Sometimes I come across problems that make me wish I was teaching High School again. I mean I don't want to grade papers or go to staff meetings or get up every morning at 5am like I used to; but the idea of watching a bright class of kids thinking about a problem that is just different enough to make them use some of the skills they have other than their great memory was always exciting, and I guess I'll never get over it.
Ever once in awhile I come across a problem that makes me want to pop them on a HS class, but since I'm retired and too lazy to go back to work full time, I thought I would share a couple of the recent ones with the folks still out there working the front lines in case you may have missed them with all the papers to grade and such.
Not too long ago James Tanton reminded me of an old problem about chess boards by giving a new one I had never seen. The classic problem I refer to is the one that asks, "Is it possible to cover an 8x8 chess board with dominoes (1x2 rectangles) if the opposite corner squares are cut off. You, and probably your clever students have already seen this, but it is still a clever problem because of the symmetry idea in the solution. Tanton threw out the question, "In tiling an 8x8 board with dominos, must there be two dominos making a 2x2 square?"
Now at this point I admit I haven't even solved the question, and I'm not sure if Professor Tanton has, (but bet he has). My first instinct is that there must, but I haven't spent the time to test the "Why?" of that. You see, that would spoil it a little. I guess I eventually will, but presenting it to a class when you DON'T know the answer makes the discovery even more exciting. You let the students work without the temptation to rush in and "guide them" to an answer. My experiences in such situations always made me proud of the kinds of thinking my kids could produce when the problem was not "textbook".
Another I saw recently was on Greg Ross' Futility Closet blog. He posted "(5/8)2 + 3/8 = (3/8)2 + 5/8."
Now giving this to students is not actually a question, unless they have a mathematical mind, in which case they will ask the question; "What does this imply?" For me the immediate question looked like two fractions a/c and b/c so that if you added either to the square of the other the results would be equal. Now the numbers a,b, and c that make that happen would be the question of interest. This might be at a slightly different level than the previous problem, but I keep thinking both would be appropriate from 7th grade to the last year of High School.
The most recent is from John Allen Paulos twitter where he posted a complete proof in the 140 characters allowed. The question was prove that there must be some irrational numbers a and b so that ab is rational. I'll give you his solution to this one because it is one of a couple that should be exposed to advanced level math students in HS so that they can see some of the beautiful proofs that are out there:
Paulos proof: expanded beyond 140 characters for greater readability, Let a and b both be sqrt2 (irrat.)
Now it may be that c=ab is rational. If it is, we are done; but if not, then c b = 2.
That is not one that will pop out as easy to most students. I played with it and decided that I would try to explain it like this:
sqrt(2) = 21/2 so sqrt(2)^sqrt(2) = 21/2sqrt(2) and using the product of powers we can get 2sqrt(2)/2. By using sqrt(2)/2 = 1/sqrt(2) we finally arrive at sqrt(2)^sqrt(2)= 21/sqrt(2).
I'm thinking that after this they will be able to see that raising that to the power of sqrt(2) will give a result of 2, a perfectly rational number.
I think at this stage in their lives many of them find rational, irrational, transcendental, imaginary and such a bit mystifying, and some controlled experiments let them gain a little confidence. I'm reminded of a recent blog I read where a teacher/researcher talking to two kids sitting across from each other asked one if she knew a name for the shape in front of her. It was a triangle arraigned so that from her view it was a nabla (∇)(had she been at all aware of the word). She replied that she only knew that from where here classmate was sitting across the table, it would be a triangle. Imagine how many times she must have seen a triangle without seeing one in different orientations. True learning spins on such delicate wheels. You can say the terms as many times as you wish, but when you get your students to describe their views of them, they may start to get a deeper understanding of what they are dealing with.
And if time allows, it is always a wonderfully amazing thing to walk students through the remarkable fact that ii is a real, and infact, is equal to e-π/2. Do point out that this is real, but not rational.
One disappointment with both of these examples is they really illustrate that there exists a number a of a certain type (irrational for instance) such that aa has a certain property (rational for instance) . I think it would be nice to have a collection of irrational pairs a,b such that ab are rational and demonstrably so at a high school level. Would love to hear your suggestions?
One final one just because it demonstrates how beautifully geometry can make some math problems visual. This one is also from Greg Ross.
The question is, "How can six people be organized into four committees so that each committee has three members, each person belongs to two committees, and no two committees have more than one person in common?" The question reminds me of Kirkman's schoolgirl problem which involved 15 girls walking in groups.
The geometric solution is easy if you start with the committees as lines in a plane so that no three are concurrent. Then each of the six intersections represents a person and the problem is solved.
OK, Just one more. A short time ago I came across a neat trick by Martin Gardner in Ivars Peterson's column, the Mathematical Tourist that I had never seen, and I thought I had read Gardner's stuff.
And since all kids love Mobius strips, and this one was even new and surprising to me, I thought I would share.
Start with a simple cross of paper (make it kind of large as some cutting is involved) as shown in the illustration at right.
Take one cross and do the usual half-twist to make a Mobius Strip. The other is just made into a conventional loop with no twist.
NOW, trisect the Mobius band, and bisect the normal loop.. shake it all out... and be amazed. Then share it with kids..
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