Intent

This activity gives students an opportunity to work with the concept of proportionality in nongeometric, real-world contexts.

Mathematics

Two similar triangles have side lengths that are proportional and angles that are equal. In other words, one similar triangle is related to another by a scale factor, but their angle measures remain invariant. Analogously, when proportions occur in the context of real life, multiplying one aspect of the situation by some factor doesn’t mean everything should be multiplied by that factor. For example, when doubling a recipe, you double the amount of each ingredient but not the cooking time. In this activity, students continue to develop their understanding of similarity by focusing on the idea of invariance.

Progression

Students explore the three contexts posed here individually and discuss their results in class.

Approximate Time

15 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

This activity requires little or no introduction.

Discussing and Debriefing the Activity

The key idea to bring out in the discussion of this activity is that when one number in a situation is multiplied by some factor, other numeric aspects of the situation may or may not be multiplied by that factor.

After reviewing each example, ask how the ideas in this activity are related to what students have been learning about similar triangles. How is this idea related to similar triangles? Students should recognize that although, for example, the lengths of the sides of one triangle may be double those of a similar triangle, the angles are not doubled.

Key Question

How is this idea related to similar triangles?

Supplemental Activity

What If They Kept Running? (extension) uses distance and rate of speed as another context for investigating proportionality.