Lesson 7

Students examine statements deciding in each case whether the statement is always true, sometimes true, or never true. Since figures are similar if one is the result of reflecting, rotating, translating, and dilating the other, and figures are congruent only as a result of the rigid transformations, congruent figures are similar but not vice versa. Though students have not studied how angles behave under dilations, students have had multiple opportunities to see that dilations influence the size of a shape but not its angles. They also know from work in grade 7 that scaled copies have corresponding angles with the same measure, and dilations have been characterized as a process that makes scaled copies.

Display all 3 statements. Ask students to decide whether each of the statements is always, sometimes, or never true and give a signal when they have reasoning to support their decision.

Discuss each statement one at a time with this structure:

• Poll the class on their answer choice and display the answers.
• If everyone agrees on one answer, ask a few students to share their reasoning, recording it for all to see.
• If there is disagreement on an answer, ask students with opposing answers to explain their reasoning to come to an agreement on an answer.

In the previous lesson, students saw that figures are similar when there is a sequence of translations, rotations, reflections, and dilations that map one figure onto the other. This activity focuses on some common misconceptions about similar figures, and students have an opportunity to critique the reasoning of others (MP3). 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 foster student thinking about when two polygons are not similar. 

Monitor for students who formulate these ideas for establishing that two figures are not similar:

• The angle measures are different (problem 1).
• The side lengths need to be multiplied by different scale factors in order to match up.

Select these students to share during the discussion.

Tell students they will look at some polygons that are claimed to be similar. What are some characteristics of similar polygons that are easy to recognize that can be used to confirm or deny the claims? Give 3 minutes of quiet work time followed by a whole-class discussion.

Students might think that checking for angle congruence alone can determine similarity. They might also think that just checking if all side lengths have the same scaled value will be enough to determine similarity. However, both the angle sizes and the scale factors must be checked together in determining similarity.

Students might think the side lengths must be different in order for two figures to be similar, but a dilation does not need to be used in the sequence of transformations to show similarity (recall the warm-up where students saw that congruent figures are always similar).

Invite selected students to share their choices and explanations, making sure that these points are highlighted:

1. If corresponding angles aren't congruent, then the figures cannot be similar (first problem).
2. If the side lengths are not all scaled by the same value, then the figures cannot be similar (second problem).

Unlike what Clare says, the same dilation (with the same scale factor) has to be performed to all parts of a figure to result in a similar figure. Another way to state the two important conclusions above is that for similar polygons:

• Corresponding angles are congruent.
• Corresponding side lengths are proportional.

In the previous activity, students learn that in order to be similar, two figures must have congruent corresponding angles and proportional corresponding side lengths. In this activity, students apply this knowledge. Each student has a card with a figure on it and they identify someone with a similar (but not congruent) figure.  

Monitor for students using these methods to identify a partner:

• Process of elimination (figures with different angle measures cannot be similar to one another)
• 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  

Distribute one card to each student. Explain that the task is for each student to find someone else in the class who has a similar (but not congruent) figure to their own, and be prepared to explain how they know the two figures are similar. 

If the number of students in class is not a multiple of 10, ensure that any unused cards are matching pairs of similar figures. If there is an odd number of students, one or more students can be responsible for two cards, or some students can be appointed as referees. 

If students have a hard time getting started, ask 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? What can you say about the angles in a similar figure?

Invite selected students to share how they found a partner. Highlight different strategies: it is possible to proceed by process of elimination (this shape is a rectangle and mine is not so it is not a match) or by actively looking for certain features (e.g. a rhombus whose angle measures are 60 degrees and 120 degrees). Next ask how they knew their polygons were similar and which scale factor they used to show similarity. 

Ask students what they looked for in searching for a partner with a similar figure. In particular ask

• “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.)

Students may notice that all squares are similar since they have 4 right angles and proportional side lengths. On the other hand, this activity provides examples of rhombuses that are not similar to one another.