Educating Risa

Yesterday the boys and I left the workbooks, and did some activities around negative numbers as Daegan has been asking to learn about them. I adapted ideas I found on two very helpful blogs, here and here. I began by writing a problem of the day on our board, leaving Daegan to mull it over:

We then read the book Less Than Zero (Canada) (US) by Stuart J Murphy, one of the MathStart series. Gareth chose to join us.

We then moved to the hallway, where earlier Gareth and I had created a number line:

I made up a story about the temperature in Calgary one day—zero when we woke, then a bit warmer, then the sun really warmed it up, then clouds cooled it down, then there was a chill wind, then snow. At each part of the story I said how much warmer or colder it had gotten, and the boys moved up and down the number line accordingly. They walked forward when it got warmer, and walked backwards as it got colder. Soon, we had a problem: it had gotten colder than zero. What to do?–Add negative numbers to our number line:

I then made up other stories about riding an elevator (get on at ground level, dentist office on third floor, car is parked in the sub-sub basement, etc.), and spending money (getting and spending their allowance, wanting to borrow extra money to buy a toy dinosaur, etc.). Gareth then asked a really interesting question: What about even numbers? Are any of them negative? (This is an AWESOME question from a 5-year-old!)

I had the boys “double jump” in the positive numbers from one even number to the next. They then “double jumped” backwards, and saw both than zero is an even number, as is –2 (and –4, etc.):

Daegan then asked how many negative numbers you could have. I asked him about the positive number line.

Mom: Does it stop at 9 like in our hallway?
Daegan: (laughing): No!
Mom: How far does it go?
Daegan: To infinity!
Mom: So how far does the negative number line go? To negative 10? Negative 100? Negative thousand? Negative million? Negative billion?…
Daegan: Oh! So it goes to negative infinity!

I also positioned the boys at different spots on the number line and asked them who was at the greater / larger number. The boys saw that while 5 is greater than 2, –5 is NOT greater than –2.

We then went back to the problem of the day, which Daegan was able to solve by drawing a vertical number line (like a thermometer). There was some interesting learning there about how to draw on hash marks, label it, etc. As an adult I take for granted how to construct this sort of number line, but this was Daegan’s first time drawing one:

I then talked about how number lines came in (at least) two forms, horizontal and vertical. I showed him Cartesian graphing, and we found and labelled various points. I said that we’d see this in math again later, but notice that this “high school level graphing” is really just about number lines with positive and negative numbers:

I wanted to make this more concrete for him, as I could see he was a bit puzzled. The book Less Than Zero involved keeping a much simpler graph of the main character’s money, as he earns or borrows. But when would you ever use a Cartesian graph? Aha! I’ll talk about flying to various dinosaur sites, and needing to be able to track the location to send palaeontologists back to do digs. So with the help of an airplane eraser, from the Origin Airport (0, 0), our plane flew up (lots of finds, like the Burgess Shale, are in mountain slopes that used to be on sea bottoms) and down (into valleys and badlands—places below in altitude / sea level measurement from Origin Airport):

For the horizontal axis, I talked about the plane flying east and west. This caught Daegan’s imagination and I was THRILLED to see him extend this concept by running to our world map in the hallway:

Of course! Why didn’t I think of that a minute ago? Our world map has latitude and longitude lines on it, and hence a (0, 0) point where the equator and Prime Meridian connect:

But instead of labelling with negative numbers, maps use directions: north (positive y-axis, or vertical number line), south (negative y-axis), east (positive x-axis, or horizontal number line) and west (negative x-axis). But these labels allow us to do the exact same thing as with Cartesian graphing, like find and label precise spots on the globe. And the numbers for the degrees of latitude / longitude work the same was as the Cartesian graph, getting larger in absolute value as you move away from the origin, point (0, 0).

We then played a bit of the card game Zero—a variant of 21—but Daegan found it too challenging. He did like 21 a lot though, which I started with to introduce him to the idea, so we’ll come back to that as a way of practicing addition facts. We tried this game on the computer, which turned out to be pretty easy for him, even on the “hard” levels. We watched this song on youtube (and did the arm motions, making – and + signs with our arms, to the number line dance. My wiggly boys loved this!):

We then wrapped it all up by watching the Cyberchase episode “Less Than Zero”: