Unit 2, Lesson 7
Similar Polygons
Grade 8
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This Math Talk focuses on congruence and similarity. It encourages students to think about the similarities and differences between the two terms and to rely on what they know about dilations and rigid transformations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students determine if two figures are similar or not.
In explaining whether a statement is always true, sometimes true, or never true, students need to be precise in their word choice and use of language (MP6).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Decide mentally whether each statement is always true, sometimes true, or never true.
If two figures are congruent, then they are similar.
If two figures are similar, then they are congruent.
If a triangle is dilated with the center of dilation at one of its vertices, the side lengths of the new triangle will change.
If a triangle is dilated with the center of dilation at one of its vertices, the angle measures of the triangle will change.
To involve more students in the conversation, consider asking:
“Who can restate
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
MLR1: Stronger and Clearer Each Time
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This activity focuses on common misconceptions about similar figures. Two polygons with proportional side lengths but different angles are not similar, and two polygons with the same angles but side lengths that are not proportional are also not similar. Reasoning through these problems will help further student thinking about when two polygons are not similar and give students an opportunity to critique the reasoning of others (MP3).
Ask students to share characteristics of similar polygons that are easy to recognize and record them for all to see. Students may say that similar polygons look like the same shape but could be different sizes (dilated by a given scale factor), or that similar polygons might be facing a different way (reflected or rotated). It is not necessary for students to discuss angle measures at this time. This discussion is meant to prime student thinking around similarity.
Arrange students in groups of 2. Give students 3–4 minutes of quiet work time followed by a partner then whole-class discussion.
Let’s look at a square and a rhombus.
The goal of this discussion is to make sure students understand that for similar figures, corresponding angles are congruent and corresponding side lengths are proportional.
Discuss with students:
“In the question about the square and the rhombus, both polygons had all the same side lengths but they still were not similar. What does this tell us about similar polygons?” (If corresponding angles are not congruent, then the figures cannot be similar.)
“In the question about the rectangles, both polygons had all the same angles but they still were not similar. What does this tell us about similar polygons?” (If the side lengths are not all scaled by the same value, then the figures cannot be similar.)
When two polygons are similar:
Every angle and side in one polygon has a corresponding angle and side in the other polygon.
All pairs of corresponding angles have the same measure.
Consider the two rectangles shown here. Are they similar?
It looks like rectangles
Here is an example that shows how sides can correspond with a scale factor of 1, but the quadrilaterals are not similar because the corresponding angles don’t have the same measure:
Help us improve by sharing suggestions or reporting issues.
Some students may think the side lengths must be different in order for 2 figures to be similar, but a dilation does not need to be used in the sequence of transformations to show similarity. Prompt students to recall the Warm-up where students saw that congruent figures are always similar.
Some students may have a hard time getting started. Prompt them to focus on properties of their figure that will be shared by a similar figure. For example, will a similar figure be a quadrilateral? Will a similar figure be square? A rectangle? A rhombus? Ask them to recall what is true about the angles in a similar figure.