Ratios and Proportions / Ratios Within and Between Scaled Shapes

Understanding

Students should recognize that lengths scale by a linear scale factor, but areas scale by the square of the scale factor that relates the lengths (if the area is measured in the unit of measurement derived from that used for length).

What to look for

Have students examine the internal ratios between each shape as another way to make a connection of the ratio to scaling shapes.

Sample Assessment

If the Rectangle S is an enlarged copy of Rectangle R, what is the scale factor?

Answer: 3

The Big Idea

Students can set up proportions in tables or equations. Students can also reason multiplicatively by 1) applying a scale factor that relates lengths in two different figures, or 2) considering the ratio of two lengths within one figure, finding a multiplicative relationship between those lengths, and applying that relationship to the ratio of the corresponding lengths in the other figure

What are the students doing?

Students can use the grid to determine measures as well as the visual representation of how a scale factor of x, which is a linear measure, produces x2 copies of the area in a scaled figure.

What is the teacher doing?

Note that the scale factor and side lengths are randomly generated so students may have different responses. Sharing their examples and thinking could help them begin to generalize about scale factors. Examining the internal ratios among the sides of each shape is another way to make a connection between the shapes.