This activity introduces important language students will apply to describe scaled copies. In particular, it introduces the important idea of corresponding parts. Students have previously analyzed corresponding sides in figures. Here they will begin to examine angles explicitly as well, understanding that corresponding angles in a figure and its scaled copy have the same measure. As students identify corresponding parts, they are making use of structure (MP7).

This activity works best when each student has access to tracing paper. If tracing paper is unavailable, consider using the digital version of the activity. In the digital version, students use an applet to drag and rotate a given fixed angle over the images of the railroad sign. Similar to using tracing paper, the applet allows students to compare the size of corresponding angles without needing to find specific numeric values for the measurements.

Select a few students to share their observations about angles. Discuss the size of corresponding angles in figures that are scaled copies and those that are not. Ask questions such as:

• “In the scaled copy, Copy 1, did the size of any angle change compared to its corresponding angle in the original sign?” (No)
• “In Copy 2, did the size of any angle change compared to its corresponding angle in the original sign?” (Yes, for example, angle has a different measure than angle .)
• “What can you say about corresponding angles in two figures that are scaled copies of one another?” (They have the same measure.)
• “What can you say about corresponding angles in two figures that are not scaled copies?” (They might not have the same measure.)

In this activity, students continue to practice identifying corresponding parts of scaled copies. By organizing corresponding lengths in a table, students see that there is a single factor that relates each length in the original triangle to its corresponding length in a copy (MP8). They learn that this number is called a scale factor.

As students work on the first question, listen to how they reason about which triangles are scaled copies. Identify groups who use side lengths and angles as the basis for deciding. (Students are not expected to reason formally yet, but should begin to look to lengths and angles for clues.)

As students identify corresponding sides and their measures in the second and third questions, look out for confusion about corresponding parts. Notice how students decide which sides of the right triangles correspond.

If students still have access to tracing paper, monitor for students who use this tool strategically (MP5).

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing students that use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—including, especially, everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

Display the image of all triangles and invite a couple of students to share how they knew which sides of the triangles correspond. Then, display a completed table in the third question for all to see.

Direct students’ attention to the reference created using Collect and Display. Ask each group to present its observations about one triangle and how the triangle has been scaled from the original. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. As students present, record or illustrate their reasoning on the table, such as by drawing arrows between rows and annotating with the operation that students are describing, as shown here.

Use the language that students use to describe the side lengths and the numerical relationships in the table to guide students toward scale factor. For example: “You explained that the lengths in Triangle F are all twice those in the original triangle, so we can write those as ‘2 times’ the original numbers. Lengths in Triangle A are half of those in the original; we can write ‘times’ the original numbers. We call those multipliers—the 2 and the —scale factors. We say that scaling Triangle O by a scale factor of 2 produces Triangle F, and that scaling Triangle O by a scale factor of  produces Triangle A.”