Lesson video

Hiya, my name's Miss Lambell.

You've made an excellent choice deciding to stop by and do some math with me today.

I'm really pleased.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is checking understanding of addition and subtraction with fractions.

And that is within our unit arithmetic procedures with fractions.

By the end of this lesson, you will be able to use the mathematical structures that underpin addition and subtraction of fractions.

Most of what you see today should be familiar to you.

Here are some keywords that we'll be using in today's lesson.

Numerator, denominator, proper fraction, and LCM.

The numerator is what is written above the fraction line.

So in this example here, that would be the 2.

The denominator is what is written below the fraction line, and in this case that would be the 3.

A proper fraction is a fraction where the numerator is less than the denominator.

And LCM is an abbreviation for lowest common multiple.

We're going to split today's lesson into two separate learning cycles.

The first of which is checking our understanding of addition and subtraction, and then we'll further improve our understanding in the second learning cycle.

Let's get going with that first learning cycle.

Alex has noticed that there are two open boxes of cereal in the cupboard.

That sometimes happens in my house.

Does it happen in yours too? One box is 2/5 full and the other box is 2/3 full.

Can he fit both boxes into one? Now, what he doesn't want to do is to just tip one into the other because if it doesn't fit, he's gonna end up with a bit of a mess with the bit that's left over, so he's actually going to use his skills with maths and fractions to see whether he can fit both boxes into one.

Here's the two boxes of cereal.

The first one, we know that 2/5 of it is full.

So the denominator is gonna tell us how many equal parts to split our box into, that's 5.

And we've got 2/5, so the numerator is telling us how many parts we've got, so in this case 2.

That's how much cereal is left in the first box.

Let's look at the second box.

We know that 2/3 of this box is full, so we're going to split this one into three equal parts, and there are still 2/3 of the box.

This is how much is left in the second box.

Can we fit one into the other? Because here I've got my boxes split into different number of equal parts, it's difficult to tell.

So what we're going to do is we're going to split each box so that it has the same number of equal parts, so the first and the second box have the same number of equal parts.

So I've added here my lines.

I now have 15 equal parts in the first box and 15 equal parts in the second box.

What does that look like? Well, we know that to create an equivalent fraction we need to multiply 2/5 by 1 because multiplying by one doesn't change its value, and we know that we want the denominator to be 15.

What do I multiply 5 by to make 15? Well that's 3.

So I need to multiply by 3/3 because remember that's 1.

So I've not changed the value of 2/5, I've just changed what it looks like, that's 6/15.

Do the same with the second box, 2/3.

We know we want the denominator to be 15, so here we're going to do 3 x 5.

So we'll multiply by 5/5 'cause we've multiplied by 1, remember, which hasn't changed the value of the 2/3, and that is 10/15.

Now, what we want to do here is to see if we can fit both boxes together, so we need to add those two fractions.

6/15 + 10/15 is 16/15.

So Alex can't fit both into the same box because we can see here our fraction is greater than one.

To be able to fit it into one box, it would have to be equal to or less than 15/15.

So to add or subtract fractions, they must have the same denominator.

Aisha and Alex are working out the answer to this question.

5/6 - 3/4.

Here are their workings.

So here's Aisha's working.

She's got 5/6 - 3/4.

She's created some equivalent fractions by multiplying by 1, which gave her 20/24 - 18/24.

She's done that subtraction, so we get 2/24.

Then she simplified that fraction by taking out a common factor of two in the numerator and the denominator, so we have multiplied by one remember, we've not changed the value, so that's 1/12.

This is what Alex does.

5/6 - 3/4.

So he's again created two equivalent fractions, and then he's got those two fractions as 10/12 and 9/12, so the answer is 1/12.

Whose method do you prefer? I don't know about you, but I prefer Alex's method because it's more efficient.

So if we can, we like to choose as mathematicians the most efficient method we can.

But why? That's important, isn't it? If we're going to try and use Alex's method, the most efficient method, we need to know why it's the most efficient.

