To teach several different algebra concepts, my school uses manipulatives called algebra tiles.

See: http://www.google.com/products/catalog?q=algebra+tiles&cid=9798463280773607865&sa=X&ei=6hOJTq6SBseXtwfjovku&ved=0CEYQ8wIwAQ

Basically, there are 3 different shapes of tiles with side lengths set up so that the tiles each have areas of 1, x, or x^2. The back of each tile is colored red to indicate -1, -x, or -x^2. This makes it very easy to see which terms are like terms. 3x+2x is obviously 5x, since if you have 3 x-pieces and 2 additional x-pieces, you obviously have 5 total x-pieces, so the answer is 5x. Similarly, you can’t add 2x + 3x^2, because the pieces are different shapes, so there is no way to simplify this.

This works really well if you are adding two positive terms or two negative terms. However, if you are trying to add two terms with opposite signs (for example 3x + -5x), you have to do some canceling. A red (negative) tile plus a positive tile of the same size and shape together make zero, so you can get rid of those pieces as you try to simplify. For that example, there would be 3 such pairs (each of the three negative-x tiles would pair with a positive-x tile) and when you slide all of those away, you have two negative-x tiles left, so the answer is -2x.

The problem is that this concept doesn’t actually emerge in any physical way from the tiles. For all of the other concepts, there is actually a clear physical reason for why the concept must be true–for example, if two tiles are different sizes, they aren’t like terms and there is no simple way of combining them. However, when you put positive tiles and negative tiles together to make zero, you just have to explain why this is the case, there is no way to see this just by using the tiles without some outside explanation of that point.

The point of using manipulatives is so that people can actual see and touch a particular concept. While these tiles do work pretty well for teaching many concepts, they really don’t add anything to the discussion of zero-pairs (except that you can then physically slide them away when you find a pair). I’d rather have some sort of manipulative where it is obvious that when you put a positive piece and a negative piece together that they actually form zero (I don’t think just having two different colors really suffices to convince anyone of this point).

Can anyone think of a design for some manipulatives that actually have this property (without losing the ability to explain any of the other important concepts)?

I haven’t been able to come up with a design, but I bet someone can create a good one….

Algebra tiles do have their limitations, and you just bumped into one. You may want to consider beginning your conversation about variables and coefficients with a review of basic multiplication. For example, 3 x 2 means 3 sets of 2′s. If you extend this idea where the coefficient means the number of set, and in this case, the variable means how many are in each set, then making alternative manipulative to algebra tiles is easy. Assume and model some arbitrary value for X (six in the set) and different value for Y (8 in the set, for example). Then each x is a row of six dots, each x^2 is a six x six square with Y’s being built in the same way. Positive variables are made with black dots and negative variables are made with red dots. As with all integers, combining one set of positives with one set of negatives make a zero pair/field and cancel each other out. This comes from years of trial and error and why it is called practice.

I hope this helps.

Dave (10 year veteran)