## Thursday, October 30, 2008

### The Sum of Two Numbers Equals Their Product... A Problem for All Grades 3-12?

A simple warmup for Grades 3-12?? Can one problem really be appropriate at many levels?

Would 3rd, 4th or 5th graders guess the obvious answers 0,0 or 2,2 provided they understand the meaning of the terms sum and product? Do youngsters immediately assume the two numbers are different? Children at that age are thinking of whole numbers, however, what if you allowed them to try 3 and 1.5 (with or without the calculator)?

For middle schoolers: After they 'guess' the obvious integer answers, what if you were to ask them: "If one of the numbers is 3, what would the other number be?" If one of the numbers were 4? 5? -1? -2? Is algebra necessary for them to "guess" the other number? Would a calculator be appropriate for this investigation? Would they begin to realize there are infinitely many solutions? What if you asked them to explain why neither number could be 1...

For Algebra students: If one of the numbers is 3, they should be able to solve the equation:
3+x = 3x; they can repeat this for other values including negatives as well. They should be able to explain algebraically why neither number could be 1. Let them run with this as far as their curiosity takes them!

For Algebra 2, Precalculus and beyond: See previous ideas. Should they be expected to solve the equation x+y = xy for y obtaining the rational function y = x/(x-1)? Analysis of this function and investigation of its graph may open new vistas for this 'innocent' problem about sums and products. Does this function really make it clear why 0,0 and 2,2 are the only integer solutions?

The original question is well-known. At any level, I would recommend that they be allowed to explore and make conjectures before more formal analysis. High schoolers enjoy coming up with 0,0 and 2,2 as much as 8 year olds! Modifying it and asking probing questions as students mature mathematically is the challenge for all of us. Have fun with this 3rd grade question!