If Not Methods: Fraction Multiplication

This is the last post (for now, at least) in our If Not Methods series about how to help children figure out tough calculations.

By the time students reach the topic of multiplying fractions, they have become well-practiced at following rules. After some of the complex procedures they’ve learned, a simple rule like “tops times tops, and bottoms times bottoms” comes as a relief.

But we know that relying on rules like that weakens understanding, just as relying on crutches weakens physical muscles.

If we want our students to think, to make sense of math, to figure things out, what can we do with a problem like ⁵/₆ × 21 ?

When Kids Say, “I Don’t Know How”

We can draw forth our children’s intuition about numbers, helping them learn to reason by using the Notice-Wonder-Create cycle.

As we explore the problem, taking turns with our students, we might say things like…

• I notice that we’re multiplying. Does that mean the answer will be bigger?
• I notice that ⁵/₆ is less than one, so ⁵/₆ × 21 must be less than 1 × 21.
• I wonder, how much less?
• I notice that ⁵/₆ is almost one, so I predict the answer will be almost 21.
• I remember hearing that “times is of,” so we can read this problem as “⁵/₆ of 21.”
• Oh, that’s easier to think about. I wonder, why does it work that way?
• I notice that sixths are a pain. It would be easier to find halves.
• I notice that half of 21 is 10 ½.
• I notice that thirds are even easier: ¹/₃ of 21 is 7.
• I wonder, would drawing a picture help?
• I wonder, what kind of picture would you draw?

And so on, until the child’s mind sparks with an “Aha!” that leads to a way forward on the path toward creating a solution.

Multiplication is “of”

It’s a quirk of the English language that we can often replace the multiplication symbol with the word “of.”

Young children often think about multiplication as “groups of” or “sets of.” With fractions, it makes more sense to think “parts of” or “chunks of,” or simply to use the word “of” by itself.

2 × 3 = 2 groups of 3
4 × 8 = 4 sets of 8
¹/₂ × 3 = ¹/₂ part of 3
⁷/₈ × 16 = ⁷/₈ of 16

This helps because our minds think in words, not math symbols. The times symbol sparks no mental connections, but when we replace it with a plain-English word, that lets us reason about what the numbers mean.

When we want to find a fractional part of some quantity, we are using the fraction as an operator.

Five meanings of a fraction

As we’ve seen in earlier posts, there are five different ways we can think about what a fraction means:

• part-whole: seeing fractions as parts of some whole thing, visualizing geometric shapes cut into equal portions
• measure: reasoning about fractions as marks on the number line, measuring out a given length
• operator: finding a fractional part of some number
• quotient: considering the numerator divided by the denominator
• ratio: thinking about a proportional relationship

Part-whole and measure are most the most helpful when making sense of addition and subtraction. For multiplication, we can still use measure, but we also want students to develop the more advanced models.

Fraction as measure: The area model

We can think about whole-number multiplication by drawing a rectangle with sides the length of the factors. In the same way, we can multiply a fraction by drawing a rectangle with a fractional side.

With a small fraction, the sides of the rectangle will not be exactly to scale. One short side and one long side can approximate our product.

Then we can split the sides into parts that are easier to calculate.

⁵/₆ = ³/₆ + ²/₆ = ¹/₂ + ¹/₃

So we want to split 21 into pieces that make it easy to find halves and thirds.

⁵/₆ × 21 = 9 + 6 + 1 + 1.5 = 17.5

Fraction as operator: Find the unit fraction

To find a fractional part of a number, we begin with the unit fraction. If we want to find ⁵/₆ of 21, we start by finding ¹/₆.

¹/₆ of 21 = half of ¹/₃ of 21
¹/₃ of 21 = 7
¹/₆ of 21 = 3.5

Then we can scale up the unit fraction to find the fraction we really wanted. ⁵/₆ = 5 × ¹/₆, so:

⁵/₆ of 21 = 5 × ¹/₆ of 21
= 5 × 3.5 = 17.5

Fraction as quotient: A fraction is a division problem

A quotient is the answer to a division calculation. When we view the fraction as a quotient, we have the division problem and its answer all in one:

⁵/₆ = 5 ÷ 6

Which tells us:

⁵/₆ × 21 = 5 ÷ 6 × 21

Now we can reorder the factors to make our calculation easier to think about:

5 ÷ 6 × 21 = 5 × 21 ÷ 6
= 105 ÷ 6 = ¹⁰⁵/₆

At the end, we can simply turn our division problem back into a fraction, which gives us the answer because a fraction is a quotient.

Don’t worry about the improper fraction. When students get to algebra, such fractions will be much more useful than their equivalent mixed numbers. So let your children get used to seeing and working with improper fractions.

Fraction as ratio: Equivalent fractions

When we view a fraction as a ratio, we can set up a series of equivalent fractions, adjusting them as needed until we get the form we want. This is called proportional thinking, keeping our numbers in the same proportional relationship as we work through our calculation.

⁵/₆ × 21 means we are trying to find the unknown number that will make the following equation true:

5: 6 = N : 21

I like to keep track of the steps in a ratio table, which is just a row of equivalent fractions side-by-side along with the transition calculations between them.

We can’t go directly from six to twenty-one, so we divide both parts by two:

5 : 6 = 2.5 : 3 = N : 21

Then we can multiply our ratio by seven to get the number we were looking for:

5 : 6 = 2.5 : 3 = 17.5 : 21

Therefore, ⁵/₆ × 21 = 17.5.

Our children don’t have to think up all the above methods for multiplying the fractions. They don’t have to use any of these methods, as long as whatever they do makes sense.

But we do want students to be able to understand methods like these when someone else uses them. As part of their mathematical growth, we want them to be able to think in many flexible ways about the patterns and relationships between numbers, so they have choices of how to solve the next math problem they face.