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Skip to 0 minutes and 7 seconds MICHAEL ANDERSON: So, let’s have a look at an example where the denominators are the same. So, in these types of questions, where the denominator is the same, it’s dead easy to add them up. We’ve got 2/7 shaded in the first diagram, 3/7 in the second diagram. So, when we combine them, 2/7 plus 3/7, we end up with 5 over 7.

Skip to 0 minutes and 29 seconds PAULA KELLY: So, let’s have a look now adding two factions together, where the denominators are not the same. So, we have here 2/7, so one piece cut into seven equal pieces. And we have a half, the same size box cut into two pieces. So, if we add those together, we get–

Skip to 0 minutes and 50 seconds MICHAEL ANDERSON: Well, it’s quite hard to tell, isn’t it? We’ve shared it in this part and this part that would represent our answer, but it’s really hard to see what fraction that would be. Because all the boxes are different sizes.

Skip to 1 minute and 0 seconds PAULA KELLY: So, what we really need is to make the denominator the same, or the numerator the same, so we can compare the same size boxes. OK, so let’s come back to adding our two fractions with different denominators. So, still, I have these 2/7, so my whole has been cut into seven equal parts. I’m going to shade in that I had two of them.

Skip to 1 minute and 26 seconds MICHAEL ANDERSON: OK, and we want to add a half to the 2/7. So, I’m going to shade in half, and I’m going to shade in the top bit this time in a different colour, in green. I’ve split the whole into two equal sections, and I’ll coloured in one of them.

Skip to 1 minute and 42 seconds PAULA KELLY: OK, so, still it’s difficult to put these together, because our pieces are different sizes. To help us, we need to have a common denominator.

Skip to 1 minute and 50 seconds MICHAEL ANDERSON: Right, OK.

Skip to 1 minute and 51 seconds PAULA KELLY: We need to have equal size parts of a whole. So, I need a number that’s a multiple of both 7 and 2.

Skip to 1 minute and 58 seconds MICHAEL ANDERSON: OK, so, if I think of the 7 times tables, it goes 7, 14. 14’s in the 2 times table. It’s an even number, so we’ll go for 14.

Skip to 2 minutes and 6 seconds PAULA KELLY: Good. So, it’s often quite good to have as small a number as possible. So, if I’m going to have a denominator of 14, I’ve had to double my 7.

Skip to 2 minutes and 16 seconds MICHAEL ANDERSON: Yeah.

Skip to 2 minutes and 17 seconds PAULA KELLY: To keep this the equivalent size, I need to double my numerator. So, 2 times 2 will give me four.

Skip to 2 minutes and 25 seconds MICHAEL ANDERSON: OK, so I’ll have to do the same with my half. So, I’m going to think of a number. When I multiply it by 2, then I get 14. So, 2 multiplied by 7 gives me 14. And, because I’ve timesed the bottom by 7, I’m going to also times the top by 7. 1 multiplied by 7 gives me 7. So a half is equivalent to 7 over 14.

Skip to 2 minutes and 47 seconds PAULA KELLY: OK, so that’s much easier now, because now our pieces are the same size. We can add them together.

Skip to 2 minutes and 52 seconds MICHAEL ANDERSON: OK.

Skip to 2 minutes and 53 seconds PAULA KELLY: So, on our whole, I’m going shade in I have 4/14. And, we can see from my diagram, it’s still the same size. All of my pieces have been cut into half. I have twice as many pieces.

Skip to 3 minutes and 8 seconds MICHAEL ANDERSON: OK, so, looking at this part, this was then split into seven parts, and this bit into seven to give us 14. I’m going to colour in the same amount from the top here. So, that’s going to be seven parts this time to represent half of the whole. But, this time, we’re dealing with 14ths.

Skip to 3 minutes and 27 seconds PAULA KELLY: OK, so now we have a common denominator. Our pieces are the same size.

Skip to 3 minutes and 32 seconds MICHAEL ANDERSON: Yeah.

Skip to 3 minutes and 32 seconds PAULA KELLY: We can add them together.

Skip to 3 minutes and 34 seconds MICHAEL ANDERSON: OK.

Skip to 3 minutes and 34 seconds PAULA KELLY: I know, if I had 4/14, you had 7/14, altogether we had 11/14.

