Injecting Surprise Into the Triangle Midline Theorem

Pi Day 2022 is now over, but I’m still thinking about a tweet from 10-K Diver: Take two random numbers X and Y between 0 and 1. What is the probability that the integer nearest to X/Y is even? The answer—spoiler ahead—is (5 – π)/4. (You can run my Web Sketchpad simulation of the problem here.)

I smiled at this result, amazed that π found its way into the answer. This element of surprise is something that I regard as mathematics’ secret weapon: Mathematics is beautiful and its results are often astonishing—why not take advantage of both aspects of the subject to stir students’ interest?

With this goal in mind, let’s see what surprises await us in the triangle midline theorem. In the Web Sketchpad model below, points D and E are the midpoints of sides AB and AC, respectively. For this particular triangle, the ratio DE/BC is 0.5. That value may raise an eyebrow since the measures of DE and BC are “messy.” Is this a coincidence or a hint of a deeper mathematical relationship?

To investigate, set the triangle in motion by dragging point A. It’s no surprise the length of BC stays the same, but it is surprising that DE‘s length remains constant as well, keeping the ratio DE/BC fixed. If instead we drag point B, the lengths of both BC and DE change, but our ratio is a steadfast 0.5.

Such experiments, with one or more numerical values remaining invariant while other change, are at the heart of many dynamic geometry investigations. Baked into these activities is an implicit assumption that the discovery of a numerical invariant, such as the ratio DE/BC, will provide a moment of surprise and wonder, lending motivation for students to prove why such invariants exist. I’m not about to discount this method of inquiry—clearly numerical data matters—but I do know that I’ve been repeatedly disappointed when students’ reactions to a numerical invariant was muted at best.

So what might we do in this midline example to inject some other element of surprise into this mix? The Connected Geometry curriculum offers a clever idea: Rather than head straight to the midline construction, start with a related construction: In the websketch below, segments BD, BE, BF, and BG all share endpoint B. Drag point B back and forth along AC and observe the traces of the midpoints of the four segments. Even though all four segments have different lengths and share only an endpoint, it appears that the traces all form segments of the same length and are parallel to AC. Wow!

You can gather more data by dragging points D, E, F, and G to new locations, clearing the traces with the Trace widget, and dragging point B again. (You can also build the model from scratch using the tools on page 2.)

This dynamic geometry experiment is less precise than the typical numerical approach: We don’t learn anything about the length of the traces and how they compare to the length AC. But I would argue that the images of the traces—all seemingly the same length and parallel to AC—create an impact beyond what can be achieved from numerical data alone.

I’ll end with an ambitious proposal: How hard would it be to examine the entire geometry curriculum and recast each topic so as to make it more surprising? And would such work simultaneously promote the beauty of mathematics, or is that a separate pedagogical goal?

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