Similarity in Light and Shadow

This lesson introduces similarity not through a definition but through two experiences that force students to reason proportionally. The first — estimating the size of Washington’s nose from a photograph of Mount Rushmore — is accessible, playful, and reveals immediately that students already understand something about similarity even before it is named. The second — the flashlight shadow investigation — builds the geometric intuition that will underpin the entire unit: scaling happens from a point, all points move along straight lines from that point, angles are preserved, and corresponding lengths change by a constant factor.

 

The shadow investigation is adapted from Matthew Engle’s Rosenthal Prize lesson “Bringing Similarity Into Light” (MoMath, 2017). The full original is available at momath.org and is worth reading before teaching this lesson. Credit should be given in the student materials.

 

A note on the opening task: some students will use scale factor instinctively, setting up a ratio of nose to head in the photograph and multiplying by 60. Others will write a proportion. Both approaches are correct. Do not push students toward one method in Lesson 6.1 — let both surface naturally. The consolidation in Lesson 6.2 is where both methods are explicitly named and compared.

 

The definitions conversation worth having here: the task begins with a question about where the nose starts and ends. This is a genuine definition problem. What counts as the top of the nose? The bottom? Two groups using the same photograph but different definitions will get different answers — and both will be right by their own definitions. This connects directly to the course-long thread about definitions as prerequisites for valid arguments. Euclid named his terms before his axioms for the same reason: without agreed definitions, measurements are not comparable. Spend a few minutes on this before students measure.

 

For the flashlight activity: productive struggle is intentional in this lesson. The task directions are deliberately open-ended, and many students will be initially unclear what to do. Let groups discuss and work it out. The discomfort is brief and productive — groups that figure out the setup themselves understand what they are measuring and why. The key physical insight to watch for: all three side ratios of the shadow triangle approximate the same value, and that value equals the ratio of the distance from light to shadow vertex to distance from light to cardboard vertex. Students who find this connection are beginning to understand dilation as a geometric transformation.