From CPM:

The Sutton family took a trip to see the mountains in Rocky Mountain National Park. Linda and her brother, Lee, kept asking,

“Are we there yet?”At one point, their mother answered,“No, but what I can tell you is that we have driven 100 miles and we are about 2/5of the way there.”Linda turned to Lee and asked,

“How long is this trip, anyway?”They each started thinking about whether they could determine the length of the trip from the information they were given.

And I like both methods, especially Linda’s.

Without using a visual, we may have students solve for *x* in the equation (2/5)(*x*) = 100 by multiplying both sides by 5/2.

But I notice two things: 1) Students don’t always remember *why* they are multiplying by the reciprocal, and 2) Students have difficulty showing Linda’s method with an equation like (9/2)(*x*) = 27.

So, I’m having the students think through the problem by answering these two questions:

- If we know that
halves of*nine**x*is 27, then what ishalf of*one**x*? - Now that we know what
half of*one**x*is, what is a*whole**x*?

As we write the fractions, we can keep our focus on the whole number numerator and treat the denominator as if it were a thing, and that thing is not changing.

Another example,

This helps us go back to finding the unit rate in the first step via division, and then find a multiple of that unit rate via multiplication.

Once students make sense of these two steps and become fluent in solving for a whole *x*, then they can work on the not-so-friendly equations — such as (5/6)(*x*) = 4 — because they are more confident and trust the process.

Sure, multiplying by the reciprocal would have solved for *x* in one step, but there’s something uniquely comforting to students when they can first find just *one* part of something.

## 5 Comments

Think of denominators as the name and the numerator as “how many” of those you have. Later, as students begin to see that the name isn’t arbitrary and corresponds to the size of the unit fraction with that name, there can be even deeper sense-making of numerators, denominators, ratio(nal) numbers, etc.

There is an analogy to the arithmetic of polynomials, where there are the same restrictions on addition/subtraction (like denominators or like terms are required for those operations to make sense) but not for multiplication/division. And that bears some serious thinking about.

Fawn, do you have any concern that using language “nine halves of x is 27) masks the operation (mult.) and suppresses the development of 9/2 as a number? It strikes me that what you describe supports children’s ability to think through the problem type you suggest. And maybe this doesn’t “hurt” for further learning… really, am just wondering

Using your 2nd example, I would take it one step further by decomposing (3/7)x=15 into (1/7)x + (1/7)x + (1/7)x=15, then it would be easier to show (1/7)x=5, and also allow students to understand what 3/7 is (three one-sevenths!)

I LOVE how you brought this up because I have never thought about a fractional coefficient this way. When I show the students conceptually, I often would have to resort to pictures (nothing wrong with that) and I like your way in a sense that it provides a verbal way to guide students.

I, too, share your concern about multiplying by the reciprocal. Instead, I encourage students to take two steps. First, multiply by the denominator so that the variable does not have a pesky fraction there anymore. Then divide by the remaining coefficient.

This seems like a perfect sort of problem to deconstruct with students via a number talk, like some of the examples Cathy Humphreys gives in chapter nine of Making Number Talks Matter.