This activity is the first time students are asked to dilate a figure in this course. Monitor for students with misconceptions, such as making the distance from to 120 units, rather than making the distance from to 120 units so that the distance from to is 80 units.

If students aren't sure how to get started, support them in understanding the definition of dilation given in the task. Draw an example using the labels in the definition, and invite them to match the labels in the task to the example.

Invite students to share ideas that they had for how to get started with the problem. (Draw a ray from through .)

Display the images, and ask students why they are incorrect. (The total distance should be 120 units. Point is not on ray . They used the wrong center of dilation, should be as long as .)

Make sure that all students understand how to draw rays from the center point through the point(s) to be dilated, and how to find the image of the points by measuring along the rays. Students will dilate more complex figures in subsequent activities.

In this activity, students dilate a figure with multiple points and multiple shapes. The goal is for them to practice dilation and begin to confirm some properties of dilation. Students will have additional opportunities to formalize properties of dilation in subsequent lessons. (A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor.)

Monitor for students who use these strategies, listed in order of precision:

• Use the two given lines to place and , and then estimate all of the other shapes and points.
• Measure distances within the figure, and try to replicate them, for example, calculating the radii of the two dilated circles and trying to place them without scaling and along rays through .
• Draw rays from through each point in the original figure, and place points in the image halfway between and the corresponding point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

The goal of this discussion is to surface some observations about dilations. 

Invite previously selected students to share their strategies for dilating points. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work.

Connect the different responses to the learning goals by asking questions, such as:

• “What is precise about each student’s method for dilating points?”
• “How did each student’s method fill in the rest of the figure after dilating a few points?”

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the relative length of segments, or measure of angles, does not come up during the conversation, ask students to discuss these ideas.

Tell students that when we write in an equation or expression it means the length of the segment . The equation means “The length of segment is half as much as the length of segment .”