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).

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.

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.)

In this activity, students practice identifying similar figures. Each student has a card with a figure on it and they identify someone with a similar (but not congruent) figure. Students work together to construct an argument for why their two figures are similar (MP3).  

Monitor for students using these methods to identify a partner:

• Process of elimination (figures with different angle measures cannot be similar to each other)

• Looking for the same kind of figure (rectangle, square, rhombus)

• Looking for a the same kind of figure with congruent corresponding angles

• Looking for a the same kind of figure with corresponding sides scaled by the same scale factor 

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.

The purpose of this discussion is to highlight different strategies for finding similar figures. For example, students can use the process of elimination (one shape is a rectangle and the other is not) or actively look for certain features (a rhombus whose angle measures are 60 degrees and 120 degrees).

Direct students' attention to the reference created using Collect and Display. Ask students to share how they determined that two figures were similar. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. 

Discuss with students:

• “Were the side lengths important?” (For some figures, such as the rectangles, the side lengths were important.)

• “Were the angles important?” (Yes. Several of the figures had different angles and that meant they were not similar.)

• “How did you know when two figures were not similar? (The shapes were not the same. Corresponding angles didn’t match. Corresponding sides were multiplied by different scale factors.) 

Some students may notice that all squares are similar since they have 4 right angles and proportional side lengths. If not mentioned by students, show how the first problem gives an example of rhombuses that are not similar to one another.