
use coordinates to find the length of a side joining points with the same
first coordinate or the same second coordinate. Apply these techniques in
the context of solving real-world and mathematical problems.
A calculator is an allowable tool for this cluster of standards. For more detail and
information on the assessment of this cluster of standards, see SBAC Grade 8 Claim 1
Item Specifications for Target G. This cluster of standards will also be assessed
through Claim 2 (Problem Solving) and Claim 3 (Communicating Reasoning).
Based on information from the Claim 1 item specifications, the majority of this
cluster will be assessed through Claim 3, Communicating Reasoning where students
will be asked to clearly and precisely construct viable arguments to support their own
reasoning and to critique the reasoning of others.
Connections to Curricular Materials
Kaleidoscopes, Hubcaps and Mirrors does not cover this standard
since dilations were introduced in Stretching and Shrinking.
Supplemental lessons will be identified in the pacing guide below and
in the additional resources section in the box to the right.
Additional Resources/Technology Resources
CMP2 Supplemental Lessons Student Edition and Teacher Edition
Modified CMP2 Supplemental Lessons and Answer Keys
Kaleidoscopes, Hubcaps and Mirrors modified lessons for investigation 5
8.G.1 – 8.G.5 Transformational Geometry PowerPoint from NJCTL
Dilation Lesson 3 from Newark Public Schools Similarity and Congruence
module
Fishing for Points (TI-84 activity)
Playing with Dilations (NVLM)
Scale Factor applet
http://www.Geogebra.org
Transmographer (Shodor.org)
3D Transmographer (Shodor.org)
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Explanations and Examples
Students identify resulting coordinates from translations, reflections, and rotations (90º, 180º and 270º both clockwise and counterclockwise) and
dilations, recognizing the relationship between the coordinates and the transformation.
Translations
Triangle ABC has been translated 7 units to the right and 3 units up. To get from A (1,5) to A’ (8,8), move A 7 units to
the right (from x = 1 to x = 8) and 3 units up (from y = 5 to y = 8). Points B and C also move in the same direction (7 units
to the right and 3 units up), resulting in the same changes to each coordinate. Students should recognize each xcoordinate increased (added) by 7 from translating right 7 and each y-coordinate increased (added) by 3 from translating
up 3.
Reflections
Students recognize that when a figure is reflected across the y-axis, the reflected x-coordinate is the opposite of the
original x-coordinate. Students should also recognize that when a figure is reflected across the x-axis, the reflected ycoordinate is the opposite of the original y-coordinate.
Rotations
Consider when triangle DEF is 180° clockwise about the origin. The coordinate of triangle DEF are D(2,5), E(2,1), and
F(8,1). When rotated 180° about the origin, the new coordinates are D’(-2,-5), E’(-2,-1) and F’(-8,-1). In this case, each
coordinate is the opposite of its original.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Dilations
A dilation is a non-rigid transformation that moves each point along a ray which starts from a fixed center and multiplies distances from this center
by a common scale factor. Dilations enlarge (scale factors greater than one) or reduce (scale factors less than one) the size of a
figure by the scale factor. In 8th grade, the center of dilation will be the origin. The dilated figure is similar to its original figure.
The coordinates of A is (2, 6) and A’ is (1, 3). The coordinates of B is (6, 4) and B’ is (3, 2). The coordinates of C is (4, 0) and
C’ is (2, 0). Each of the image coordinates is ½ the value of the original coordinates indicating a scale factor of ½.
The scale factor will be evident in the length of the line segments using the ratio:
.
Students recognize the relationship between the coordinates of the original, the image and the scale factor for a dilation from the origin. Using the
coordinates, students are able to identify the scale factor (image length/original length).
Example:
Students identify the transformation based on given coordinates. For example, the original coordinates of a triangle are A(4, 5), B(3, 7), and C(5, 7).
The image coordinates are A’(-4, 5), B’(-3, 7), and C’(-5, 7). What transformation occurred?
Solution: The x-coordinate for each vertex changed from a positive value to a negative value. This means the triangle was reflected across the yaxis.
13
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Domain: Geometry (8.G)
Big Idea (Cluster): 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.
See Grade 8 Flip Book page 39 for more explanations and examples.
Learning Objectives
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Describe a sequence that exhibits the similarity between two shapes and understand that the angle measures are unchanged (SBAC ALD Level 4)
Understand and distinguish between congruency and similarity
Recognize and describe a sequence of transformations and justify if there is similarity or congruence
Understand a dilation is the only transformation where congruency is not preserved
Understand the relationships of angles and side lengths of similar figures
Understand that any combination of transformations will result in similar figures
Explain how transformations can be used to prove two figures are similar
Describe the sequence of transformations needed to show how one figure is similar to another
Justify two figures are similar, congruent, or neither following a series of transformations
SBAC Required Evidence (Claim 1)

