Students may multiply the area of the bottom floor by the scale factor, , 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.

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).

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.

The goal is to make sure students understand that area scales by the square of the scale factor. 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 by the original triangle’s area.)
• “What are some different ways to deal with the scale factor ?” (We can square the fraction to get , and then multiply that fraction by the area, which gives us  . Or, we can divide 3 by 4 to get 0.75, square that to get 0.5625, then multiply that by 100, which gives us 56.25.)

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.

The goal of this discussion is to describe the graph representing . Here are some questions for discussion:

• “What equation represents the relationship between the square feet covered by the primer () and the scale factor ()?” ( )
• “What is the domain of this function?” (Only -values greater than or equal to 0 make sense in a square root function.)

Tell students, “While a dilation of 0 doesn’t make sense for the painting, it is in the domain of the square root function. Add the point with an -coordinate of 0 to your graph, and connect it with a smooth curve.”

• “You calculated the rate of change for an additional 10 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 11 than it was between 16 and 26.)
• “What would the rate of change for the perimeter graph be over those same intervals?” (That graph is a line with a constant slope of .)
• “How do the different rates of change relate to the shape of the square root graph?” (The graph is steeper at first then gets flatter.)