Meanwhile, since your students are clearly excellent at making conjectures and investigating problems, I have a problem for them.

   Farmer John has long weighed his hay bales by using a balance beam scale and a rock. The rock is exactly forty pounds, which perfectly balances the right amount of hay for bales placed on the other pan of the beam.

   Farmer Joe comes by one day and is very impressed with the rock and asks Farmer John if he can borrow it. Farmer John says, Sure...

   But you know what happened next, I'll bet: Farmer Joe borrowed the rock and broke it by accident, into three pieces.

   Farmer Joe profusely apologizes to Farmer John for breaking the rock, but as it turns out it's all for the good: A day later Farmer John reports to Farmer Joe that he can now weigh ANY item accurately that weighs between one and forty pounds using only the three rock pieces (and the balance beam scale)!