If Not Methods: Dividing Fractions

As I said in an earlier post, we don’t want to give our children a method because that acts as a crutch to keep them from making sense of math.

But what if our children get stumped on a tough fraction calculation like 1 ¹/₂ ÷ ³/₈?

When Kids Say, “I Don’t Know How”

We can teach without crippling a child’s understanding if we follow the Notice-Wonder-Create cycle:

• Notice everything about the problem, all the details about the numbers, shapes, or story problem situation.
• Wonder about the possibilities, posing questions related to the things you’ve noticed.
• Create something new: a solution to the problem, an explanation of how to solve it, or perhaps a journal entry.

I like to take turns with my students making Notice and Wonder statements, digging deep into a math problem.

“Notice, Wonder, Create” is not a three-step method for solving math problems. It’s the natural, spiraling cycle by which our minds learn anything.

For Example: 1 ¹/₂ ÷ ³/₈

As we notice and wonder together, we might say things like…

• I notice that 1 ¹/₂ is more than one but less than two.
• I wonder, will the answer be more or less than one?
• I notice that we’re dividing, so that should make the answer less.
• I wonder, does dividing always make the answer less?
• I notice division means we need to find out how many pieces of size ³/₈ are in a total of 1 ¹/₂.
• I notice that ³/₈ is a bit less than half.
• Oh, wait! I notice that 1 ¹/₂ is THREE halves. So if ³/₈ is less than half, there must be MORE than three of them in 1 ¹/₂.
• Hmph. So that means division doesn’t always make numbers smaller.
• I wonder, would this be easier if the fraction pieces had matching denominators?
• I wonder, would drawing a picture help?
• I wonder, could we make up a story problem to help us imagine what the numbers are doing?

And so on, until the child’s mind sparks with an “Aha!” that leads to a deeper understanding and progress toward a solution.

How To Solve It

So how might a child think their way through this calculation?

We have two ways to think about division: fair sharing, or measurement.

Fair sharing is also called partitive division because we imagine splitting the original amount into a certain number of parts. That’s often the first way children think about division, but it’s hard to make sense of splitting something into 3/8 parts.

Measurement division is like scooping raisins or cutting cloth, creating same-size chunks and finding how many of them we get. It’s also called quotative division because it matches the quotient in long division: How many times does this amount “go into” our total?

For fractions, the “goes into” idea of quotative division makes intuitive sense, so that’s usually the best place to start.

Measurement: Scooping Raisins

We can imagine pouring raisins into small packages that hold ³/₈ cup each.

• The first package gets ³/₈ cup.
• Two packages need ⁶/₈ = ³/₄ cup.
• Three packages take ⁹/₈ = 1 ¹/₈ cup.
• Four packages hold ¹²/₈ = 1 ¹/₂ cup.

Therefore, 1 ¹/₂ ÷ ³/₈ = 4.

Measurement: Draw a Picture

We can draw a picture of one shape plus a half-shape, and then cut those into chunks of size ³/₈.

Since we’ll be cutting eighths, it makes sense to start with a rectangle on graph paper that’s eight squares total.

• One whole = 8 squares, plus half of the next rectangle = 4 squares.
• Now split that into chunks of 3 squares (³/₈) each.
• How many chunks do we get?

1 ¹/₂ ÷ ³/₈ = 4 chunks

Measurement: Common Denominator

How many ³/₈s are there in 1 ¹/₂? First, let’s find a common denominator so we can compare the fractions more easily:

1 ¹/₂ ÷ ³/₈
= 1 ⁴/₈ ÷ ³/₈
= ¹²/₈ ÷ ³/₈
= 12 ÷ 3 = 4

Because 12 of anything (in this case, eighths) divided into groups of 3 of that thing must always make 4 groups.

Measurement: Think One

Before we try to figure out how many ³/₈s are in 1 1/2, let’s first find how many ¹/₈s there are.

We have eight ¹/₈s in one whole thing, and four more ¹/₈s in the extra half, so:

1 ¹/₂ ÷ ¹/₈ = 12

But we don’t really want single eighths. We want them in sets of three.

So that means…

1 ¹/₂ ÷ ³/₈
= 12 ÷ 3 = 4

Fair Sharing: A Fraction of a Part

Fair sharing isn’t intuitive with fractions, but sometimes it can be a fast path to an answer.

If 1 ¹/₂ is ³/₈ of something, how much is the whole thing?

We know 1 ¹/₂ is three halves, so each half must be ¹/₈ of the whole amount.

That means our whole thing must be eight halves:

1 ¹/₂ ÷ ³/₈
(1 ¹/₂ ÷ 3) × 8
= ¹/₂ × 8 = 4

Rote Method: Flip and Multiply

The standard “flip and multiply” method does produce correct answers, as long as students remember which numbers they are supposed to flip. But it does nothing to build anyone’s intuitive understanding of fractions.

And children almost never remember this rule correctly. I’ve had algebra students who were convinced that every time they needed to multiply fractions, the first step was to flip them all upside-down.

Students do need to know the standard method in order to work with algebraic fractions, but it’s the most confusing procedure in all of elementary arithmetic.

The rule should only come after kids have plenty of experience making sense of fraction division using intuition.

Rather than “flip and multiply” or other mnemonics, I prefer to give students a true principle of mathematics: “To divide by any number, you multiply by that number’s reciprocal.”

• 6 ÷ 2 = 6 × ¹/₂
• 9 ÷ 3 = 9 × ¹/₃
• 4 ÷ ¹/₂ = “How many halves in 4?” = 4 × 2
• to divide by ³/₈, you have to multiply by ⁸/₃

1 ¹/₂ ÷ ³/₈
= 1 ¹/₂ × ⁸/₃
= ³/₂ × ⁸/₃ = 4

It’s the Journey that Matters

There may be only one right answer to a math problem, but there’s never only one right way to get that answer.

Your children don’t have to solve math problems by the same method you would use, nor by following the procedure in the book. They can use any method that makes sense to them, as long as they can explain why it’s true.

What matters in math is the journey. How do your children make sense of the problem and reason their way to that answer?

As always, real math is not about the answers but the thinking.