Friday, September 26, 2008

Geometry Investigation: Combining Similarity, Reflection, Paper Folding in an SAT-Type Problem

The problem/investigation below lends itself to a variety of approaches:
Similar triangles
Coordinate methods
Relationships Among ⊥ Bisector, Reflection, Paper Folding (Origami), and Symmetry

Note: Parts of the problem below are appropriate for middle schoolers.

Part I
Show that the length of the ⊥ bis segment EF is 7.5.
(1) Suggested Questions or Tips To Get Started:
(a) How does the length of EF compare to the diagonal? What would the rectangle look like if this length were equal to the diagonal?
(b) Mark (label) all known segments. Even if you cannot immediately determine a method for finding EF, what other segments can be more easily determined?
(c) Label congruent angles! This is critical!!

(2) Of course, the instructor doesn't have to give the answer away, but if the focus is on methods, you may want to consider this.

(3) Students usually struggle with recognizing similar triangles, particularly if they're not looking for them. You may want to give this as a hint after a few minutes. Even some strong honors students may be challenged. Whenever I make these kinds of claims, you can be assured I have already tried this kind of question and observed the reactions! However, coordinate methods are also powerful here and provide excellent review. It's more cumbersome than synthetic methods (Euclidean), but definitely worth discussing and is often a method of choice when other methods are not obvious. There are other methods as well using congruent right triangles, Pythagorean relationships and algebra. Then of course there are the wrong assumption methods: Assuming 3-4-5 triangles are the same as 30-60-90 triangles! As you circulate, see if anyone comes up with an answer involving √3!!

(4) For students preparing for SATs who have not had geometry for a year or more, this problem can be an excellent review even though the difficulty level is somewhat above the SATs. IMO, working a bit beyond the level of an assessment is usually the best way to prepare most students. Anyone agree?

(5) Problems that require thinking 'outside the box' and not mechanically are the most challenging for many students who view mathematics as algorithmically driven (procedures to follow).

Part II
Surely, group, you don't believe we will stop here! Generalize your result to determine an expression for the length of segment EF for an arbitrary rectangle whose dimensions are L and W. Your result will naturally be in terms of these two parameters (L and W).

Part III
Time to "reflect" on what you've done!
(a) Since EF is the ⊥ bis of diagonal AC, it follows that __ and __ are reflection images of each other.
(b) Take a large index card (ordinary filler paper is alright too but the stiffer the better). Fold the paper or card so that A and C coincide. What does this have to do with the original problem?
[Students need to "see" that the length of the fold or crease is the length of EF in the original problem!]
The mathematics of paper-folding (origami) is fascinating. The problem in this post is just an initial view of some of the underlying concepts. Students should have the opportunity to PLAY with the index card. Have them label all congruent segments and angles. Have them identify which segments or figures are reflections of each other. Have them look for a rhombus (explain why it is!), isosceles triangles, congruent right triangles, etc. Folding and unfolding the card is fascinating and instructive!

Note: Using geometry software is also instructive and should be considered as a supplement to the physical paper-folding here. There are also excellent sources for all this on the web. One of the best is the Math Forum at Drexel. An excellent investigation from the Forum can be found here. This is appropriate for both middle schoolers and high schoolers and is the kind of activity that is promoted on this blog. Check out the links at the bottom of this activity (some require subscription). It utilizes special software but can be modified. IMO, nothing replaces the need for students to hold an object in their hands.


Eric Jablow said...

In one important manner, origami takes you exactly one step up from ordinary compass and straightedge constructions. As we all know, a number is constructible by compass and straightedge (and a fixed segment of length 1) if it can be obtained from integers by addition, subtraction, multiplication, division, and square roots. You might want to show how one generates the square root of a number using circles.

A paper by James King (then of McGill),, gives a proof (originally by Kalle Karu), that a number is constructible by origami if it can be generated by those operations and by solving cubic equations with constructible coefficients. After reading that, you might want to show how one computes a cube root with origami.

I wonder how useful that is, however. I remember an article in a long-ago Scientific American on catapults that described a simple mechanical linkage that could be used to compute cube roots; the Romans used that to get the proper torsion on onagers.

Jonathan said...

I am surprised this problem did not generate more comments - it is fantastic. It is exactly the sort of problem I could drop on a class - watch this carefully - not while studying similar triangles, but months later, or even in the next course. Use it to test the knowledge after it's simmered, or use it to stir the pot.

Me? I went off in a non-useful direction. Extend EF in both directions, extends the sides of the rectangle to read it... Looking for the wrong similar triangles... and then in the process of playing, seeing something better.

And that's what I want the kids to do. Not rushed. Not simple. Force them to play a little. Let them be wrong. Let them find themselves in dead ends. There is value in grabbing a problem, struggling with it. And this one is good for that.


Dave Marain said...

Thank you Jonathan. I knew you would appreciate this problem as much as I do. Besides, the paper-folding aspect is just plain fun to play with. Of course, what I find to be "fun" may not be a sentiment shared by normal people!

Let me know how your students respond to this question if you decide to use it. The feedback is what makes this blog worthwhile for me.