options: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

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.

Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.

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.

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.
problem_activity: Optional activity
The goal of this activity is to verify, via angle calculations, that equilateral triangles (and hence) regular hexagons can be used to make regular tessellations of the plane. Students have encountered the equilateral triangle plane tessellations earlier in grade 8 when working on an isometric grid. In order to complete their investigation of regular tessellations of the plane, it remains to be shown that no other polygons work. This will be done in the next activity.
Students are required to reason abstractly and quantitatively (MP2) in this activity. Tracing paper indicates that six equilateral triangles can be put together sharing a single vertex. Showing that this is true for abstract equilateral triangles requires careful reasoning about angle measures.
Launch
In the previous task, equilateral triangles, squares, and hexagons appeared to make regular tessellations of the plane. Tell students that the goal of this activity is to use geometry to verify that they do.
Refer students to regular polygons printed in the previous activity for a visual representation of an equilateral triangle.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Some students may benefit by using virtual or concrete manipulatives (geogebra, tracing paper, or equilateral triangle cut-outs) to connect abstract concepts to concrete objects.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
What is the measure of each angle in an equilateral triangle? How do you know?
How many triangles can you fit together at one vertex? Explain why there is no space between the triangles.
Explain why you can continue the pattern of triangles to tessellate the plane.
How can you use your triangular tessellation of the plane to show that regular hexagons can be used to give a regular tessellation of the plane?
Student Response
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Anticipated Misconceptions
Students may know that an equilateral triangle has 60-degree angles but may not be able to explain why. Consider prompting these students for the sum of the three angles in an equilateral triangle.
Students may not see a pattern of hexagons within the triangle tessellation. Consider asking these students what shape they get when they put 6 equilateral triangles together at a single vertex.
Activity Synthesis
Consider asking the following questions to lead the discussion of this activity:
“How did you find the angle measures in an equilateral triangle?” (The sum of the angles is 180 degrees, and they are all congruent so each is 60 degrees.)
“Why is there no space between six triangles meeting at a vertex?” (The angles total 360 degrees, which is a full circle.)
“How does your tessellation with triangles relate to hexagons?” (You can group the triangles meeting at certain vertices into hexagons, which tessellate the plane.)
“Are there other tessellations of the plane with triangles?” (Yes. You can make infinite rows of triangles that can be placed on top of one another—and displaced relative to one another.)
Consider showing students an isometric grid, used earlier in grade 8 for experimenting with transformations, and ask them how this relates to tessellations. (It shows a tessellation with equilateral triangles.)
Point out that this activity provides a mathematical justification for the “yes” in the table for triangles and hexagons.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a response to one of the discussion questions is shared with the class, invite students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. This will provide more students with an opportunity to practice using language about tessellations.
Design Principle(s): Support sense-making


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