Skip to 3 minutes and 44 seconds MICHAEL ANDERSON: So, 2/7 plus 1/2 is equal to 11 over 14.

Skip to 3 minutes and 50 seconds PAULA KELLY: So, let’s have a look at another example of adding fractions where the denominators are different.

Skip to 3 minutes and 55 seconds MICHAEL ANDERSON: OK.

Skip to 3 minutes and 56 seconds PAULA KELLY: So, we have here– I have a half. You have a third. At the moment, we can’t add them together. We can see, from our diagram, our parts of our whole are different sizes.

Skip to 4 minutes and 7 seconds MICHAEL ANDERSON: OK.

Skip to 4 minutes and 7 seconds PAULA KELLY: So, I have a half. My whole has been cut into two. So, if I just shade in, that’s my half.

Skip to 4 minutes and 17 seconds MICHAEL ANDERSON: Yeah. And, for my third– well, I’ve got one out of three, so I’m going to colour in from the top, one piece out of three.

Skip to 4 minutes and 27 seconds PAULA KELLY: OK, all right. So, we can see our pieces are different sizes.

Skip to 4 minutes and 32 seconds MICHAEL ANDERSON: Yeah, so we can’t add them yet.

Skip to 4 minutes and 33 seconds PAULA KELLY: Can’t add them yet– need a common denominator– we need a number that’s a multiple of both 2 and 3.

Skip to 4 minutes and 39 seconds MICHAEL ANDERSON: Yeah.

Skip to 4 minutes and 40 seconds PAULA KELLY: OK, lots of options we could have– our smallest option, if we listed our 2 times table and our 3 times table, is to have 6.

Skip to 4 minutes and 48 seconds MICHAEL ANDERSON: OK.

Skip to 4 minutes and 49 seconds PAULA KELLY: So, I’ll make my fraction have a denominator of 6. To go from 2 to 6, I multiply by 3. From 1, to find out what’s here, to keep it the same, I multiply it by 3 again to give me 3.

Skip to 5 minutes and 5 seconds MICHAEL ANDERSON: OK, so, 3 over 6 is the same as a half. I’m, then, going to change the 1 over 3 into an equivalent fraction where the denominator is 6. So, to take 3, I’m going to multiply that by 2 to get 6. I’m going to do the exact same to the top. 1 multiplied by 2 gives me 2, so I end up with 2 over 6.

Skip to 5 minutes and 27 seconds PAULA KELLY: So, we now have the same denominator. Our whole has been cut into the same equally sized pieces. We have six pieces.

Skip to 5 minutes and 33 seconds MICHAEL ANDERSON: OK.

Skip to 5 minutes and 35 seconds PAULA KELLY: I’m going to have three of them, so I shade in here. And I can see the area mine that’s remained the same.

Skip to 5 minutes and 44 seconds MICHAEL ANDERSON: And then the same, going across the same area, is going to be these two squares out of the six in total.

Skip to 5 minutes and 53 seconds PAULA KELLY: So, all together, our whole has been put into six equal parts. We’ve put our 3/6 and our 2/6 together. We’ve shaded, now, 5/6.

Skip to 6 minutes and 4 seconds MICHAEL ANDERSON: Well, so, the diagrams really give us a great way of seeing what’s going on here because, often, I’ve seen students just use this working to get to 5/6. Is that OK?

Skip to 6 minutes and 14 seconds PAULA KELLY: Absolutely, yes. A misconception would be to have 2/5. And we can see now why that wouldn’t work, why it’s important that our pieces are cut the same size, our common denominator. Then we can put those together.

Adding fractions

Whist adding and subtracting decimal numbers appears straightforward, students often encounter problems when asked to find the sum of two or more fractions.

Some of the common problems include:

  • Just adding or subtracting numerators and denominators.
  • Difficulty deciding what is the common denominator.
  • Inaccuracies when finding equivalent fractions.
  • Leaving answers without simplifying.

In this video, simple diagrams are used to help students understand the need to express each fraction using the same denominator before the fractions can be added together. Paula and Michael go on to explain how similar diagrams can be used to develop an understanding of the process when applied to subtracting fractions.

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This video is from the free online course:

Maths Subject Knowledge: Fractions, Decimals, and Percentages

National STEM Learning Centre