Multiplication of fractions: role of unit fractions

In Math is Figure-Out-Able podcast, Pam Harris and Kim Montague have been discussing multiplication of fractions in the recent episodes (Episodes 206, 207, and 208). In those episodes, they really stress the importance of multiplying a unit fraction by a unit fraction. In the Common Core State Standards, they also discuss the idea dividing a unit fraction by a whole number and a whole number by a unit fraction in Grade 5 before discussing fraction division more generally in Grade 6.

In the US, fractions were often introduced using the part-whole interpretation. So, when the CCSS discussed the idea of thinking of a non-unit fraction as a collection of unit fractions, that made many US teachers to shift their teaching of early fractions. This way of thinking about fractions is very common in Japan – and I suspect in some other East Asian countries, too. When I first taught a fraction course at KSU and brought in this idea (15+ years ago), I was surprised to read so many students writing in the course evaluations, how this way of looking at fractions made fractions easier to understand. So, unit fractions play a very important role in earl fraction teaching and learning.

Interestingly, though, in the Japanese curriculum, unit fractions do not play a particularly special role at all in fraction multiplication and division. Rather, the emphasis is more on understanding (or expanding) the meaning of multiplication and division.

As is the case in the US (and many other places), multiplication in Japan is introduced in equal group situations. Thus, the meaning of multiplication students first develop is that it is an operation to find the total amount when we have so many equally sized groups. Thus, there is the relationship, Total = Group Size × Number of Groups [in Japan, we write the group size, multiplicand, first, then the number of groups, multiplier]. When we try to extend the study of multiplication and division to fractions and decimal numbers, we need to expand the way we think about multiplication (thus division). In particular, when the multiplier (number of groups) become a fraction or a decimal number, multiplication based on equal group doesn’t work because the number of groups is a counting number (natural number). So, when it is a fraction, it doesn’t necessarily make sense. It is not much of a problem if the group size becomes something other than a whole number because we can still think of so many groups of fractional amounts.

This is the reason the Japanese curriculum – and the CCSS – first think about multiplying fractions by whole numbers. So, if you have 3 groups of 4/5, for example, you can think of it as having 3 groups of (4 1/5s – so unit fraction does play a role). This way of thinking parallels 3 groups of 40 can be thought of as 3 groups of 4 10s. In the Japanese curriculum, the main emphasis on teaching of multiplication and division of fractions and decimal numbers is helping students to extend their understanding of multiplication as an operation. In the CCSS, this is the idea of understanding multiplication as scaling, which was the topic of my last post. That part wasn’t discussed much in any of the episodes. It seems like they just rushed over that idea by saying “of means to multiply.” Although I have heard many US teachers say that phrase, I am really not sure exactly when students learn “of means to multiply.” To me, it sounds like a mnemonic device like PEMDAS or “keep change flip.”

Once students learned to interpret multiplication by a fraction using the idea of multiplication as scaling, another important idea students need to understand is that to multiply a number by a fraction means it is the combination of multiplication (by the numerator) and division (by the denominator). So, to multiply by 3/4 means to multiply by 3 and divide by 4, and this is the idea discussed in the CCSS standard 5.NF.4.a, although I sometime think this is the most ignored standard in the document. By the way, this combination of multiplication and division can be done in either order. That is, you can think of multiplying 2/5 by 3/4 as either multiplying 2/5 by 3 then divide the result by 4, or divide 2/5 by 4 then multiply the result by 3.

If you put this in a concrete situation, it might make better sense. Let’s say 1 m of iron pipe weighs 2/5 kg. We want to know how much 3/4 m of the same pipe would weigh.

We can think about how much 1/4 m of the pipe weigh by dividing 2/5 by 4 (since 1 m = 4 1/4 m). Since 3/4 m is made of 3 1/4s, you then multiply the quotient by 3. So, we divided by the denominator first, then multiplied by the numerator.

Another way of thinking about this problem is to think about how much 3 m (4 x 3/4m) will weigh by multiplying 2/5 by 3. But since we know that 3 m is 4 times as long as 3/4 m, to find the weight of 3/4 m, you have to divide by 4. So, we multiplied by the numerator first, then divided by the denominator.

When you understand this relationship, you can make sense of division of fractions in a similar manner. If 3/4 m of an iron pipe weighs 2/5 kg, how much does 1 m of the pipe weigh? This is 2/5 ÷ 3/4. Just as the case with multiplication, we can think about how much 1/4 m will weigh by dividing 2/5 by 3, then multiply it by 4 to get the weight of 1 m. Or, we can think about how much 3 m (4 times of 3/4 m) will weigh by multiplying 2/5 by 4, then divide it by 3 to find out how much 1 m will weigh. Either way, dividing by 3/4 meant the combination of multiplying by 4 and dividing by 3 – which means it is the same as multiplying by 4/3.

This is how the Japanese curriculum discuss division of fractions. However, the CCSS seems to be sidetracked by discussing dividing a unit fraction by a whole number or a whole number by a unit fraction.

Anyway, multiplication and division of fractions seem to remain as a most challenging ideas in middle grades mathematics curriculum. To me, the key in this process is extending the meaning of multiplication – and refining the interpretations of division using that interpretation of multiplication. Unless that is done carefully, I think multiplication and division of fractions will continue to be a bit mysterious to students. But that’s my opinion. I would love to hear what others think.