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Skip to 0 minutes and 8 seconds PAULA KELLY: OK, so if we look at multiplying together two factions, there is a rule that we can use. You may be familiar with this rule. But what’s really important– that you know where it comes from and why it actually works. So if you return back to our unit square, like we had with our multiplying decimals, if we have it split into fifths and also into thirds, if we’ve shaded three of our fifths, we’ve then shaded 2/3 of these 3/5. So if you notice now, we have 15 rectangles, of which 6 are shaded. So our answer is 6/15. But let’s see where it comes from, if our rule reinforces that.

Skip to 0 minutes and 50 seconds So if we had our 3/5 multiplied by 2/3, we know we’re going to end up with 6/15 from our diagram. That appears to have come from our 3 multiplied by 2, and our 5 multiplied by 3.

Skip to 1 minute and 9 seconds MICHAEL ANDERSON: And you can simplify this as well. So 16/15– they’re both in the 3 times tables.

Skip to 1 minute and 16 seconds PAULA KELLY: So we’d have two 3’s in there, and five 3’s in there.

Skip to 1 minute and 21 seconds MICHAEL ANDERSON: It’s quite interesting, looking at that diagram, to notice that the 2/3 that were shaded– the overlap represents 3/5 of those, and then vice versa with the 3/5 that were shaded and the overlap is 2/3 of them.

Skip to 1 minute and 32 seconds PAULA KELLY: Yeah, because with multiplying, our order doesn’t matter. Other way around, OK.

Skip to 1 minute and 36 seconds MICHAEL ANDERSON: And does this rule always work when multiplying the tops together and then the bottoms together?

Skip to 1 minute and 41 seconds PAULA KELLY: Well, before we had our quarter of 12, didn’t we?

Skip to 1 minute and 43 seconds MICHAEL ANDERSON: Yeah.

Skip to 1 minute and 43 seconds PAULA KELLY: And again, we can write that as 12 multiplied by a quarter, or a quarter multiplied by 12. Let’s say 12 multiplied by a quarter. Now, we do also need to have, a denominator here, 12 whole ones. We can write as 12 over 1. We can multiply together our numerators, as we did before, to give us 12. And our1 times 4, which gives us 4. Finally, though, we wouldn’t leave it like that. We can simplify. We’ve got an improper fraction here. So how many 4’s go into 12? We know it’s just three.

Skip to 2 minutes and 17 seconds MICHAEL ANDERSON: Yeah, it’s the same. Cool.

Skip to 2 minutes and 21 seconds PAULA KELLY: OK, so we’ve got our rule for multiplying our numerators together and our denominators. It’ll work for two fractions, three fractions– however many you have. So our numerators– if we did 2 multiplied by 4, it would give us 8. By 3 is 24. And the same in my denominator. So 5 times 7, 35. Times that by 8, we get 280. And again, the order we multiply wouldn’t make any difference. Not in its simplest form, so we wouldn’t leave it there. Both even numbers. You can half them. So we have 12 and 140. Again, both even. We could do it in fewer stages, but just to be clear, we have 6 out of 70. Still both even.

Skip to 3 minutes and 11 seconds Finally we go down to 3 and 35. Then we haven’t got a number that goes into 3 and 35, so whenever we finish, we’re fully simplified.

Skip to 3 minutes and 24 seconds MICHAEL ANDERSON: So in this example, the numbers got quite big quite quickly, because we’re multiplying lots of numbers together. In the next step, we’re going to look at a strategy to make these types of calculations more simple.

Finding a fraction of a fraction

When finding the product of two fractions the rule multiply the numerators and multiply the denominators is well known.

When teaching this topic it is worth developing the idea that what we are doing when multiplying two fractions is in fact finding a fraction of a fraction.

In this video, Paula and Michael answer what is \(\frac{3}{5} \times \frac{2}{3}\)? They look at the underlying mathematical structure of what is happening when we find a fraction of a fraction to help understand why the rule for multiplying fractions works.

The same method can be used for any number of fractions being multiplied together. If you are multiplying by a whole number, remember that this should be represented as that number over one. For example 12 needs to be represented as \(\frac{12}{1}\).

Problem worksheet

Now complete question 4 from this week’s worksheet.

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Maths Subject Knowledge: Fractions, Decimals, and Percentages

National STEM Learning Centre