Students learned how generate equivalent fractions in grade 4, and are doing just what their teacher has told them to do:

"Whatever you do to the top, you have to do to the bottom." Whenever I hear this, I think of The Golden Rule. Do unto the numerator what you would do unto the denominator. Something like that. |

There's a page to complete. It encourages the students to apply

*The Multiplication Rule for Equivalent Fractions,*which, in case they forget, is written in the middle of the worksheet:

**When the numerator and the denominator of a fraction are multiplied by the same number, the result is a fraction which is equivalent to the original fraction.**

There are lots of opportunities to practice, and there are fraction circles available so students can model what they've created.

I approach a student and start talking to him:

**Me:**Hi! What are you doing?

**Student:**Making equivalent fractions.

**Me:**Great! How do you do that?

**Student:**Whatever you do to the top, you have to do to the bottom.

**Me:**Say more about that. What do you mean exactly?

**Student:**So if I have 1/2, and I want to make an equivalent fraction, I have to multiply the numerator by 3 and the denominator by 3 and that will make 3/6.

**Me:**Sounds like fun! What would happen if I multiplied the numerator and the denominator by different numbers?

**Student:**You'd get the wrong answer.

**Me:**(As I write out 1/2 X 3/5 = 3/10 on a piece of paper) So if I multiply 1 x 3 and 2 x 5 and get 3/10, that would be wrong?

**Student:**(politely, blithely, but somewhat exasperated) All I know is that the teacher said, "Whatever you do to the top, you have to do to the bottom." I never really worried about why.

Later, the math coach and I talked about the interaction. We filled up a whiteboard with our own equations and visual models, explaining to each other what we understood, or thought we understood, about what was going on in that grade 5 class. There's lots happening underneath the deceptively simple, oft-repeated phrase

*Whatever you do to the top, you have to do to the bottom,*just as there is underneath the student's reflection that, "

*I never really worried about why."*

More than the math, it's the

*I never really worried about why*that's had me thinking. Here's what I've been asking myself:

- Is there a compelling reason that the student should have to worry about why? A reason not that
*we*think is important, but that the*student*thinks is important? - Is there a difference between being
*worried about*why and*wondering about*why? What exactly did the student mean? - We already know what might make a student
*worry*about why: It's going to be on the test! You'll have trouble next year if you don't know! But what has to happen in a classroom to make a student*wonder*why? - Is it always bad just to follow a rote procedure without understanding, wondering, or worrying about why? Maybe that needs time to develop. Maybe it will come later.
- What routines or rote procedures do I follow without worrying about why? Should I be worried about them? Should I be more curious about them?

On

*your*timetable, not 5.NF.A.1's.

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