Isosceles EGADs: Functions, Geometry, and Modeling

In trying to come up with more activities for EGADs (Exploring (or maybe Enriching) Geometry and Algebra though Data), the following dropped into my lap. Because it’s so simple and so interesting, I’d better write it down…

Everybody get a sheet of paper and draw an isosceles triangle. Try to make your triangle big enough to kinda fill the page, but also try to make it different from those around you. Make your triangle pretty carefully, but don’t measure and don’t use a straightedge.

Individuals can do this too, but I’m writing this as if it’s a class activity. The idea is to get a wide variety of shapes. It is not vital that these just be sketched, but (a) I think that makes the data more interesting, (b) it opens the possibility to drawing more carefully later, and (c) it’s much faster.

Measure the base angles and the vertex angle, and write them on the page.

If you need to introduce vocabulary, do it here. By the way, we don’t assume that these students know that the base angles should be the same. Also, we all know that measuring angles is hard, right?

We’re going to plot the measurements from the whole class. So write your angle measurements on the board.

You may need to help organize this. Will we plot both base angles? Up to you. If so, consider having each kid make two entries in the T-table or whatever.

Now make a graph. Put vertex angle on the horizontal axis and base angle on the vertical. Think about the range of values before you make your axes!

You may want to discuss what goes on which axis. Without having done this with kids, I bet most of us think of the vertex angle as the independent variable and base angle as the dependent. I, at least, think of the vertex angle as the defining angle in an isosceles triangle. This also has the happy consequence of requiring a change of axes in order to get the coolest version of the formula.

At any rate, the graph should look linear. Address outliers (probably due to bad measurement).

Draw the line you think best approximates the data. Find its equation.

Be ready to present your data and line, and explain as much as you can about the line. In particular, why does it have that slope and intercept?

In the spirit of SERP “Poster problems” this could be a poster-plus-gallery-walk event.

Things I like about this:

• Even though I adore tech-assisted graphing, this can clearly be done by hand.
• I had never thought to graph these quantities.
• In programming and general quantitative thinking, I have to come up with formulas like these all the time.
• Looking at limiting cases is rewarding, both symbolically and spatially. Good habit of mind pays off.
• Even though we may never have written these formulas in this way, they’re not mysterious.
• Even though they’re not mysterious, they’re a completely different way of thinking about these angle relations.
• If students do not know these relationships, this can give them an empirical start.
• If students do know them, this can help connect algebra to geometry.
• The two different forms of the equation (depending on what you solve for) have very different feels. The way we set it up, you get . Which is OK but not easy. The “other” one, however, , is great because you can rearrange it to and discover angle sums.