The student describes sequences of rotations, reflections,
translations, and dilations that can verify whether two-dimensional
figures are similar or congruent to each other (required evidence
#2).
A calculator is an allowable tool for this cluster of standards. For more detail and
information on the assessment of this cluster of standards, see SBAC Grade 8 Claim 1
Item Specifications for Target G. This cluster of standards will also be assessed
through Claim 2 (Problem Solving) and Claim 3 (Communicating Reasoning).
Based on information from the Claim 1 item specifications, the majority of this
cluster will be assessed through Claim 3, Communicating Reasoning where students
will be asked to clearly and precisely construct viable arguments to support their own
reasoning and to critique the reasoning of others.
Connections to Prior Learning
Standard 7.G.2 (see above), Standard 7.G.5 (see above), Standard
8.G.1 and Standard 8.G.2 (see above)
Standard 6.G.3
Draw polygons in the coordinate plane given coordinates for the vertices;
use coordinates to find the length of a side joining points with the same
first coordinate or the same second coordinate. Apply these techniques in
the context of solving real-world and mathematical problems.
14
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Connections to Curricular Materials
Additional Resources/Technology Resources
Kaleidoscopes, Hubcaps and Mirrors does not cover this standard
since dilations were introduced in Stretching and Shrinking.
Supplemental lessons will be identified in the pacing guide below and
in the additional resources section in the box to the right.
CMP2 Supplemental Lessons Student Edition and Teacher Edition
Modified CMP2 Supplemental Lessons and Answer Keys
8.G.1 – 8.G.5 Transformational Geometry PowerPoint from NJCTL
http://www.Geogebra.org
Understanding Congruence, Similarity, and Symmetry Using
Transformations and Interactive Figures
Understanding Congruence, Similarity, and Symmetry Using
Transformations and Interactive Figures: Visualizing Transformations
Explanations and Examples
Similar figures and similarity are first introduced in the 8th grade. Students understand similar figures have congruent angles and sides that are
proportional. Similar figures are produced from dilations. Students describe the sequence that would produce similar figures, including the scale
factors. Students understand that a scale factor greater than one will produce an enlargement in the figure, while a scale factor less than one will
produce a reduction in size. Within the SBAC item specifications, the students will have at most three transformations in a sequence of
transformations.
Example:
Is Figure A similar to Figure A’? Explain how you know.
Solution:
Figure A’ was dilated from the original Figure A using a scale factor of ½. The side lengths of the original figure were each 4
units and the side lengths of the image were 2 units. The angles stayed right angles after the dilation and the figure is the same
shape. Therefore, the figures are similar and the scale factor from original to image is ½. Figure A was also reflected across the
x-axis which can be identified by the change of the y-coordinate to the opposite value.
Example:
Describe the sequence of transformations that results in the transformation of Figure A to Figure A’.
15
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Solution:
Figure A was rotated 180 , dilated ½ from the original figure and then translated 4 right and 2 up. The coordinates after the rotation became (2,2),
(0,0), and (-2,2). Following the dilation, the coordinates (1,1), (0,0), and (-1,1). After the translation of 4 right and 2 up, the coordinates for Figure
A’ became (5,3), (4, 2), and (3,3).
16
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Domain: Geometry (8.G)
Big Idea (Cluster): 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 explain, in terms of transversals why this is so.
See Grade 8 Flip Book pages 40-41 for more explanations and examples.
Learning Objectives
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Understand that that corresponding angles, alternate interior angles, and alternate exterior angles of parallel lines are equal.
Understand that when corresponding angles, alternate interior angles, and alternate exterior angles are equal, then lines are parallel.
Understand that corresponding angles of parallel lines are equal because of properties related to translation.
Understand that alternate interior angles of parallel lines are equal because of properties related to rotation.
Construct an informal argument to draw conclusions about angles formed when parallel lines are cut by a transversal.
Understand the Angle Sum Theorem for triangles; the sum of the interior angles of a triangle is always 180˚.
Construct an informal argument to draw conclusions about the angle sum of a triangle.
SBAC Required Evidence (Claim 1)
SBAC does not provide any information on how this standard within
the cluster of 8.G.1 – 8.G.5 will be assessed.
A calculator is an allowable tool for this cluster of standards.
Connections to Curricular Materials
Kaleidoscopes, Hubcaps and Mirrors does not cover this standard.
