It's True

Yes, I put a $20 bill on the whiteboard and told the students it belonged to anyone who could disprove the axiom.

An axiom is a proposition that we can't prove is true but we've never found a case that hasn't worked.

This is the axiom:
If we take a set of equals and add them to a set of equals, the sums will be equals.

So, take two sets of addition facts:

4 + 2 = 6
5 + 5 = 10
_______

Add them together and see what happens:

4 + 2 = 6
5 + 5 = 10
_______
9 + 7 =

Add the result (Add 9 + 7):

4 + 2 = 6
5 + 5 = 10
_______
9 + 7 = 16

Finally, notice what happens when you add the right-hand side of the original sets (Add + 10)!

What the kids are learning is the algebraic assumption that left-hand side must always equal right-hand side.

So, even though the $20 bill isn't on the board anymore, the offer still stands: If anyone can find a case where this doesn't work, I'll give them the money. Students can try this at their own developmental level, using fractions, decimals, integers...