- Topic: Triangular Numbers. Book: The Number Devil by Enzenberger, Chapter: “The Fifth Night”, pages 89 – 101.
- Topic: Triangular Numbers. Compute the 100th triangular number.
- Topic: Big Number Guessing. Think of a number between 1 and 1,000,000, how many guesses does it take to get it.
How Did it Go?
My daughter whined when she heard we were reading this book, but a couple other kids cheered, so it’s a mixed review. This chapter was about triangular numbers. Here is wikipedia’s picture of the first six triangular numbers:
In the book, Robert learned that he could compute the 12 triangular number (1 + 2 + 3 + 4 + …+ 12) as:
Each column in this table adds up to 13, so the 12 triangular number is 13 * 6 = 78.
The kids all enjoyed seeing the pattern of how to compute the next triangular number from the previous (i.e. the 12 triangular number is the 11th + 12 more).
The 100th Triangular Number
After the book, I asked the kids what they thought the 100th triangular number was. The kids said the 10th triangular number was 55, so the 100th should be 55 * 10 = 550. I said we should compute it similarly to the the way they figured out the 12th in the book.
The kids then suggested I should group the numbers into sets that add up to 13. Instead I said we should pair up the smallest and the largest numbers.
1 with 100, 2 with 99, etc. I asked how many pairs of numbers we would have? The kids at first said 100, or said they couldn’t figure it out. Then I showed how there had been 6 pairs when there were 12 numbers, and one girl said “Oh it should be 50!” and explained that she divided 100 by 2.
Next we checked how much each pair added up to: 101. So the 100th triangular number should be 101 * 50. One girl tried to compute this long-hand, but got immediately frustrated because she had only done 2 digits times one digit in school. She felt that this would be impossible until her school taught it.
I suggested first calculating 100 * 50. Eventually someone said that was 5000. But we were supposed to compute 101 * 50, so how many more 50s do we need to add? My daughter said one more, so the 100th is 5050.
Big Number Guessing
First we warmed by up having the kids guess my number between and 1 and 20. This was easy. Then I had one kid write down a number between 1 and 1,000,000. He chose 456, 276. The other kids thought it would take 800 guesses to find his number. I gave them 25. They got down to 456,267…456,299 before they ran out of guesses. This was definitely NOT easy for them to think of a number between 461000 and 447,000.
Next I claimed that I could guess their number between one and one million in just 25 guesses. They wrote down a secret number, and I started guessing, doing binary search.
It went well at first, but I wasted a few guesses by getting mixed up with whether they had said higher or lower. Then when I had 10 guesses left, I got mixed up again and wasted about 6 guesses going the wrong direction. I hit 25 guesses and did NOT get their number. My daughter kindly gave me two extra guesses, and I got it, but I totally lost. The kids enjoyed that 🙂
This was a good activity because it works on their sense of big numbers. But it was hard enough that some kids (including my daughter), didn’t really want to participate, so I had 2 – 3 very interested kids and a couple more who were busy drawing.