Supplemental lessons will be identified in the pacing guide below and
in the additional resources section in the box to the right.
Connections to Prior Learning
In Grade 7, students will have used “facts about supplementary,
complementary, vertical and adjacent angles in a multi-step problem to
write and solve simple equations for an unknown angle in a figure”
(7.G.5). This will be backfill until 2015-2016.
Additional Resources/Technology Resources
CMP2 Supplemental Lessons Student Edition and Teacher Edition
Engage NY Grade 8 Module 2 Lessons 12-14 (properties of angles and
triangle angle sum)
8.G.1 – 8.G.5 Transformational Geometry PowerPoint from NJCTL
Lunch Lines task and Window “Pain” task from Georgia DOE
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Grade 8 Transformational Geometry Unit Pacing Guide for 2014-2015
using Kaleidoscopes, Hubcaps, and Mirrors and supplemental lessons
The concern with this unit is that the investigations are primarily off grid and students will only be presented with transformations in the coordinate
plane during an assessment. Modifications should be made at the building level to provide more practice with transformations in the coordinate
plane given the parameters on page 2 which are also found in the SBAC item specifications for Claim 1 Target G.
Please read the alignment notes when teaching the CMP2 supplemental lessons 3.1-3.7. There are some questions that will need to be
modified based on the Smarter Balanced Assessment Consortium item specifications for reflections, rotations, and dilations (reflections are
across the x- or y-axis, rotations are in multiples of 90 degrees, and dilations are from the origin).
Kaleidoscopes, Hubcaps, and
Mirrors (KHM) and CCSS-M
Aligned Lessons
KHM Problem 2.1
Problem
Set
A-C, E-F
KHM Problem 2.2
A-C, E
KHM Problem 2.3
A and D
Optional Extension task: MAP
formative lesson “Representing and
Combining Transformations”
KHM Problem 3.1
A-E
Alignment Notes
Move away from language of symmetry and focus more on measuring angles,
distance between corresponding vertices, lengths of segments of original and
image, develop relationship between pre-image and image. The focus of 8.G.1
is lines taken to lines, angles stay same measure, and parallel lines taken to
parallel lines.
Alter degree rotations to be multiples of 90 degrees for CCSS-M specifications.
Change language away from rotation symmetry and keep focus on
corresponding parts, their measurements, and relationships between their side
length and angle measures. Continue to develop connections between original
and image. The focus of 8.G.1 is lines taken to lines, angles stay same measure,
and parallel lines taken to parallel lines.
Continue to measure side lengths, angles, and make connections with
corresponding parts. Students should begin to describe the transformation
using the words right, left, up, and down. The “vector of translation” made be
introduced in this investigation. The focus of 8.G.1 is lines taken to lines,
angles stay same measure, and parallel lines taken to parallel lines.
Optional task that may be used as an extension. The lesson extends student
thinking to reflect across lines beyond x- and y-axis and rotate around points
other than the origin which are an extension beyond common core standards.
Investigation 3.1 and 3.2 will introduce the notion and notation of congruence
between figures which should be used as students move to the CMP2
CCSS-M
8.G.1
8.G.1
8.G.1
8.G.2
8.G.2, 8.G.4
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
KHM Problem 3.2
CMP2 Supplement Problem 3.1
CMP2 Supplement Problem 3.2
CMP2 Supplement Problem 3.3
CMP2 Supplement Problem 3.4
CMP2 Supplement Problem 3.5
A, C, D,
E
A-D
supplemental lessons Problem 3.1 – 3.7. Make sure to download the student
edition and teacher edition.
Continue to measure line segments, angles, and check for sets of parallel lines
8.G.2, 8.G.4
to support mathematical evidence of congruence.
Additional questions for students to answer:
8.G.2
 Write a description of the translation from
to
.
 Are
and
congruent? What evidence demonstrates the
triangles are congruent?
 How do the coordinates of the corresponding vertices compare to each
other? Is there any mathematical connection between the corresponding
vertices given the translation?
This investigation needs to be modified so the reflection of the line and polygon 8.G.2
occurs across either the x- or y-axis. Students will need more opportunity to
practice reflecting across the x- and y-axis along with a sequence of
transformations.
Additional questions for students to answer:
 Write a description of the reflection.
 Are
and
congruent? What evidence demonstrates the
triangles are congruent?
 How do the coordinates of the corresponding vertices compare to each
other? Is there any mathematical connection between the corresponding
vertices given the reflection across axis?
Provide students with grid transparencies or patty paper to help visualize
8.G.2
rotations. Students will need increased experience with clockwise and
counterclockwise rotations of multiples of 90 along with combining rotations
with other transformations. Have students continue to measure side lengths and
angles for corresponding parts of the original and image to verify congruency
