Connecting arithmetic and geometry

Last week we did a short project on approximations (based on a section in Mazur and Stein’s new book in primes):

Looking at Mazur and Stein’s new book about primes, part 2

I thought it might be fun to revisit approximations this weekend. My idea was to look at the two sums 1 + 2 + 3 + 4 + . . . + n , and 1^2 + 2^2 + 3^2 + . . . + n^2. The idea would be to use some ideas from geometry to approximate these sums.

We started by looking at 1 + 2 + 3 + 4 + 5 and thinking of ways to approximate this sum. The boys were a little confused at the beginning because they already new the value of the sum, but eventually they landed on the idea that we could approximate the value of the sum as the area of a triangle.

Next we looked at a few more sums of consecutive integers from 1 to n to see if we always had a “good” approximation according to the definition we’d seen last week in Mazur and Stein’s book.

Now we moved on to adding up perfect squares from 1 to n^2. We started with 1^2 + 2^2 + 3^2 + 4^2 + 5^2. Initially the 5 snap cube squares were arranged sort of in the shape of a right triangle. The kids had a hard time seeing how to use this triangle, though – that surprised me a little. Eventually they discovered that the squares could be arranged in a pyramid-like shape, and we used that pyramid as the basis of our approximation.

One difference between the sum of squares and the sum of integers is that the approximation we used for the sum of squares was not “good”.

For the last part of the project I had the boys build two more snap cube pyramids. They new from some prior project that pyramids could form a cube, so they tried to see if our approximate pyramid could also form a cube. A few samples of those prior project are here:

Summing up Squares

Pyramids and Count Like and Egyptian

It turned out that our three of our approximate pyramids do not form a cube, but the shape they do form provides some additional insight into the actual sum. In fact, it gives the exact formula 🙂

So, a fun way for kids to see a connection between arithmetic and geometry. It was also a nice way to see some examples where the “good approximation” idea from Mazur and Stein worked and didn’t work. Nice little Saturday morning project 🙂

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