Students sort different images during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

As students work, encourage them to justify their decisions about which triangles are similar using more precise language (MP6).

If students need help visualizing the similar triangles, ask them what they know stays the same between similar triangles (angle measures). Then, encourage them to rotate the cards to line up corresponding angles to better see the corresponding sides and the scale factor.

The purpose of this discussion is for students to revisit how to find scale factors between similar triangles and use them to find side lengths.

For each pair of similar triangles that students found, ask a group what they wrote for their similarity statement. Then ask the whole class if anyone could suggest a different way of writing the similarity statement. (The letters can be in a different permutation as long as corresponding vertices are in the same order.)

Then invite a group to share what they found for the scale factor between the similar triangles and how they found the value. Follow their response by asking the whole class if there could be another scale factor that applies or another way to get the scale factor. (The reciprocal of the scale factor would also work. Because there are many ratios that are equivalent in similar triangles, students may use different pairs of values in different orders.)

This lesson gives students an opportunity to practice using the Pythagorean Theorem along with similar triangles in a real-world situation. In later lessons students will prove the Pythagorean Theorem using some of the ideas from this activity.

If students struggled to use the Pythagorean Theorem in a previous lesson:

• Begin by asking students what they notice and wonder about the image from this activity.
• Draw students’ attention to the right triangles, and ask them what must be true about the side lengths.
• Provide some possible lengths for sides, and have students practice finding an unknown side length in an isosceles right triangle to allow students to practice working with the square root of 2.

A key point to emphasize in this discussion is that ratios in which each ratio compares side lengths within each figure are also equivalent across similar figures. That is, for a triangle with side lengths , and a similar triangle with corresponding side lengths , we have equivalent ratios such as  .

Invite students to share strategies for scaling the figure up from a baby blanket to a quilt for a queen-size bed.

Here are some questions for discussion:

• “What would it look like if the pattern continued with 1 larger set of triangles? What is the scale factor from these new triangles to the previous largest ones?” (A larger red triangle on top, a green one on the right, a yellow one on the bottom, and a blue one on the left. The scale factor would be another .)
• “What other considerations would a quilter need to think about before making this quilt?” (How much of each color fabric would they need to buy? Will this quilt get stuffed?)