Lesson 5

Students apply the concepts of scaling and area to the cross sections of a pyramid-shaped building.

Monitor for students who find the dimensions of the top floor then multiply to find the area, and for those who multiply the original area by the squared scale factor.

Arrange students in groups of 2. Provide access to calculators. Give students quiet work time and then time to share their work with a partner.

Students may multiply the area of the bottom floor by the scale factor, \(k\), arriving at an area of 746 square meters. Ask these students to check their answer by finding the dimensions of the top floor and multiplying them to find the area.

Invite previously selected students to share their reasoning in this order: first, a student who found the dimensions of the top floor; second, a student who squared the scale factor.

Ask students how the scale factor is used in each approach. In the first approach, the scale factor is multiplied by each dimension, then the resulting dimensions are multiplied by each other:

\((0.32 \boldcdot 53)(0.32 \boldcdot 44)\)

\(16.96 \boldcdot 14.08\)

\(238.7968\)

The second approach uses the same values, but in a different order. First, the scale factor is multiplied by itself, then by the area of the original rectangle:

\((0.32 \boldcdot 0.32)(53 \boldcdot 44)\)

\((0.32^2)(53 \boldcdot 44)\)

\((0.1024)(2332)\)

\(238.7968\)

Each approach arrives at the same answer.

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

Students may believe they can’t calculate the area of the dilated triangle if they don’t have its dimensions. Remind them of the two approaches highlighted in the warm-up and ask if either of those applies here.

Here are some questions for discussion:

• “Why didn’t we need the dimensions of the original triangle to calculate the area of the dilated triangle?” (We could multiply \(k^2\) by the original triangle’s area.)
• “What are some different ways to deal with the scale factor \(\frac34\)?” (We can square the fraction to get \(\left(\frac34 \right)^2 = \frac{9}{16}\), then multiply that fraction by the area. Or, we can divide 3 by 4 to get 0.75, square that to get 0.5625, then multiply that by 100.)

In this activity, students create a graph representing the relationship between dilated area and scale factor, and use it to answer questions. They analyze the average rate of change at different parts of the graph, noting that the rate of change is not constant. As students use the graph to solve problems, they are reasoning abstractly and quantitatively (MP2).

The goal is to draw conclusions about the shape of the graph representing \(y=\sqrt{x}\). Here are some questions for discussion:

• “What equation represents the relationship between the square feet covered by the paint (\(x\)) and the scale factor (\(y\))?” (\(y=\sqrt{x}\))
• “You calculated the rate of change for an additional 2 square feet of paint at different parts of the graph. Were those rates the same?” (No. The rate of change was larger between 1 and 3 than it was between 14 and 16.)
• “How do the different rates of change relate to the shape of the graph?” (The graph is steeper between 1 and 3 than it is between 14 and 16.)