So I'm just gonna give you a moment to think about why is Alex's method more efficient? Because he has used the LCM, the lowest common multiple, of 6 and 4 to create his common denominator of 12, whereas Aisha used 24, which was not the lowest common multiple.

So as Aisha's method was correct, it got the right answer, but she had to do a little bit more work.

So if we use the lowest common multiple, we'll always use the most efficient method for adding or subtracting our fractions.

Gonna have a go now at one together on the left, and then you'll be ready to have a go at the one on the right by yourself.

We're going to look at 3/8 + 5/6.

The lowest common multiple of 8 and 6 is 24, so I create my equivalent fractions.

Remember we need to make the denominator 24, 8 x 3 is 24 and 6 x 4 is 24.

Remember that your numerator and denominator must be the same so that we're multiplying by 1 and therefore not changing the value of the original fraction.

I end up with 9/24 + 20/24, and that is 29/24.

We can then separate that 29/24 out.

We know that 1 is 24/24 and there'll be 5/24 left, so this is 1 5/24.

I'd like you now to have a go at this question by yourself.

So you're going to pause the video now, and when you've got your answer, you're gonna come back.

Remember no calculators, and if you can, try and use the LCM because it's the most efficient and it's gonna save you a bit of work.

Good luck.

I'll see you in a bit.

Well done.

Let's have a look and see if you've got this question right, but I'm sure you have.

So the lowest common multiple of 6 and 4 is 12.

Did you get that? If you've got 24, it doesn't matter.

If you've got 48, you may have a different denominator, it won't matter, but I'm going to use 12 as that is the lowest common multiple and therefore the most efficient method.

So I've created my equivalent fractions knowing that I want my denominator to be 12.

That gives me 10/12 + 9/12, which is 19/12.

I know that 12/12 is 1 and I have 7/12 left over, so the answer to this question is 1 7/12.

Did you get that right? Well done.

Now I'm gonna have a look at this.

We've got 2/7 + 2/3 and 2/9 + 2/3.

Is it necessary to calculate the exact answer? Alex says, "Of course we will have to." Is that what you thought, that we will need to work out the exact answer? And Aisha says, "No we don't." Who did you agree with? Why might Aisha say that? Well, 2/7 we know is larger than 2/9, and you are adding 2/3 in both, so therefore the one on the left hand side has to be the greater of the two because we've got the same fraction, 2/3.

In the left one, we are adding 2/7, which is greater than 2/9.

It's not always necessary to work out an exact answer if all we're doing is making the comparison as to which is greater or smaller.

What number is the arrow pointing at? How are we going to work out what number is the arrow pointing at? Well, we can see on the right hand side of my number line I've got 2/7, and we can see that the distance between the arrow and 2/7 is 3/4, so we need to calculate 2/7 - 3/4.

First thing we need to do then is find our lowest common multiple of 7 and 4, and that's 28.

We create our equivalent fractions, which you're expert at now because you've been doing that a lot recently.

We end up with 8/28 - 21/28.

When we subtract, we're going to add the additive inverse remember, so we end up with 8/28 + -21/28.

If we calculate that, 8 + -21 is -13, so we end up with -13/28.

That's what the arrow is pointing to.

Notice here we have a negative answer.

It's okay to have a negative fraction.

Let's do this one together.

Similar to the first problem, what number is the arrow pointing at? So we know that the arrow is 5/6 less than 3/8.

So this is our calculation we want to do.

Let's find the lowest common multiple of 8 and 6, which is 24.

Create our equivalent fractions, and we end up with 9/24 - 20/24.

Change that subtract to add in the additive inverse, and we end up with -11/24.

I'd like you to have a go at what the arrow is pointing at in this question.

So pause the video.

Good luck.

Come back when you're ready, and we'll check your answer.

Great work.

Let's have a look.

So the calculation we needed to do was 4/9 - 5/7, and I've decided to use the lowest common multiple remember 'cause that's most efficient.

