enVision Mathematics Topic 6
8th Grade; February – March (6 weeks); 3rd Quarter
Embedded Files
Topic Title(s):
Congruence and Similarity
Prepared Graduates:
MP3. Construct viable arguments and critique the reasoning of others.
MP5. Use appropriate tools strategically.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.
Standard(s):
4. Geometry
The highlighted evidence outcomes are the priority for all students, serving as the essential concepts and skills. It is recommended that the remaining evidence outcomes listed be addressed as time allows, representing the full breadth of the curriculum.
Students Can (Evidence Outcomes):
8.G.A. Geometry: Understand congruence and similarity using physical models, transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and translations: (CCSS: 8.G.A.1)
Lines are taken to lines, and line segments to line segments of the same length. (CCSS: 8.G.A.1.a)
Angles are taken to angles of the same measure. (CCSS: 8.G.A.1.b)
Parallel lines are taken to parallel lines. (CCSS: 8.G.A.1.c)
Demonstrate 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. (CCSS: 8.G.A.2)
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (CCSS: 8.G.A.3)
Demonstrate 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. (CCSS: 8.G.A.4)
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. (CCSS: 8.G.A.5)
Think about how rotations, reflections, and translations of a geometric figure preserve congruence as similar to how properties of operations such as the associative, commutative, and distributive properties preserve equivalence of arithmetic and algebraic expressions. (Entrepreneurial Skills: Critical Thinking/Problem Solving and Inquiry/Analysis)
Explain a sequence of transformations that results in a congruent or similar triangle. (MP3)
Use physical models, transparencies, geometric software, or other appropriate tools to explore the relationships between transformations and congruence and similarity. (MP5)
Use the structure of the coordinate system to describe the locations of figures obtained with rotations, reflections, and translations. (MP7)
Reason that since any one rotation, reflection, or translation of a figure preserves congruence, then any sequence of those transformations must also preserve congruence. (MP8)
Inquiry Questions
How are properties of rotations, reflections, translations, and dilations connected to congruence?
How are properties of rotations, reflections, translations, and dilations connected to similarity?
Why are angle measures significant regarding the similarity of two figures?
Coherence Connections
This expectation represents major work of the grade.
In previous grades, students solve problems involving angle measure, area, surface area, and volume, and draw, construct, and also describe geometrical figures and the relationships between them.
In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations.
In high school, students extend their work with transformations, apply the concepts of transformations to prove geometric theorems, and use similarity to define trigonometric functions.
Academic Vocabulary & Language Expectations:
Transformation, translation, image, reflection, line of reflection, rotation, angle of rotation, center of rotation, congruent, dilation, scale factor, enlargement, reduction, similar, transversal, corresponding angles, alternate interior angles, same-side interior angles, remote interior angles, exterior angle of a triangle
Assessments:
Instructional Resources & Notes:
enVision Mathematics Topic 6
Let's Investigate! Tasks
Let's Investigate! Copy That (TE) (relates to Lesson 6-1)
Tier 1 Intervention & Supports (i-Ready Tools for Instruction):
Tier 1 Intervention: Rigid Transformations, 180° Rotations of Polygons, Transformations and Congruence, Sequences of Transformations, Dilations, Dilation and Similar Figures
enVision Mathematics 6-8 & Number Worlds Connections (for SVVSD Special Education teachers only)
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