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Adding Fractions

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Mathematics has got to be the most interesting of all subjects. As I was telling the wonderful math faculty at Berkshire Community College, even arithmetic is fascinating. Addition and multiplication are commutative: 4+7=7+4 and 4x7=7x4. I recall MIT Professor Michael Artin saying:

When I ask my kids what's 4x7, they answer "28."
7x4? "28."
4+7? "11."
7+4? "You already asked us that."

Commutativity of multiplication is true, but commutativity of addition is obvious. Later on those youngsters will find that for matrices, addition remains commutative but multiplication does not.

"Now if you could only make fractions interesting," the math faculty responded. The thing about adding fractions such as 1/3 + 1/4 is that nothing sensible works. If you add the numerators, what do you do with the denominators? Add them? Doesn't work. Multiply them? Doesn't work. Of course if the denominators are the same, it's easy: 1/5 + 2/5 = 3/5 just as 1 apple + 2 apples = 3 apples. But if the denominators are different, you apparently can't add the fractions any more than you can add 1 apple + 2 oranges. There's no such difficulty with multiplication; you just multiply the numerators and multiply the denominators: 2/3 x 2/5 = 4/15. OK, there is a way to add fractions by finding a common denominator, but this is not the time to tell the students that. Here's how their test should go:

Question: 1/5 + 2/5. Correct answer: 3/5.

Question: 1/3 + 1/4. Correct answer: Can't be done.

After all, this is how we teach square roots. √4 = 2, √-4 can't be done. Only much later do we thrill them with √-4 = 2i.

Then, later, when you note that 1/3 = 4/12 and 1/4 = 3/12, a bright rebel student will say, "So we can add them!" At that moment they are ready to delight in doing the impossible.