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## Friday, November 11, 2011

### When the Problem Does the Teaching

This recent Geometry lesson is a good example of setting the kids in pursuit of a problem, where they have to learn the thing you want them to learn anyway in the process. (That wasn't that eloquent, sorry, I will illustrate.) On Tuesday, we developed the rule for the sum of the angles in a polygon by the chopping-into-triangles technique that many of you are probably familiar with. The next day I wanted them to be able to find the degree measure of one angle in any regular polygon, so I set them this task, which I stole from a PCMI problem set:

I did not include that first question when I did this in class, and many students stumbled over restricting their search to regular polygons. So I added it after the fact for next time I give this problem.

There are lots of these triplets to find, so all the kids met with some success pretty quickly. It is also a little like finding a pearl in an oyster, so they were rewarded and motivated to keep looking. Regular polygons are hard to draw, so with a little reminding and prodding, they started to find the degree measure of one angle in a regular pentagon, hexagon, octagon, etc (the whole, covert point of the activity, anyway! Yay!) I had them add their finds to a whiteboard everyone could see as they were discovered. They also wanted to verify by using the Smartboard to render regular polygons perfectly, and fit them together like puzzle pieces, which I was happy to allow them to do. This was actually a pretty great class - some kids conjecturing likely candidates, some kids armed with calculators cranking out angle measures, some kids organizing all their finds, some kids going up to the smartboard in groups of two or three for visual/spatial verification. And when I assessed them the next day, no one had any trouble understanding the question or coming up with correct angle measures. This problem is a keeper.

1. Makes me wish I taught geometry!

2. I love the title. I've been using the term learning THROUGH problem-solving versus learning ABOUT problem solving, but I think this better describes "setting the kids in pursuit of a problem, where they have to learn the thing you want them to learn anyway in the process."

I've been wondering if the term learning THROUGH problem-solving was clear. I know the difference between freedom of/from religion, but I have to stop and think when assessment of/for/as is mentioned.

Nice to see a good use of a smartboard - students sharing solutions and confirming using accurate shapes. Too many times I see it used as a glorified whiteboard - improved productivity for teachers, but most kids remaining spectators.

Thanks for sharing this lesson. On my blog, I shared an idea for the previous lesson that you mentioned
(interior angle sum) using the opening minutes of Flatland: The Movie.

3. I love this problem too - I try to teach it every year in my geometry class as well.

If you're interested, this is a great resource for this problem, just to have in the back pocket: http://en.wikipedia.org/wiki/List_of_uniform_tilings

4. Hey! Quit stealing problems that we legitimately stole from elsewhere :)

Nice job. It's awesome how much mileage you can get from a deep, challenging question. I feel like kids then think the "real topic" is super easy.

Sue: there are other ways to ask this same question so it becomes about arithmetic or algebra instead...

Thanks and good luck!

5. @pip, I'd love to hear more of your thoughts on that.

6. Were there any particularly interesting responses?

Jonathan

7. To Sue: there are a few ways to proceed. One way (in the 2011 PCMI problem sets) is to start with this question:

Find all possible integer dimensions of a box whose surface area, in square units, equals its volume in cubic units.

There are a few to find: 6-by-6-by-6, 4-by-8-by-8, among others, but also a general formula:

2bc + 2ac + 2ab = abc

Manipulating this formula may lead to some insights on the minimum values of the three variables, and some other potential constraints.

The more direct correspondence to the geometry problem presented here uses the formula for the interior angle of a polygon:

180(a-2)/a + 180(b-2)/b + 180(c-2)/c = 360

There is much cleaning that can be done here, and a simpler version is available by using the formula for the EXTERIOR angle and recognizing that the exterior angles must add to 180 degrees:

360/a + 360/b + 360/c = 180

It's pretty fun to try and manipulate each of those three equations into one another. (Three equations??)

- Bowen

8. Can you give a few examples of some 3 regular polygon shapes that "fit".? Your post has inspired me to try and do the sum and individual angles on seperate days. I used to teach them on the same day. Mike

9. A shortcut is if unit fractions using the numbers of sides sum to 1/2. For example, 5,5,10 and notice, 1/5+1/5+1/10=1/2. Some others are 8,8,4 and 3,12,12 and 20,5,4.

10. Thank you for the quick response! That makes sense. I am nervous that my students may not be able to do this. The 8,8,4 they may stand a chance at but not sure the others...

11. There are more than those! Somebody will figure one out. They will. Have them demonstrate how they found it, or how they know it works. It will give the others ideas of things to try.

12. I wonder if there is a website with a list of the possibilities. Thanks for the push! Monday it will go down. Very lucky I stumbled upon your and Dan's blog. Big impact on my energy level to teach.

Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.