homeschooling,  mathematics

Triangular numbers, Pascal, and how everything interconnects

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Triangular numbers, Pascal's Triangle, the 3 Musketeers... everything interconnects at some point.One of my son’s math lessons on square numbers led to me looking up triangular numbers. Triangular numbers are the numbers you get if you add consecutive numbers together. (Think of adding 1 + 2 + 3 + 4.) I found a helpful webpage and then demonstrated the ideas to my oldest son using little plastic tiles. It took him a few minutes to grasp the idea and I wasn’t sure it had stuck until he announced the next morning that 1 + 2 + 3… to 10 = 55 and explained to me how he had done it. Then later in the evening we were reading a book called “The Aunts Go Marching…” (where 55 aunts bring their musical instruments as they visit their brother), and I could ask him how many aunts had gone into the house at different points, and he could tell me. Yay!

Reading about triangular numbers got me curious also about Pascal’s triangle, so I looked that up too, and had my sons looking at that. Filling in a big chart with the numbers from Pascal’s triangle gave him addition practice, but it also encouraged him to look for patterns and explain the reasoning behind them. After he had about twelve rows filled in, I got him to mark off the multiples of three, and then the multiples of two, so he could see the patterns and we talked about why the patterns worked.

The next day I followed that up with having him color in a big triangular chart using a base system other than 10. I had him color using base 5. When numbers reached a second digit, we discarded all but the last digit. So in this system 3 + 5 = 3 (properly it would be 13 or one group of five plus 3 ones, but we’re discarding all but the last digit), and 3 + 4 = 2. I started coloring my copy with base 2, but we switched part way through so that we could both explore both modes. After a few false starts we discovered it was easiest to do if we assigned the colors according to their order in the rainbow and for the mode 5 chart we wrote the numbers lightly in pencil first so we didn’t get confused.

  • red = 1
  • yellow = 2
  • green = 3
  • blue = 4
  • purple = 5

When we finished coloring two beautiful patterns, my son wanted to mark off the multiples of 3 on my bicolor pattern. We couldn’t really do that because non of the spaces had numbers marked off, so I printed out a pre-numbered copy of Pascal’s triangle and let him color on that one. He marked of multiples of three and multiples of ten, and it gave me a chance to show him how to determine if a number is divisible by three.

Then I was looking up a biography of Pascal, and read that Pascal’s father had to flee Paris because of his opposition to Cardinal Richelieu’s fiscal policy. I had to laugh at that, because Cardinal Richelieu is part of the story of the Three Musketeers, and we recently read a children’s version of that together. Everything connects.

Triangular numbers connects with graph theory, which we have also been looking at together a bit. So one of these days we’ll sit down and have a math lesson around the handshake dilemma and how many handshakes would happen if everyone in a room shook hands once with everyone else.

And then just to make my interweaving of different lessons complete, I discovered today that the song “The ants go marching…” is mentioned in the poetry book we’re about to start using, as an example of iambic rhythm. So I’ll have to show my son that part of the book before we return the Aunt’s book back to the library, and show him how the stressed syllables are demonstrated in the piano music. (I had gotten the Aunt’s book out of the library so as to reinforce a recent spelling lesson on the difference between aunt and ant.) Then we can tie piano in with poetry, math, spelling and everything else. Oh, and Greek, because I noticed the poetry book has mention of Greek letters in it, and my children have been enjoying learning to recognize letters of the Greek alphabet. Everything interconnects.

Every lesson leads to new lessons. Everything connects. I knew I was doing something right today when my son was actually insistent on doing more math than I had planned on us doing. He wanted to explore more patterns.

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