I've kept a question on my similar triangles unit exam over the past three years. While the spirit has generally been the same, I've tweaked it to address what seems most important about this kind of task:
My students are generally pretty solid when it comes to seeing a proportion in a triangle and solving for an unknown side. A picture of a tree with a shadow and a triangle already drawn on it is not a modeling task - it is a similar triangles task. The following two elements of the similar triangles modeling concept seem most important to me in the long run:
- Certain conditions make it possible to use similar triangles to make measurements. These conditions are the same conditions that make two triangles similar. I want my students to be able to use their knowledge of similarity theorems and postulates to complete the statement: "These triangles in the diagram I drew are similar because..."
- Seeing similar triangles in a situation is a learned skill. Dan Meyer presented on this a year ago, and emphasized that a traditional approach rushes the abstraction of this concept without building a need for it. The heavy lifting for students is seeing the triangles, not solving the proportions.
If I can train students to see triangles around them (difficult), wonder if they are similar (more difficult), and then have confidence in knowing they can/can't use them to find unknown measurements, I've done what I set out to do here. What still seems to be missing in this year's version is the question of whether or not they actually are similar, or under what conditions are they similar. I assessed this elsewhere on the test, but it is so important to the concept of mathematical modeling as a lifestyle that I wish I had included it here.