And the lowest common multiple of 9 and 6 is 18.

This creates this pair of fractions, 8/18 - 15/18.

Let's rewrite that subtraction as an add in the additive inverse, and we end up with -17/18.

Did you get that? Well done.

If you didn't get that exact fraction, it may be that your fraction could be simplified, so just check does your answer simplify to 7/18? Okay, now you're ready to have a go at some questions independently.

In this first question, I'd like you to place the symbols greater than or less than into the following, and try to only work out the exact answer if it's not obvious which is the greater.

Think back to that question we looked at earlier with the 2/7 + 2/3 and the 2/9 + 2/3.

It was obvious which was the greater.

Good luck with this.

Pause the video now, and then come back when you're ready.

Well done.

Now let's have a look at question number two.

Question number two, so we've done some questions like this, is looking at what the question mark is.

So on the number line, given the information, I'd like you to work out, please, what each of the question marks are in each of these.

Remember here, you will need to work out your answer.

Try to use the LCM so that you are using the most efficient method, and I'd like you to give your answer in its simplest form.

So remember, if you've not used the LCM, you will need to simplify your answer, but if you've used the LCM, you will already have your answer in its simplest form.

Good luck with these, and then come back when you're ready.

Pause the video now.

Well done, and there's a third question in Task A.

Let's take a look at it.

And the third question here is working out the missing digits.

This is a lot more challenging than the questions you've done previously, but I have every confidence that you'll be able to work these out.

Good luck with this.

Come back when you're ready.

You can pause the video now.

Super work.

Well done.

Let's have a check of our answers then.

1a, we should have the symbol pointing to the left, b, to the right, and c, to the left.

Question 2, the answer to a was 1/6, b was 3/2, well done if you've simplified that to 1 1/2, c was zero, d was 5/3, which was 1 2/3, e, 1/3, and f was 4/3, which was 1 1/3.

And then the final question.

Brilliant work if you've got these right.

Like I said, it was a little bit more challenging, but I knew you were up for it.

a, the missing number was 1, b, it was 5, c was 2, and d was 9.

Well done.

We can now move on to furthering our understanding of addition and subtraction.

We're ready for that now.

We're now going to take a look at this question.

1/3 - 3/5 + 7/10.

Aisha says, "I know that 3/5 is greater than 1/3.

So if I deal with the first two fractions, I will get a negative fraction, which I'm not yet confident with yet." So Aisha's not very confident if she's gonna get a negative fraction.

Would you be confident with that? Let's see what Alex has got to say in return.

Alex says, "That's OK, add 3/5 and 7/10 and then subtract this from 1/3." Is Alex right? Can you do that? No, Alex is not right.

We need to change the subtraction of 3/5 to the additive inverse, so add -3/5, and then we can change the order using commutativity of addition.

So it is possible to change the order, but we have to make sure that we have changed the subtraction.

How else could you write 1/4 - 3/8 + 5/16? Pause the video, decide which of these are right, and then come back when you're ready.

Here, the bottom two, b and c, both of those were correct ways of rewriting the calculation we have at the top of the page.

a is incorrect.

The calculation is no longer subtracting 3/8 and then adding 5/16.

And c is the same as b, but adding a negative has been written as a subtraction.

We're now gonna look at this with some integers.

So 23 - 48 + 62.

What would be the most efficient way to answer this question? Have a think for me.

I would say reorder.

23 + 62 - 48.

We know that 23 + 62 is 85, and we know that 85 - 48 is 37.

And we can do the same with our fractions.

2/3 - 5/6 + 5/9.

So we're gonna rearrange it.

So we've got the 2/3 + 5/9 - 5/6.

Now we know how to do that.

We've just done that in the first learning cycle.

So we're going to change our fractions so that they have a common denominator.

The lowest common multiple of 3,9, and 6 is 18.

So here are my three equivalent fractions.

We know that 12 + 10 is 22, so 12/18 + 10/18 is 22/18.

Those two things are equivalent.