following a transformation.
Provide students with transparencies or patty paper to help visualize rotations.
8.G.2
Have students continue to measure side lengths and angles for corresponding
parts of the original and image to verify congruency following a transformation.
Students should measure angles to verify angle measures stay the same. Begin
making connections between original to image scale factor and then image to
8.G.3
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
original scale factor as the reciprocal.
Modified CMP2 Supplement 3.6
Modified CMP2 Supplement 3.7
CMP2 Supplement 4.1
CMP2 Supplement 4.2
CMP2 Supplement 4.3
Modified KHM Problem 5.1
Modified KHM Problem 5.2
Additional questions for students to answer:
 What is the scale factor from
back to
?
 What is the scale factor from
back to
? Students may
find the reciprocal of 1.5 challenging. Encourage students to re-write
1.5 in fraction form.
Supplement 3.6 Teacher Guide may still be used to Launch, Explore, and
8.G.3, 8.G.4
Summarize the lesson. The modified lesson, located in the Math 8 binder,
helps build connections between coordinate points during a dilation. The
modified lesson also helps students justify how a dilation transforms the
original into a similar figure.
Supplement 3.7 Teacher Guide may still be used to Launch, Explore, and
8.G.3, 8.G.4
Summarize the lesson. Use the modified Problem 3.7 located in Math 8
binder. Problem 3.7 was modified so the center of dilation was the origin and
accurate connections with scale factor could be found by students.
CMP2 Supplement 4.1-4.3 will help students establish facts about the angle
8.G.5
sum and exterior angles of triangles. This set of problems will also help
students establish facts about the angles created when parallel lines are cut by a
transversal. Make sure to download the student edition and teacher edition.
8.G.5
Following this lesson, students should extend their investigation of the angle
8.G.5
sum of a triangle. Engage NY Grade 8 Module 2 Lesson 13 and 14 may be a
resource to support this extension into triangle angle sum.
Problem 5.1 Teacher Guide may still be used to Launch, Explore, and
8.G.3
Summarize the lesson. Continue to have students make connections between
congruence, distance of corresponding vertices and side length measures, shape
movement, corresponding parts, and descriptions of the transformation both
numerically using a coordinate rule and with words.
For the Common Core, students will have to make conjectures and justify
whether the coordinate rule is always true for each reflection.
Continue to have students make connections between congruence, distance of
corresponding vertices and side length measures, movement of parallel lines,
8.G.3
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
shape movement, corresponding parts, and descriptions of the transformation
both numerically using a coordinate rule and with words.
KHM Problem 5.3
KHM Problem 5.4
Transformational Geometry
Common Assessment (8.G.1 –
8.G.5)
For the Common Core, students will have to make conjectures and justify
whether the coordinate rule is always true for each translation.
Continue to have students make connections between congruence, distance of
corresponding vertices and side length measures, movement of parallel lines,
shape movement, corresponding parts, and descriptions of the transformation
both numerically using a coordinate rule and with words.
For the Common Core, students will have to make conjectures and justify
whether the coordinate rule is always true for each rotation.
Following this lesson, students will need continued practice with describing
sequences of transformations and the effect of the transformation on
coordinates from original to image.
A calculator is an allowable tool for this cluster of standards. For
understanding the content required to be assessed, see the rubrics at the end of
the unit guide which is based on the SBAC Grade 8 Claim 1 item specifications
for Target G.
8.G.3
8.G.3
8.G.1 – 8.G.5
Hard copies of the lessons and materials can be found at http://staff.rentonschools.us/renton/secondary-math/math-8-ccss-m-resources .
*SBAC ALD Level 3 means Smarter Balanced Assessment Consortium Achievement Level Descriptors Level 3 criterion for the SBAC assessment.
The following resources were used to create this curriculum guide: SBAC Claim 1 Target G Item Specifications and 8th Grade Common Core State
Standards Flip Book compiled by Melisa Hancock.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
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.
8.G.1
4
Within the SBAC Claim 1 item
specifications for Target G, this
standard is a basic skill to gain
access to the remaining cluster
standards. This individual
standard mastery is actually
assessed at a Level 1 for the
entire cluster (8.G.1 – 8.G.5).
Therefore, assessing 8.G.1
individually as a level 4 would
be inappropriate since a student
would only receive a Level 1 on
the SBAC for being proficient
on this individual standard
when assessed as an entire
cluster. SBAC summative
assessment assesses primarily
at the cluster level unless
specifically called out to be
assessed as an individual
standard.



