Spinning Out Sine and Cosine

A nice animation from the Wolfram Demonstrations Project showing the relationship between a circle and the graphs of sine and cosine. I like the way the cosine graph is show on its side. I'm going to try using it (after editing it in Mathematica to remove the sine and cosine labels) as an introduction both to the unit circle and the trig graphs. It looks like a good candidate for the technique of showing an animation with no commentary and then asking for student questions, writing them on the board, and asking for answers to/discussion of any of the questions.

Constructing the Area of a Circle

A beautifully designed applet that walks you through finding the area of a circle by dividing it up into sectors and rearranging them to form a shape which becomes a rectangle as the number of sectors goes to infinity. I recommend using it in conjunction with Steven Strogatz's fabulous article, "Take it to the Limit". (If you like this, you might also like this applet which approaches the same problem in a slightly different way.)

An actually fun fraction game!

Refraction is truly a game that teaches mathematical concepts. It's fun. And whether you know the underlying concepts well or not, you'll stick with it because, as my ten-year-old said, "it's addicting." I came across it through a series of twitter posts from Dan Meyer (@ddymeyer). How could I not try it when he was saying things like "The summer Olympics are coming up, right? I'm probably going to medal in this fraction game" and "Yeah, I should go pro at this game. I've found my calling."  While it's obviously meant to teach fraction concepts, it also requires lots of spatial thinking. In the first couple of rounds, there's nothing that looks like a fraction, you just figure out how to use the tools provided to bend light beams in the right way to power a ship. In typical video game fashion, you're introduced to harder puzzles and new tools that help you solve them as you move through the levels.

My ten-year old was watching me play an early level, and despite the fact that her math-teacher mom was doing something with fractions all over the screen, she was intrigued and asked to play. She kept at it until we called her to dinner in half-an-hour or so, and she went back to it after dinner. Later, after she'd made it far into the game, she watched me working on a level that was a little higher than she had reached (because, as I said, it's actually fun and I wanted to play, too!) and had ideas that she really wanted to try, so I gave it back and let her at it. It was interesting to listen to her talk to herself as she worked through the puzzles. She was whispering things like ("the denominators have to be the same" and "yeah, 8 is a factor of 24"). At one point, when she was doing a level where one of the targets was something like 5/4, I said something to the effect of "OK, so you just need a full power and a quarter power" and she said "I never would have thought of it that way!", but she absolutely understood it in the context of the game. Other quick teachable moments like this arose from time to time. I surreptitiously watched her while I gave the impression that I was watching the Phillies game. I was mostly quiet, enjoying watching her mind work, but occasionally I'd say something. For example, after a couple of rounds where she used lots of trial and error to solve the puzzles when there were fractions of different denominators that needed to be added together (though there were no + signs anywhere) I said something like "Oh, you need to get 7/6 and they give you 1/3 which is 2/6 and 1/2 which is 3/6 ." That made perfect sense to her in the context of the game and, after that brief comment at a time when she was primed for it, she began the next level by doing in-her-head conversions to a common denominator of the input fractions so that she could figure out how to combine them into the target fractions.

In addition to bending light beams to power spaceships, you can optionally chose to collect coins (you get to keep them in your trophy room). As I played, I always tried to get the coins, which can make a level quite a bit harder in some cases. My daughter tended to ignore them unless it was really obvious to her how to get them. Occasionally, when she thought she'd missed an easy coin, she'd retry the level immediately upon completion. I expect that after she's made it through all the levels, if she's still interested in playing the game she'll go back and do it again trying to get all the coins. (I discovered after observing this that there's actually a research article on the game's web page at the University of Washington's Center for Game Science about how different players respond to the presence of coins and what educational game designers need to think about when incorporating such "secondary game objectives.")

Equation of a quadratic

This sleek GeoGebra applet by Michael Borcherds provides practice in determining the equations of quadratic functions from their graphs. Some of the generated parabolas are easier to write equations for in factored form while others are easier to do in (h, k) form. All have leading coefficients of 1. After each function you type in, the function you have given is graphed for comparison with the original and you're told how many you've gotten right and the total amount of time it has taken you.