When I got home, I spent some time searching for tools to develop an applet for an idea I had. If you search on the web, you can find a whole bunch of virtual manipulatives of the two-pan balance. But I could not find any that allowed me to add a negative number to one side. If you use a balance with shapes, imagine the negative numbers as helium-filled balloons. So I want to develop or adapt an applet that can add balloons. (Scuba divers will easily appreciate how bouyancy and ballast interact. Thanks for explaining, Kaylan)
But then I discovered this thing I had never heard about before. The four-pan balance.
In the picture above, the inner pans represent adding negative numbers; the outer pans positive numbers. The difficult-to-see white pointer on the central support points to the heavier side. Adding negative numbers moves the pointer away from your side - i.e. your side just got lighter.
There are all sorts of games one can make up using this really nifty gadget. Here's one I am mulling over for two players. (Warning: it is not a beginnners' version)
1. Each player gets 20 poker/checker counters. 10 black, 10 red. You also need a 6-sided dice.
2. A black counter is +1, a red counter is -1. So your starting score is 0.
3. On your roll, you have to move that many counters on or off your side of the balance. For example, if you roll a '4', then you add or remove 4 blacks (+4,or -4). Or add or remove 4 reds (-4, or +4). Or you can add 3 reds and 1 black (-3 + 1=-2). Or you can add 3 reds and remove 1 black (-3 - 1 = -4). etc.
4. Black counters may only be placed on the outer pans. Red counters only on the inner pans.
5. On your turn, if you equalize both the sides, then you get to take all the counters in the 4 pans, or give them all to your opponent.
6. First person to a predetermined score (say 10) wins.
Feel free to make or suggest your own rules.
I once listened to Denise Mewborn (head of the math education dept at UGA, the best program in the country) give a talk on Theoretical Frameworks. Specifically, she said that the week before she had to submit her dissertation proposal (which was all written with data collected), she still hadn't found a TF. She sat on the floor of her apartment with all the reseearch books she could find and searched through them looking for stuff in tables (because a TF should be in a table :) ) and basically kept at it until she found something she could fit her research into.
ReplyDeleteThat struck me because, essentially, that is what I did also. I don't it is uncommon. I think we all move toward our understanding of a TF we subscribe to, given the research we are doing. But that it isn't a steady state but a constant, moving target to shoot for.