3
Verify that rigid
transformations (rotation,
reflect and translation)
preserve distance and angle
measure
Understand and describe the
impact of a transformation on
a figure and its component
parts by describing the side
length and angle measure of
corresponding parts between
image and pre-image
Constructs lines, line
segments, angles and parallel
lines after the original has
been rotated, reflected or
translated.


2
Measure accurately the
corresponding angles and
side lengths of two figures
after a rigid transformation
Identify whether the image
is a reflection, translation
and/or rotation of the preimage

1
With help, minimal
success identifying a
reflection, rotation and/or
translation.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics. For more detail on the assessment of the standards, read the SBAC Item Specifications-Target G.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
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.
4
In addition to being proficient
on the standard, students can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.



8.G.2
3
Describe a sequence of

transformations (rotations,
reflections and translations)
that can verify whether two
dimensional figures are
congruent to each other.
Describe the impact of a
transformation on a figure and 
its corresponding parts
without coordinates.
Construct a new figure after
the original figure has been
rotated, reflected and
translated in a coordinate
plane.
2
Construct reflections and
translations in a coordinate
plane.
Measure accurately the
corresponding angles and
side lengths of two figures
after a rigid transformation
Describe how to verify
whether two figures are
congruent.

1
With help, minimal
success constructing
reflections and
translations in a
coordinate plane.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics. For more detail on the assessment of the standards, read the SBAC Item Specifications-Target G.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
Standard 8.G.3 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
8.G.3
4
In addition to being proficient
on the standard, students can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.



3
Describe a sequence of

transformations (rotations,
reflections, translations and

dilations) that can verify
whether two-dimensional
figures are similar to each
other.
Construct a new figure after
the original figure is dilated,
rotated, reflected and/or
translated in a coordinate
plane.
Describe the effect of
dilations, translations,
rotations, and reflections on
two-dimensional figures using
coordinates.
2
Identify a dilation in the
coordinate plane.
Describe the results of a
dilation on a figure.

1
With help, minimal
success describing the
results of a dilation on a
figure.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics. For more detail on the assessment of the standards, read the SBAC Item Specifications-Target G.
24
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
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.
8.G.4
4
In addition to being proficient
on the standard, students can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.

3
Describe a sequence that
exhibits the similarity
between two shapes and
understand that the angle
measures are unchanged


2
Describe the results of a
dilation on a figure.
Describe how to verify
whether two figures are
similar.

1
With help, minimal
success describing the
results of a dilation on a
figure.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics. For more detail on the assessment of the standards, read the SBAC Item Specifications-Target G.
25
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)
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 explain, in terms of transversals why this is so.
8.G.5
4
In addition to being proficient
on the standard, students can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.



3
Construct an informal
argument to draw conclusions
about angles formed when
parallel lines are cut by a
transversal.
Construct an informal
argument to draw conclusions
about the angle sum of a
triangle.
Construct an informal
argument to draw conclusions
about the exterior angles of
triangles


2
Understand that that
corresponding angles,
alternate interior angles, and
alternate exterior angles of
parallel lines are equal.
Understand the Angle Sum
Theorem for triangles.

1
With help, minimal
success identifying angle
measures when parallel
lines are cut by a
transversal.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics. For more detail on the assessment of the standards, read the SBAC Item Specifications-Target G.
26
This document is a draft and will continue to develop as we learn more about the Common Core State Standards for Mathematics and the SBAC Assessment. (8.G.1-8.G.5)