Week of Inspirational Math Extension
The week of inspirational math
For one of the first weeks of math class, we explored a few problems to help develop an understanding of the types of problems we will be presented with in class. We also watched a few videos about how making mistakes or taking a lot of time on a problem in math class isn’t actually harmful, but instead beneficial. According to the videos, when you make a mistake, your brain grows and learns more than when you get the right answer the first time.
The first problem we worked on was a puzzle involving tiling a rectangle. The goal was to fill the 11 by 13 rectangle with as little squares as possible. The general consensus was that six squares is the least that can fill the entire rectangle.
Another problem we worked on was the “Squares to Stairs” problem which gave us a pattern with triangular staircases made of squares. The first figure had one square, the second had three, the third had six, and the fourth had ten. Our goal was to figure out how many squares would be in a 10 square high staircase, or even one with 55 steps.
The other problem dealt with Hailstone Sequences, or sequences of numbers that go up and down like hailstones. The sequence starts with any positive whole number. If the number is even, divide it by two. If it’s odd, triple it and add one. Repeat this condition until the sequence starts to loop. For example, the sequence [6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1…]. We found out after a bit that once the number hit four, two, or one, it would start infinitely looping. Because the number is cut in half every time it’s even, any power of two will eventually get down to 4, 2, and 1 and loops the sequence.
The extension
After we worked on all three problems, we chose a problem to extend in some way and do a write up about. I chose the hailstone sequences problem because it was my favorite of them all. For the extension, I decided to see what would happen if I let the numbers in the sequence go into the decimals. I came up with new rules for the sequences that ignored the odd or even condition and instead focused on whether the number was a whole number or a decimal number. If the number is whole, divide by two. If the number is a decimal, triple it and add 0.5.
I found that the new rules gave similar patterns. Eventually, the new sequences would loop like the originals. However, instead of 4, 2, 1... the sequence looped with 2, 1, 0.5, 2, 1, 0.5…. I tested the new rules with three starting numbers: 10, 26, and 39:
[10, 5, 2.5, 8, 4, 2, 1, 0.5, 2, 1, 0.5…] [26, 13, 6.5, 20, 10, 5, 2.5, 8, 4, 2, 1, 0.5, 2, 1, 0.5…] [39, 19.5, 59, 29.5, 89, 44.5, 134, 67, 33.5, 101, 50.5, 152, 76, 38, 19, 9.5, 29, 14.5, 44, 22, 11, 5.5, 17, 8.5, 26, 13, 6.5, 20, 10, 5, 2.5, 8, 4, 2, 1, 0.5, 2, 1, 0.5…]
Overall, I really enjoyed this week of inspirational math and the problems and activities we did throughout.