## Thursday, October 28, 2010

### Exponential!

Every so often, you just let your 6-year-old do whatever she wants to during math time, and she ends up calculating 2^30.

I love homeschooling.

At breakfast time on Tuesday, P started with "Two ones is 2, two 2s is 4, two 4s is 8, and two 8s is 16." E then asked me, "What are two 100s?" I doubled for him until we got to 409 600, whereupon folding the laundry and feeding the baby took up too much of my attention.

The interest both P and E had showed in doubling suggested to me that I abandon my original plans for math (which weren't anything special, anyway) and let P practice addition by doubling until she lost interest. She didn't. We started by writing the doubles on the chalkboard, but soon ran out of space. I copied the answers we had so far onto a piece of paper. P struggled to write neatly enough to line up the problems exactly, so I set up each subsequent problem for her on the chalkboard, she solved it, and I wrote it on the paper. By the time we got to 262 144, P was not only not losing interest, she was excited. "Mommy, math is so much fun!" I began to think of possible strategies for ever stopping her, because the boys were getting bored (B was trying to eat the chalk each time P put it down).

Half a year ago or so, we read a picture book in which a girl tricks a greedy king by asking for a grain of rice as a reward, doubled daily for a month. So I suggested that P calculate how much rice the king gave the girl on the 31st day, and then stop. She added excitedly, finally concluding that the total was 1 073 741 824 grains of rice. While P worked, E kept commenting, "That king must be getting worried!" I only pointed out 2 minor errors during the course of this monumental calculation, which P corrected herself.

While P was busy, I had been reflecting on the fact that Sonlight's Core K, which we're using this year, contains a longer book with the same basic storyline. So, once P was done, we read A Grain Of Rice and thoroughly enjoyed it. The story was well told and both P and E followed it with enthusiasm. We compared the amount of rice the emperor was having to give the peasant each day with P's calculations, and it was fun to see them line up perfectly.

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## Saturday, October 23, 2010

### 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...)

## Saturday, October 16, 2010

### Spelling

P wrote me a note:
"I Like sgoL
MoMMy.
SbeshaLy
riting.
I Am Happy."

Once I realized that "sgoL" refers to school, the note made sense.
Although the spelling is not perfect, it shows progress in topics I haven't made an issue of. For example, she correctly used the silent e in "like" and the "ing" in "riting". Since she isn't reading fluently yet, I'm not expecting her to spell well, but it's fun to see that she's picking up some rules without being formally taught them.

I'm also glad that she likes school, and is happy. Given that she usually complains about copywork, I'm also strengthened in my resolve to make her do things that are good for her even if she doesn't appear to like them at the time.

## Friday, October 1, 2010

### Binary

I wasn't planning on teaching the kids about binary numbers yet, but Ari and I ended up doing it yesterday. It was Barry Simon's fault.

Here's how it happened. We were discussing CBS (Community Bible Study, which the kids and I attend on Thursday mornings) at lunch time. E is never able to remember the story he has heard at CBS - I'm not sure if he just isn't aware that they're telling a story, or if he's decided that forgetting the story is easier than telling us about it. In any case, this led to a discussion of good and bad teachers. And Ari and I started talking about Barry Simon.

When I was a frosh (insert quavery voice) at Caltech, Barry Simon taught Math 1a. He was brilliant, I'm told. So brilliant that there's no way he could stoop to teaching the elementary concept of epsilon-delta proofs in such a way that the average Caltech freshman could understand them. Instead, he found special cases and exceptions, and talked exclusively about those. At least, I think that's what he did. I never really understood anything in Barry Simon's class. I spent one evening determined to understand the topic of the next day's lecture before it happened, so I studied the subject over and over until I was sure I grasped it. I entered the class knowing how it worked; I left class hopelessly confused. The joke was that one day Barry Simon would teach us to breathe, and we'd all suffocate.

As Ari and I discussed his teaching style, I wondered aloud how he would teach a 6-year-old how to add with regrouping. I figured the first thing he would do would be to convert 28+37 into binary, without mentioning that he was doing anything of the sort, or explaining to the 6-year-old that such a thing as binary even existed. I was unsure of whether P was ready to learn binary, but when Ari was done laughing, he decided to prove me wrong.

This actually turned out to be a lot of fun. Ari labeled a piece of paper with columns for 8s, 4s, 2s, and 1s, and we explained that in binary you're only allowed to use 1s and 0s. Ari started by having P identify the values of 0000 (B's age), 0100 (E's age), 0110 (P's age), and 0111. E then wrote a few 1s and 0s at random, and it ended up being 00101. P's ability to easily figure out that that was 5 made me wonder if I really could show her how to add in binary without and with regrouping. We started out with 1+2=3 (01+10=11), and then I explained how 2+3 (10+11) had 2 in the 2s place, meaning 4, so we had to write a 1 in the 4s place instead: 10+11=101. She thought this was really neat - and I suspect that it'll help her grasp regrouping more easily when adding in base 10.

All of a sudden, I started wondering: What does the Fibonacci sequence look like in binary? What about skip counting - what patterns are there? How do you do long division in binary? I can imagine myself forgetting about what I'm trying to teach P and starting to explore this sort of question myself during our math lesson, leaving her hopelessly confused. Please say it ain't so: Could I ever turn into Barry Simon?