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Topic: Action Research -- Balances

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Subject:   A balance surprise
Author: George
Date: Oct 29 2003
Last week I visited an elementary school and worked for a while with students at
various grade levels with one of the balance Math Tools.

A pair of fourth-graders did something that really surprised me. They worked
for a bit with two shapes, trying to figure out how to use the balance to
determine the "weight" of one if they were given weight of the other. They did
it by guess-and-check, which the applet allows. They didn't really make much
use of the balance itself.

Then they wanted do a harder one. They went straight to the balance with four
shapes with the value of only one being given. To my surprise, they found a
guess-and-check method that allowed them to find an answer with what
appeared at first to be a silly approach. They took numerous versions of the
shape they new and put them on the left tray; then they took at least one of
each of the shapes they didn't know and put them on the right tray. They kept
placing shapes on the right side, while putting knowns on the left side until
they got a balance. Then they would reason their way to an answer. A screen
capture of one of these approaches is at
On the left are two stars. They know each weighs 4, so the total is 8. They then
put other objects on the left side until they get a balance and start reasoning.
For example, they try
Square = 1
Triangle = 2 and
Pentagon = 1
They come to this by agreeing between them that this will add up to 8. Then hit
the check button.

If that doesn't work, they try something else
Square = 1 (some discussion of why it can't be 2)
Triangle = 1 and
Pentagon = 3

What was surprising to me was that they hadn't developed a strategy for finding
the weights one at a time, which I thought would be easier and a stepping stone
to a more sophisticated approach. This was part of the intent in having
increasing numbers of weights. But instead of doing this, they jumped right to
mulitiple weights and found a way to get the answer.

Has anyone else worked with these balance tools?

I have another related question. The balance is used as a metaphor for equation
solving, especially for the "Hands-on Equations" program which doesn't use a
real balance. I wonder if students who used that program would have an easier
time with the balance applets. Be interested to hear from a teacher who uses
that program.


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