Constructing Equal-Area Triangles

The origins of this week’s Web Sketchpad model date back to the  Connected Geometry curriculum from the mid 1990s. I was one of the co-authors of the curriculum, working at Education Development Center with a  wonderful team of math educators (Al Cuoco, Paul Goldenberg, and June Mark, among others) to develop a habits of mind approach to learning geometry.

Mathematical habits of mind are similar to the Common Core’s Standards for Mathematical Practice in that they emphasize the process of doing mathematics rather than being a recipient of the content. As Cuoco, Goldenberg, and Mark explain: “A curriculum organized around habits of mind tries to close the gap between what the users and makers of mathematics do and what they say. Such a curriculum lets students in on the process of creating, inventing, conjecturing, and experimenting; it lets them experience what goes on behind the study door before new results are polished and presented. It is a curriculum that encourages false starts, calculations, experiments, and special cases. Students develop the habit of reducing things to lemmas for which they have no proofs, suspending work on these lemmas and on other details until they see if assuming the lemmas will help. It helps students look for logical and heuristic connections between new ideas and old ones. A habits of mind curriculum is devoted to giving students a genuine research experience.” [Habits of Mind, p. 2]

Here’s a nice problem from Connected Geometry that takes familiar content (triangle area) but spins it in a way to invite experimentation, persistence, collaboration, organization, and—best of all—creative problem solving:

Find as many ways as you can to divide an arbitrary triangle into four equal-area triangles.

I first wrote about this problem in the October 2000 issue of NCTM’s Mathematics Teacher, but I had no way include an interactive model with the article. Now, I’m able to provide a Web Sketchpad model where you can divide the triangle into smaller triangles using construction tools.

The websketch comes with three tools. To bisect a segment, tap the Midpoint tool. You’ll see a preview of a bisected  segment. The segment is glowing, indicating that it needs to be placed. Tap the segment you’d like to bisect, and the glowing segment will merge with it.  Your tool is done. The Thirds works the same way, dividing any segment into three equal parts.

To construct a segment, tap the Segment tool. You’ll see a preview of a segment with one of its endpoints glowing, indicating that it needs to be placed. Tap the point you’d like to merge it to. The other endpoint of the segment will then glow. Tap to merge it. Your tool is done.

If you make a mistake, you can use the left-pointing arrow above the three tools to back up as many steps as you like.

How many ways do you think exist to solve the problem? You might be surprised! I’ve provided six pages of triangles for your students to display their answers, and that probably is not enough. To move from page to page, just use the arrows in the bottom-right corner of the sketch.

When your students are done, try this slight variation to the problem that removes the restriction of each piece being a triangle:

Find as many ways as you can to divide an arbitrary triangle into four equal-area pieces.

It’s entirely possible your students may think of a solution that requires a tool other than the three that I’ve provided. If so, let me know!

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