A World Without 9
One of my favorite verses:
Once upon a time in our solar system,
We couldn’t make due without 9.
But Pluto’s not a planet now —
So 8 will due fine.
So sing the Barenaked Ladies in their wonderful song 7 8 9.
This got me to thinking: What would life be like if we didn’t have the number 9? It led to two questions that would be great fun to ask in a classroom:
Imagine if 7 really had eaten 9, and the digit 9 were removed completely from the number system. That is, not only is the number 9 gone, but you couldn’t make 19, 92, or 1,090,394, either. What would our number system look like?
Imagine if 7 had eaten only the number 9, but the digit 9 (or a suitable replacement) could still be used to make other numbers. That is, only the number between 8 and 10 is removed from the counting numbers. (Well, I guess technically we’d have to say that the entire interval [9, 10) is removed from the real number line, so that 8.999… = 10, but such a definition would go over the heads of most kids to whom I’d want to present this thought puzzle.) What would happen then? To think about it, calculate the results of the following problems:
- 8 + 7 =
- 8 + 14 =
- 12 + 22 =
- 4 × 5 =
- 6 × 10 =
- 11 × 12 =
As it turns out, my sons love this song. In fact, they love every song on Snacktime by BNL, and I have to admit that their daddy loves the album, too. Happily, my sons also love math, and they have shown a lot of interest in multiplication recently. A few months ago, Eli used the caculator to discover that 3 + 3 + 3 + 3 + 3 = 15, and he declared, “Daddy, if you add 3 five times, you get 15.”
“That’s right!” I said, and I explained that sometimes, instead of saying “add 3 five times,” we can just say “3 times 5.” Well, he thought this was extra cool. He entered 2 + 2 + 2 + 2 = 8 on the calculator and said, “Look, 2 times 4 is 8!” He has since converted thousands of addition problems into multiplication facts.
Just a few days ago, he made a statement that made my educator heart leap into my throat. He said, “Multiplication is just like adding a lot of times.” Of course, Keith Devlin might think this is a bad thing, but I thought it was a wonderful connection for Eli to make.
With their interest in multiplication sparked, we played with the Times Table on Illuminations. However, we did not use this as a modern-day set of flash cards with which my kids were supposed to memorize a bunch of unrelated facts. Instead, we talked about the patterns within the multiplication table, such as how the even numbers occur in the “times 2″ row and that each number is found by adding 2 to the previous number. We also examined the square numbers, which occur along the diagonal.
Yesterday, I showed them the trick for multiplying by 9 on their fingers.
- Make two fists. From the left, count as many fingers as the number you want to multiply by 9, and raise that finger.
- Count the number of folded fingers to the left of the raised finger. This is the tens digit of the product.
- Count the number of folded fingers to the right of the raised finger. This is the units digit of the product.
The example below shows the result when multiplying 9 × 7. The seventh finger is raised. There are six folded fingers to the left of the raised finger, and there are three folded fingers to the right of the raised finger. Consequently, 9 × 7 = 63.