From the Archives: Cookie Connections (on fractions and division)

Originally posted January 2007

One of the common themes you hear around homeschoolers is that so many lessons can be taught through real life. One that irritates me is "teach writing by writing thank-you notes." Unless we're getting married or having a baby, most of us just don't have to write that many thank-yous, and I assume that most of our children are in neither position. The other one is fractions-by-cookies. "Make cookies. Look at what they're learning."

Well, yes. But I have an issue with the "just make cookies" idea, besides dental objections. It's true I had my own first exposure to fractions by making cookies with my mother. ("It says put in 1, funny line, 2 cups of peanut butter. What's that mean?") But there's more to math than just recognizing what 3/8 of a cup looks like, or even that if you put two of those together you get 3/4 of a cup. There seem to be an awful lot of excellent cooks out there who still get nervous around fractions. Some of them are homeschooling moms, and that worries me.

One of the most interesting arithmetic concepts--that my teachers somehow forgot to point out in school until we did algebra--is that multiplication, division, and fractions are all interchangeable. Connected. Once you understand this, arithmetic gets so much easier.

Consider 2/3 of 5.

In Miquon Math you learn that the word "of" can be written "x." As in "times." If something doesn't make sense to you with the word "times," try substituting "of." Or the other way around. So 2/3 x 5 is the same as 2/3 of 5. If you don't know what 2/3 of 5 is, you can figure it out with multiplication. Everybody knows how to multiply fractions, right? (much easier than learning to add them) So 10/3, or 3 1/3. Simple. Little kids can get "of." You write "1/2 x 10," and they say 5. They've just multiplied fractions.

And then there's that cancelling-out maneuver. When you add this to your arsenal, you have some powerful arithmetic tools going for you. You know what I mean, right?

Like 3/10 x 5/9. Of course you can multiply the tops and the bottoms, and you end up with 15/90. And then you can fool around reducing, and you get 1/6. But sometimes that's a lot of work. So you can cancel out the numbers that criss-cross; and you know why, don't you? Because

3/10 x 5/9 is the same as 3 x 5 over 10 x 9 (I'm not sure how to get those to line up properly).

And you could write that 5 x 3 over 10 x 9; and you could split those back up and write 5/10 x 3/9 . And if you reduced the fractions before you multiplied, you'd have 1/2 x 1/3 = 1/6.

Well, just in case you need a reminder on this--you don't need to go through all that moving around. You can do the same cancelling out by checking the numbers that are criss-cross with each other in the original equation. The 3 and the 9 cancel out, and the 5 and the 10.

The third point I wish my teacher had remembered to pass on is that fractions are also division. The "funny line" is not just a fraction marker, it's a division sign. 2/3 means 2 divided by 3, or how many 3's in 2, or how much pizza do 3 people get if they split 2 pizzas? Obviously they each get 2/3 (you could have figured that out even without doing fractions), but isn't that still kind of mind-boggling? You say 2 divided by 3, you write 2/3, and you already have your answer.

And what's 5/3 of 2? Obviously, still 10/3. 3 1/3.

What's 5 divided by 3? How many 3's in 5? 5/3, or 1 2/3.

What's 10 divided by 3? How many 3's in 10? 10/3, or 3 1/3.

What's 1/3 of 10? 10/3.

Multiply 1/3 x 10/1. 10/3.

Fractions? Division? How come we're doing all this multiplying all of a sudden? Zing: connections.

This is why I like Miquon Math [and other programs that teach this way]. I like bigger ways of looking at things than the "this year we do multiplication, next year we do division" approach. And that's why I think you do need to go beyond cookies--unless you have an awful lot of them and a very sharp knife.