Recently, I tried the following exercise with really great, enthusiastic, results with sixth and seventh graders.
For the last class of "The Great Math Adventure" at the Museum of Science Boston, I baked a brownie for the kids. The catch was that no one would get a piece until they had determined a way of cutting the brownie into equal pieces.
We had sixteen people, and the brownie shape I used was a trapezoid with height 9 inches, bases 6 and 10 inches, and left and right excesses 1 inch and 3 inches respectively.
There was great enthusiasm, and in about forty five minutes, six members of the class produced four different ways of cutting with each method exhibiting a very nice property of areas.
For example, one student divided the trapezoid into two right triangles and a 6 X 9 rectangle. He computed that everyone should get 4.5 square inches of brownie. That showed that the right triangle with base 3 and height 9 had to be divided equally into thirds. He did this by dividing the base into thirds and connecting the division points to the vertex. This demonstrates a nice fact about the areas of triangles and parallelograms and amounts to the fact that one can add any multiple of a column of a matrix to another column without changing the value of the determinant.
Two students produced what I consider the most elegant solution. And to my surprise and satisfaction, when we voted as a class on which of the four ways to use, most voted for this elegant solution...including some of the ones who found other solutions!
I think this is a very practical way of getting kids to understand areas. For different class sizes, it would be necessary to use different shapes for the brownie. Rectangles are too easy.
Also, the concept of mathematical elegance is made more concrete because of the strong desirability of not having to use too many cuts, and not wanting to eat a strange sliver of brownie.
If anyone tries this with their class, please let me know of the results.
(I'm also willing to suggest shapes for a given class size.)