Monitor for different ways students apply the concepts of scaling and area to the cross-sections of a pyramid-shaped building. Here are some approaches students might take, from more common to less common:  

• Find each dimension of the top floor, and then find the area using those dimensions.
• Multiply the original area by the squared scale factor.

Plan to have students present in this order to support the understanding that multiplying by comes from both the base and the height being scaled by .

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).

The purpose of this discussion is to emphasize that two different dimensions are multiplied by the scale factor, so the area must be multiplied by the square of the scale factor.

Invite previously selected students to share their reasoning. Sequence the discussion of the approaches by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

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

• “How is the scale factor used in each approach?” (In the first approach, the scale factor is multiplied by each dimension, and then the resulting dimensions are multiplied by each other. The second approach uses the same values, but in a different order. First, the scale factor is multiplied by itself, and then the squared scale factor is multiplied by the area of the original rectangle.)
• “Which approach do you prefer?” (I can see the steps of the first method more easily in the diagram. It depends on the numbers.)

If need be, display these calculations to support students making the connections between the approaches:
 

This optional activity addresses the common misconception that if a figure is dilated by a factor of , the image’s area also changes by a factor of . Use this activity if students are still unsure about this idea. As students decide which response (if either) they agree with, they are critiquing the reasoning of others (MP3).

This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured partner meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions, such as, “What did you mean by ?” and “Can you say that another way?” Finally, students write a second draft of their response, reflecting ideas from partners and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.

In this activity, students create a graph representing the relationship between dilated area and scale factor, and use it to answer questions. As students use the graph to solve problems, they are reasoning abstractly and quantitatively (MP2).

In the digital version of the activity, students use an applet to complete a table that shows the relationship between area and scale factor. The applet allows students to use a slider to change the area of a square and observe how the square’s dimensions change. The digital version may help students make the connection between the dilated area and scale factor more quickly by seeing the relationship in a dynamic way.