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.
7 comments:
Makes me wish I taught geometry!
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.
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
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!
@pip, I'd love to hear more of your thoughts on that.
Were there any particularly interesting responses?
Jonathan
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
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