If Not Methods – Subtracting Fractions

We’re continuing our series of posts on how to build robust thinking skills instead of forcing our children to walk with crutches.

When we say, “Use this method, follow these steps,” we teach kids to be mathematical cripples.

If your student’s reasoning is, “I followed the teacher’s or textbook’s steps and out popped this answer,” then they’re not doing real math. Real mathematical thinking says, “I know this and that are both true, and when I put them together, I can figure out the answer.”

But what if our kids get stumped on a fraction calculation like 7/8 − 1/6?

When kids say, “I don’t know how”

Notice-Wonder-Create is the natural, spiraling cycle by which our minds learn anything. Take turns with your students making Notice and Wonder statements, building a deeper understanding of the math problem.

• I notice subtraction, and we’re starting with a fraction less than one, so the answer will be even smaller.
• I wonder, does subtraction always make the answer smaller?
• I notice that sixths and eighths don’t play well together. Sixths and thirds would be easier, or fourths and eighths.
• I notice that 7/8 is almost 8/8, which would be one whole thing, and 1/6 is small. I bet the answer will be more than a half.
• I notice the 7/8 is the same as seven 1/8’s.
• I notice that 1/6 is more than 1/8 but less than 2/8, because 2/8 is the same as a fourth.
• So that means the answer must be more than 7/8 − 2/8 = 5/8.
• Hey, I was right! It is more than a half.
• I notice this would be easier if both fractions had the same size of pieces, the same denominator.
• I wonder, would drawing a picture help?

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

(Children will also notice or wonder things that go radically off-topic. A little of that is fine, but don’t hesitate to direct their attention back to the problem at hand.)

Five meanings of a fraction

There are five different ways we can think about what a fraction means:

• part-whole
• measure
• operator
• quotient
• ratio

Children begin by seeing fractions as parts of some whole thing, visualizing geometric shapes cut into equal portions. Then they learn to reason about fractions as marks on the number line, measuring out a given length.

Either of these models offers a way to begin understanding subtraction.

(We’ll talk about the other fraction concepts in future posts.)

Part-whole: Subtraction as take-away

We have 7/8, which is just a bit less than one whole thing.

We need to take away 1/6.

But the size of the fraction pieces doesn’t match, so we have to cut them into smaller pieces until we can get both fractions with the same denominator. Then we’ll be able to count down the pieces we take away and see how much remains.

When we cut fraction parts into smaller pieces, the number of pieces goes up by the same proportion as the size of the pieces goes down.

If we cut each of the 1/8 pieces into three, the smaller parts will be 1/24ths. Since each piece became three smaller parts, we now have three times as many pieces.

7/8 = 21/24

If we cut 1/6 in half, that gives us two 1/12ths. And if we cut those pieces each in half, we get four 1/24ths.

1/6 = 4/24

Now it’s easy to take pieces away.

21 /24 − 4/24 = 17/24

Measure: Subtraction as distance

How far apart are 1/6 and 7/8? What is the distance between them?

We can find 1/6 and 7/8 on the number line, but how can we figure out the distance between them?

From 1/6, we can move up 2/6 more to get to 1/2. And since 1/2 is 4/8, we need 3/8 more to reach 7/8.

So…

7/8 − 1/6 = 2/6 + 3/8

We’re still dealing with sixths and eighths, but many people find it easier to make sense of addition than subtraction.

A note about common denominators

We turned our subtraction problem into addition. But we still need to convert those fractions into same-size pieces so we can add up the total.

We must find a common denominator.

Despite what our textbooks taught us, we really don’t have to worry about finding the smallest (or least) common denominator. ANY common denominator will give us fractions with same-size pieces that we can add or subtract.

For example, you can always scale each fraction by the other fraction’s denominator to get what I call the easiest common denominator. It’s easy because the numbers automatically come out equal.

If we cut the sixths into eight pieces each, that gives us 16/48. And if we cut the eighths into six pieces each, that makes 18/48.

2/6 + 3/8 = 16/48 + 18/48= 34/48

And that is a fine answer.

A note about simplifying fractions

If students happen to notice that the numerator and denominator are both even, they can divide out the common factor:

34/48 = (34 ÷ 2) / (48 ÷ 2) = 17/24

But that is only a cosmetic change, no more than skin-deep. Both fractions name the same amount of stuff. They are equivalent, so either answer is correct.