Talking about nothing

                                

 My Mojojo likes to talk to me while she plays video games.  She really likes for me to give her my undivided attention, but honestly, watching someone play video games is not challenging enough to keep me interested on a Friday night.  So tonight, when she wanted to fight some salmonids in Splatoon 3, I asked if we could talk numbers.  She was on-board.

Now Mojojo is only 11, and I usually tutor and teach kids who are older than she is, but she has a pretty advanced conceptual understanding of math, so sometimes I talk to her about how to teach things older kids have trouble with.  Tonight, because of one of my tutees, I asked her about negative numbers.  She has a basic understanding of what they are, because I started her on the idea a few years ago when she asked what the last number was and what the first number was.  (Infinity is a fascinating topic, and I'm sure we'll get into that more later.)

For now, I started by asking her what she thought 5 + -2 was.  She thought for a moment and said 3.  I tried a few more with her: -4 + 1, 13 + -6, and so on.  And then I asked her if she thought she could come up with a rule for adding positive and negative numbers.  Right away, she said, "well, you subtract."  So I asked how she knew the sign of the answer.  At first, she told me that the sign of the first number gave you the sign of the answer.  So I asked her, since -5 + 1 is -4, what's 1 + -5.  She knew that answer had to be the same, so she eventually came up with taking the sign of the larger number.  She said a few things about the number line here and how that led her to know she needed to subtract, because the numbers went in opposite directions.

I figured this was a good start, and I asked about adding two negatives, like -7 + -2.  Right away, she told me that because they go the same way on the number line the answer would be "more negative".  (I really liked that phrasing, because some of my older kids have trouble with the idea that negative numbers are ordered the opposite direction.  ie -8 is bigger than -10.)

From there we got into subtraction.  She had no problem at all with subtracting positives and negatives.  When I asked how she knew, she told me, "well it's like a sock, and subtracting a negative flips it around".  That doesn't make tons of sense to me, but it worked really well for her.  And that analogy carried her into a negative subtracting a negative as well.

So then I asked if we could kick it up a notch and try multiplication.  She knew that -2 * 1 had to be -2 and that -1 * 2 had to be -2, but she faltered a little on -2*-1.  She said, "It's either 2 or -2," but she couldn't quite decide which.  After going back and forth, I asked if she wanted to try playing with some patterns.  She said "sure, I'm good at patterns!"

So we did this:

4 * 3,  4 * 2, 4 * 1, 4 * 0, 4 * -1, 4 * -2 and then once we were sure that a positive times a negative is negative, we said, ok how about -4*2, -4 * 1, -4, * 0, -4 * -1.  From there, we were pretty convinced.  Mojojo says "ok, so with multiplication it's a double sock flip,"  She says "because if you go to the edge of the sock, you have to go up the inside, so you have to flip it around, and if you get to the edge again you have to flip it around again."  We didn't do as much with division, but we're in agreement that since division and multiplication undo each other, the rules must work the same way.

And even though it doesn't make tons of sense, I like her sock analogy, because she's clearly thinking of numbers as continuous (infinitely divisible) instead of discrete (countable).  Because each number connects to the ones next to it, like a sock!  In hearing her reason through it, it was clear that she was not thinking about counting numbers, and that's interesting to me because I hadn't realized that negative integers make much less intuitive sense than negative measurements.  I had viewed discrete and countable numbers the same way, but Mojojo straightened me out.