In this activity, students match graphs to rules in coordinate transformation notation. Then they analyze both the rules and images to decide which represent similarity transformations and which represent rigid transformations.

Students will use a variety of strategies to explain which figures are similar or congruent (counting, the Pythagorean Theorem, recognizing right angles, properties of isosceles triangles, and trigonometry). Monitor for students who use these different methods.

If students suggest that figures are congruent or similar simply because they look that way, tell them that they need to provide more backing for their answer. What are some ways we can verify that two figures are congruent or similar? Remind students that for triangles, they’ve learned some shortcuts that they can use here.

Select previously identified students to share their reasoning. Ask them how they calculated side lengths or angle measures. The key point to emphasize is that similar figures have congruent angles and proportional sides, while congruent figures have both congruent angles and congruent sides. Students know some shortcuts for triangles, but for the rectangles there is no shortcut.

In this activity, students work backward to figure out a transformation rule. They are given the opportunity to come up with their own method of organizing their work. Monitor for students who have methods of clearly documenting information they are given as well as students who record their attempts at a rule. After students write a rule, they analyze the figures to decide if the transformation takes shapes to congruent shapes.

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

If students struggle to write a rule, ask them to start by writing out, in words, the pattern that they see. For example, they may write, “The -coordinate stays the same and the -coordinate doubles.” Then ask how they could put those words into coordinate transformation notation.

The goal is to use the language of distance and angle-preserving moves to describe the two transformations. Here are some questions for discussion:

• “Look at the corresponding side lengths and angles in the 2 pairs of triangles. How do they compare?” (In triangles and , all sets of corresponding sides and all sets of corresponding angles are congruent. In triangles and , neither the corresponding sides nor the corresponding angles are congruent.)
• “What would a transformation look like if it kept the angles the same but not the side lengths?” (It could be a dilation. The corresponding sides' lengths would need to be proportional for this to be true.)
• “Is it possible for a transformation to keep side lengths the same but not keep the angles the same?” (This isn’t possible in a triangle. For a square, it could be transformed into a rhombus. However, none of our standard transformations (translation, reflection, rotation, or dilation) would accomplish this.)