Skip to 0 minutes and 8 seconds PAULA KELLY: OK. So we’ve seen how with our multiplying, our numbers can get rapidly quite big and quite difficult to simplify. So an easier method can be to do some simplifying before we do our multiplying. So with our example here, if we know we’re multiplying, our order doesn’t matter. So we could rewrite this as our 3 times 2, or divided by our 5 times 3. I’m going to write this 3 here because then, if you notice, you’ve actually got a whole 1 here
Skip to 0 minutes and 39 seconds MICHAEL ANDERSON: Oh, 3 over 3–
Skip to 0 minutes and 41 seconds PAULA KELLY: –is one whole 1. So that just gives our 2/5.
Skip to 0 minutes and 46 seconds MICHAEL ANDERSON: Ah, brill.
Skip to 0 minutes and 47 seconds PAULA KELLY: OK. So if we look here, we’ll do the same thing again. Our 2 times 4 times 3 doesn’t give us any common numbers. We could pair these two together to get our 8. So effectively, we have our 8 multiplied by 3. And then on our denominator, we have our 8 already, and our 5 and our 7.
Skip to 1 minute and 13 seconds MICHAEL ANDERSON: Ah, an 8 over 8 is just 1, so they effectively disappear.
Skip to 1 minute and 16 seconds PAULA KELLY: Fantastic. That just gives us now just our 3 over our 5 times 7, so our 35, which is much simpler than our cancelling we did earlier.
Skip to 1 minute and 27 seconds MICHAEL ANDERSON: Oh, nice.
Skip to 1 minute and 28 seconds PAULA KELLY: If we’re understanding where that comes from, we might be more familiar with– what’s slightly more straightforward is– that keeps them common factors without rewriting the single fraction. So if we notice, we have some common factors here, so they’re both divisible by 5.
Skip to 1 minute and 48 seconds MICHAEL ANDERSON: OK. And replacing those with 1’s instead of 5’s? Brill.
Skip to 1 minute and 51 seconds PAULA KELLY: Yep. We have one 5 in here, one 5 in here. Not as straightforward here. Looking for a common factor– 3 is a common factor. So we could say we have one 3 in there and four 3’s in there.
Skip to 2 minutes and 3 seconds MICHAEL ANDERSON: Ah, so divide the 3 by 3 to get 1, and the 12 by 3 to get 4. Brill?
Skip to 2 minutes and 8 seconds PAULA KELLY: Absolutely perfect. So same again– we multiply our numerators just to get 1, and our denominators just get 4, so a quarter.
Skip to 2 minutes and 17 seconds MICHAEL ANDERSON: Excellent.
Simplifying before multiplying
When completing calculations involving fractions, the question will often ask for the answer to be given in its ‘simplest form’. This means cancelling the fraction down so that the numerator and the denominator do not have any common factors.
In most of the examples the cancelling down process has been performed at the end of the calculation. In this video, Paula and Michael consider the advantage of performing the cancelling down before performing the calculation.
Rules in the wrong situation
A problem encountered when students are taught procedurally, and without understanding, is applying rules in the wrong situation. An activity which is worth using is to ask students to use their calculators to investigate in which situations you can and cannot ‘cancel first’.
Most scientific calculators have fraction functions. It is worth investing some time in ensuring that students understand how to use this function on their calculator. Students can be given a series of worked examples where cancelling has been performed before completing the calculation. Include examples where cancelling has, incorrectly, taken place i.e. before adding fractions, subtracting and dividing fractions, as well as examples which do work i.e. when multiplying fractions.
The aim of the exercise is for students to understand that cancelling before calculating only works when multiplying fractions.
Now complete question 5 from this week’s worksheet.
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