Finding Patterns

I noticed that P was having trouble with subtraction. Mainly, she was counting on her fingers inefficiently, and didn't seem to have realized that subtraction is simply the inverse of addition. So I decided to introduce that concept this week. I used Cuisenaire Rods to show her that if 4+2=6, then 2+4=6, 6-4=2, and 6-2=4. We worked on the concept for several consecutive days, and she really seemed to grasp it.

To reinforce this concept and others, on Friday I gave her a blank addition table: a 10x10 chart with the numbers from 1 to 10 on the top and left hand sides. She needed to fill in the sum at each intersection point. At first she just enjoyed the mechanics of pointing to a random space, figuring out which two numbers to add, and writing the sum in the space. Then, all of a sudden, she realized that diagonals had the same answer: 3+5 = 4+4 = 5+3 = 6+2 = 8. She looked up at me, bright-eyed, and said, "Mommy, I love this! There are so many patterns in this!" When the table was filled in, I showed her that if you want to subtract, you find the number you are subtracting (the smaller number) at the top, go down to the number you are taking it away from (the bigger number), and go left to find the answer. I promised that for all future math problems, she may use the table. Since she generated it all by herself, it isn't cheating, and using it will give her practice with the math facts.

While P was busy with her addition table, I got bored. I had been starting to teach her about perimeter, and she did great with measuring the sides of squares or rectangles and adding them together. But to mix it up, I wanted to make her some right triangles (so that I can get all 3 sides the right length - I can't draw a 60 degree angle by eye). So I wondered what the set of all possible integer side lengths for right triangles was (integer because she isn't measuring half inches yet, let alone whatever the square root of 2 is closest to on a ruler). This turned into a fun math problem, which I solved. E wanted me to use his pencil, because he would like to wear it down more quickly (he finds short pencils more attractive, I believe), so every time I stopped to think, he encouraged me with, "Do more math, Mommy!"

If you're wondering what my solution was, leave a comment and I'll give a summary of my reasoning and results. Or, have some fun with it yourself first! (Mwa-ha-ha... trying to infect the world with mathematical recreation...)