We're then going to add the additive inverse of 15/18, which is -15/18.

And we end up with our final answer because 22 - 15 is 7, and we are working with 18ths, so 7/18.

Would rearranging the calculation be useful here? We know 3/5 is larger than 2/10, so actually we don't need to.

We're not gonna end up with that negative fraction when we do that first part of the calculation.

We would rewrite it as 3/5 + -3/10 + 4/15.

Aisha and Alex are working on this question, so the question we've just looked at.

So we've already decided that we don't need to reorder them because 3/5 is greater than 3/10.

Aisha decides to change 3/5 and 3/10 into decimals.

Aisha knows that 3/5 is equal to 0.

6 and that 3/10 is equal to 0.

3.

So she does 0.

6 - 0.

3 + 4/15.

0.

6 - 0.

3 is 0.

3.

So she's converted that back to a fraction, which is 3/10, and then we're adding on the 4/15.

Create our equivalent fractions with a common denominator, in this case 30, and then we can add those together giving us 17/30.

Alex converts the fractions so that they have a common denominator.

So Alex has gone straight to common denominator of 30, and he gets the answer of 17/30.

Either of those ways is an appropriate way to answer this question.

Sometimes changing the fractions into decimals can be quite useful.

Let's have a look at this question together, and then you'll be ready to have a go at one independently.

Looking at 1/3 - 5/6 + 7/9.

In this step, I've just rearranged the calculation.

I've then created my equivalent fractions with a common denominator of 18, which gives me 5/18.

Here is an alternative method.

I'm going to deal firstly with the first pair of fractions, so 1/3 - 5/6.

1/3 is 2/6, and remember I'm going to change that subtraction into adding the additive inverse of 5/6, which is -5/6.

So I end up then with -3/6 + 7/9.

Now I'm going to create my equivalent fractions with a denominator of 18, and we can see that this gives the same answer.

So it doesn't matter whether we decide to rearrange or whether we decide to deal with the first pair of fractions first.

Either way is an appropriate method for solving this problem.

Its personal preference as to which you prefer.

You're ready now to have a go at this question.

2/3 - 7/8 + 5/12.

You can pause the video now.

Come back when you're ready.

Remember, no calculator please.

Great work.

Wondering whether you decided to go for the top method, the rearranging, or whether you decided to go for the bottom method dealing with the first two first.

16/24 + 10/24 - 21/24 is 5/24.

And you might have different steps, like I said, but your final answer should be the same as long as you've remembered to simplify it if you didn't use the lowest common multiple.

Now we're ready for Task B.

Great work so far.

Let's keep going.

In this task, you're going to calculate the answers to the following.

So think of the methods that we've just been through.

Remember, there are multiple methods of solving problems and it doesn't matter which you choose, but if you can, try to use the most efficient.

So pause the video, give these questions a good go, and then you can come back when you're ready.

And question two, so very similar to Task A.

The final question was more challenging because you had to find missing numbers, which I think is a little bit harder, but you've got all of the skills to be successful at this.

So pause the video now, and I'll look forward to seeing you when you come back.

Good luck.

Great work.

Let's check those answers.

1a is -1/6, b, 7/10, and c, -32/45.

And then question number two.

Our missing numbers in a was 7, b was 3, c, the first missing number was 1 and the second missing number was 5.

Well done with those.

Did you get them all right? You did? Amazing.

We're now ready to summarise our learning from today's lesson.

So it's not always necessary to calculate exact answers when adding and subtracting fractions if you're only comparing their size.

So that's the example that we looked at there.

When adding or subtracting fractions, they must have a common denominator, must have.

The most efficient way is to use the LCM.

Remember, that means lowest common multiple, and there's an example there.

When adding and subtracting multiple fractions, they can be reordered to make the calculation easier.

So sometimes reordering makes the calculation easier, but we did look at alternative methods too.

You've done fantastically well today.

There was a lot of really challenging stuff in there, so well done for sticking with me right through to the end.

Thank you, and I look forward to seeing you again.