I want to post about some of the resources I’ve found recently about exponents. There’s a couple of kids books that include them. One is called Minnie’s Diner: A Multiplying Menu by Dayle Ann Dodds and John Manders. A family of boys goes one at a time into the diner each ordering double what the previous one had. The other is called Sixteen Runaway Pumpkins by Dianne Ochiltree. It also deals with exponents of two, and is nice and seasonal for October. Neither of the books mentions exponents by name but we could talk about that separately. Another good book about exponents is One Grain of Rice: A Mathematical Folktale by Demi. I wrote a bit about it on an older blogpost, and Karyn of Teach Beside Me has some great activities based on it.

Of course I don’t want the kids to think that exponents are just about doubling. So I made up a few stories for them. One story went something like this:

Far far away there are three stars quite close to one another, and each of those three stars has three planets circling them. (Can you write an equation that would tell us how many planets there are?) Each of these three planets has three moons, and each of these three moons has three giant craters in them. Now if you looked closely at all of these craters you would see spaceships hiding in them…. three spaceships per crater. Can you draw a picture to show the stars, planets, moons, craters and spaceships? How many spaceships are there?

Now what if we looked closely inside a spaceship. Each spaceship has three astronauts, each with three suitcases. Can you write an equation showing how many suitcases are in each spaceship? Can you write it as an exponent? Now let’s multiply that exponent by the number of spaceships. How could we do that?

The computer makes it easy to make pictures with duplicates, so we took advantage of that and made some pictures. The one to the side is 4^5.

Then there is the Vi Hart video Connecting the Dots which really is not about exponents but mentions parabolas. We had a lot of fun drawing pictures based on her doodles and talking about the idea of math being a creative thing. Then a few days later my son started drawing what he called parabolas to try to learn something about them and a few days after that we started working on learning about logarithms and because we’ve been busy we didn’t really follow up more or parabolas. Until a few days ago when I knew I wanted to ask my seven year old to graph  y = 2 ^x  without telling him that it would be a parabola.

The first thing I needed to check was whether he knew how to multiply negative numbers. He did but we reviewed it with a simple game where we rolled two special dice made out of little squares of wood I picked up at a craft store. On one die was the numbers 0, 1, 2, 3, -2, -3 and on the other was -1, -2, 3, 1, 2, 3. We took turns rolling two dice, multiplying the results and then moving that number of squares on a little homemade board game. It wasn’t the best game in the world but it did its job.

Having confirmed that my son knew how to multiply negative numbers I presented him with some graph paper and instructions on what I wanted him to graph. He graphed it and on his own decided he wanted to graph y = 3^x  and y = 3x. Yes! Success! Success is when he find something he wants to explore himself.

The other moment of greatest success in this exploration was when the seven year old announced that 2^{4 =} 4^2. Then we had to check if 3⁶ and 6³ were equal to each other. They aren’t, but 3⁴ = 9^2.  So it isn’t x^y = y^x, like he had first guessed but x⁴ = (x²)².

My next goal was to introduce negative exponents. We used the book The Lion’s Share previously but this time we talked about it  in terms of  2^-x.