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April 2nd, 2014, 06:22 AM   #1
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5th Grade Number Theory Time Again!

I will once again soon be visiting a friend's fifth grade class to get them all abuzz about math as best I can. Last time, we did a fair amount of factoring tricks and a little on number sequences. I want to do more on number sequences for some prominent classes of figurate numbers, with special focus on how they interrelate.

Of course, we'll do triangular numbers and squares, how they interrelate, how each can be derived from serial addition (all numbers for triangular numbers and odds for squares). I will then "dissolve" triangular numbers into four smaller triangular numbers of a special sort via the various interrelations.

I will move to the "3D extensions" of triangular numbers and squares, ie tetrahedral and square pyramidal numbers and how they interrelate via triangular dipyramidal numbers.

This is likely more than enough, especially since we will be building these numbers with styrofoam balls so that the kids can FEEL how they work. But if anyone has some suggestions for further adventures, let me know!
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April 2nd, 2014, 06:34 AM   #2
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Nice typo in the subject line, no?
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April 2nd, 2014, 08:35 AM   #3
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Originally Posted by johnr View Post
Nice typo in the subject line, no?
John, the standard penalty for you: 12 minutes in the corner...
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April 2nd, 2014, 08:56 AM   #4
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Originally Posted by johnr View Post
But if anyone has some suggestions for further adventures, let me know!
I've always been amused by the idea that if you can fit figurate numbers to a polynomial of the appropriate degree, then it's the right one. I don't know if you can work that in? At a higher level this would lead to Bernoulli numbers and the Faulhaber formula.
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April 2nd, 2014, 09:13 AM   #5
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I've always been amused by the idea that if you can fit figurate numbers to a polynomial of the appropriate degree, then it's the right one. I don't know if you can work that in? At a higher level this would lead to Bernoulli numbers and the Faulhaber formula.
Can you tell me a bit more, or where to look into this more? This sounds like just the sort of thing to bring up towards then end to keep that sense of wonder and continuing adventure going. (I was so pleased to report that some of the kids were apparently messing with numbers the rest of the day last time I presented.)

Two things I hope to touch on beyond what I mentioned are my own observation that the finite diagonals on the multiplication table add up to tetrahedral numbers. MMF member icemanfan gave a proof. I will just lead the kids to spot the pattern. Second thing will be the known thing about square pyramidal numbers defining how many squares are in a n x n square grid. Again, I'll let them count up a few and form their own conjecture once they know the number sequences and leave it as something for those especially interested in to look into further.
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April 2nd, 2014, 09:59 AM   #6
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Nice typo in the subject line, no?
I fixed it.

I think few things are as rewarding as leading a young mind to the wonders of math and numbers. Let us know how it goes!
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April 2nd, 2014, 10:03 AM   #7
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I fixed it.

I think few things are as rewarding as leading a young mind to the wonders of math and numbers. Let us know how it goes!
I will! And thanks for editing the subject line. I tried to, but couldn't.
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April 2nd, 2014, 10:29 AM   #8
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Can you tell me a bit more, or where to look into this more?
OK.

A polygonal number clearly must grow as (some constant) * (some linear measure)^2 + (lower-order terms) since it describes a two-dimensional shape. (If you scale it up by a factor of 2 in all linear dimensions, its area must quadruple.) So if you could describe the number of points in a k-polygonal figure by a polynomial, it would have to be quadratic. This is 'obvious' (though not to 5th graders, I imagine!).

But for any k, you can represent it as a quadratic polynomial. This is not obvious at all!

But knowing that it is a quadratic, you might not -- should not -- be satisfied. Which one? The n-th square is n^2, clearly, and you may know that the n-th triangular number is n(n+1)/2. But what is the formula for a pentagonal number, for example?

Well, the smallest pentagonal number is n = 1 which has only a single point, and the next one at n = 2 has 5 points, naturally enough. With a bit of effort you can see that the third pentagon has 12 points, or you can use a trick: the 0-th pentagonal number is 0.

Either way you have three points. Let's say the equation for the n-th pentagonal number is a*n^2 + b*n + c for some numbers -- we don't know which! -- a, b, and c. Then we need a*n^2 + b*n + c to be 0 when n is 0, so
a*0^2 + b*0 + c = 0
We also need
a*1^2 + b*1 + c = 1
and
a*2^2 + b*2 + c = 5
and these three can be simplified to
c = 0
a + b + c = 1
4a + 2b + c = 5
Knowing that c is 0 we can get
a + b = 1
4a + 2b = 5
and a bit of magic subtraction
4a + 2b - (a + b) = 5 - (a + b)
4a + 2b - (a + b) - (a + b) = 5 - (a + b) - (a + b)
3a + b - (a + b) = 5 - (a + b) - (a + b)
2a = 5 - (a + b) - (a + b)
But we know that a + b = 1, so this is
2a = 5 - 1 - 1 = 3
and this gives a = 3/2. Then since a + b = 1 we have 3/2 + b = 2 and so b = 1/2 and we get the equation
an^2 + bn + c
is just
(3/2)n^2 + (1/2)n
which gives the n-th pentagonal number (modulo mistakes!).

The same is true of higher-dimensional shapes as well.

But I don't think I'd work through this with them, just mention that it is possible. If they can find a polynomial that works on three points (by trial and error, say), then it works for all possible choices. Amazing!
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Last edited by CRGreathouse; April 2nd, 2014 at 10:40 AM.
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April 2nd, 2014, 12:15 PM   #9
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Here is an amusing little pattern . . .

$\displaystyle \;\;\;\begin{array}{ccccc} 1 &=& 1 &=& 1^3 \\
3 + 5 &=& 8 &=& 2^3 \\ 7+9+11 &=& 27 &=& 3^3 \\ 13+15+17+19 &=& 64 &=& 4^3 \\ 21+23+25+27+29 &=& 125 &=& 5^4 \\ \vdots && \vdots && \vdots \end{array}$


And a mathematical joke . . .

$\displaystyle \;\;\;\begin{array}{ccc} 3^2 + 4^3 &=& 5^2 \\
3^3 + 4^3 + 5^3 &=& 6^3 \\ \vdots && \vdots \end{array}$

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April 2nd, 2014, 12:26 PM   #10
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Nice!
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