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Math transforms:

1. Additive and multiplicative commutativity: ab <-> ba, a+b <-> b+a.
2. Additive and multiplicative inverses: a+b = c <-> a = c-b and similarly for *,/.
3. Distributivity: a(b+c) <-> ab+ac.

Then you want a couple of "automatic" simplification rules. If you move an equation into a simplifiable form manually, the system should automatically make these changes. However, it should "show its work", probably by leaving the unsimplified version visible. Here's an initial list:

1. Identities for all operations that have them: 0+a -> a, 1*a -> a, a/1 -> a.
2. Cancellation: a-a -> 0, a/a -> 1.
3. Distributivity: -(a+b) <-> -a-b
4. Double negation: --a -> a