Lesson video

Hello everyone.

My name is Mr. Southall, and I'm going to be teaching you today about adding and subtracting fractions with the same denominator.

Now, before we begin, please try and make sure that you are in a quiet space away from any distractions, such as making sure that your mobile phone is on silent.

You will need a pencil and paper.

And when you've gathered those things, please pause this video and then we will begin.

So today we're going to be looking at adding and subtracting fractions with the same denominator.

And we're going to be looking at a couple of different phases in this lesson.

We'll start by looking at adding and subtracting fractions.

Then we'll look at fact families that we can use with fractions.

Then we'll look at crossing barriers where we will extend our knowledge of adding and subtracting fractions into things like mixed numbers.

And then we will do a quiz at the end of the lesson.

And as usual, you will need a pencil and paper for this lesson to take notes and carry out the exercises that we'll look at.

To access this lesson, we'll need to look at some key words associated with fractions.

So here they are.

We have fraction, denominator, numerator, proper fraction, improper fraction, mixed number, equivalent fraction, vinculum, and simplify a fraction.

So we'll take these in turn and have a look at what they mean.

Okay, so here we have a fraction on the right hand side, and that is 3/4.

And a fraction just means parts of a whole.

So in this example, the part is three and the whole is four, three equal parts out of four equal parts.

The numerator is the number of parts in the whole.

So that refers to the three in this example.

The denominator is the number of parts the whole is split into, in this case, it's the four; and the vinculum is the line between the numerator and the denominator.

So we can highlight these like this.

Here we go.

And the vinculum is this line in between.

Similarly, we have a proper fraction, and that's when the numerator is less than the denominator.

In this example, three is less than five.

We have an improper fraction, where numerator is greater than the denominator.

In this example, we have seven, which is greater than two.

And an improper fraction means that, it's going to be greater than or equal to one.

So this fraction is greater than one whole.

And we have a mixed number, where we write the fraction as a whole and a fraction.

In this case, we have two wholes, the number two, and the fraction is 3/5.

We also have these terms, equivalent fraction.

And that means that fractions are represented in two different ways.

So the numbers might appear different, but the value of the fraction is the same.

So in this example, we have 3/4 is equivalent, It's an equivalent fraction to 6/8, which is also an equivalent fraction to 9/12.

So all of those fractions are the same amount.

They just look different, and you can see that to using this fraction wall here on the left-hand side.

You can see that all three of those rows in blue are worth the same, but one represents 3/4, one represents 6/8, and one represents 9/12.

And we can see that here.

This is a 1/4, and a 1/4, and a 1/4.

We have one more quarter here, and these are all eighths.

Like this.

We got two there, but they're not part of this fraction.

We have 6/8 highlighted in blue.

And then finally, we have twelfths at the bottom.

And underneath that we have, simplify a fraction, which is to reduce the numerator and denominator.

So this is essentially working backwards with our example here.

We could simplify 9/12 to 3/4.

We're not changing the value of the fraction, we're just putting it in its simplest term.

Okay, so let's have a little revision of some fractions.

Here we have four different representations of fractions.

These are all pictorial.

And I want you to pause this video and see if you can identify what fraction is being represented in each picture.

So pause the video and when you are ready, press Play to continue.

Okay, So if we have a look at these, first of all, we have this example here, and this is a glass of which is half full or half empty, depending on how you're feeling.

Second of all, we have this example of pizza, and you can see that we're looking at this one here, which is one slice out of one, two, three, four, five, six slices there, okay? Each slice is equal, so this is 1/6, 1/6 of pizza.

In the third example, we have a kind of bar model here, and we have three equal pieces; one, two, three, and two of them are highlighted, so we can call this 2/3.

And the final example, we have a kind of pie chart and each of the slices of pie are equal.

So this is very similar to the pizza model.

And we have one, two, three, four, five, six, seven, eight, nine, 10 equal pieces.

So if you're looking at the white piece, that's one out of 10.

And if you were looking at the blue shaded pieces, that would be nine out of 10, 9/10.

So our fractions are 1/2, 1/6, 2/3, 1/10, or 9/10, if you were looking at the shaded part of the circle.

Now, here we have two different bar models.

And the length of the blue bar in both examples is worth one, so that's one whole.

What I want you to do is, in a moment, pause the video and have a think about how much, what fraction of the blue bar the yellow part represents.

So we're trying to decide what the length of the yellow part is in the left diagram.

And we can use the information about the blue bar to help us.

That blue bar is worth one unit.

And similarly now on the right-hand side, we have the same blue bar, which is worth one unit, and we're trying to decide what is the length of the red part.

Okay, so pause the video and when you're ready to continue, press Play.

Now, if we have a look at this diagram, we can see here that we have three equal yellow parts.

Okay? And we have two white parts.

Now they're all the same size, they're all equal pieces.

So we can represent the three, that's three equal pieces out of five equal pieces.

Okay, so 3/5 of the blue bar is yellow.

The yellow part is the same as 3/5 of the blue part.

If we look at the diagram on the right-hand side, we have one piece here and we have a further three pieces here and they're all equal size.

So we can say that this is one out of four equal pieces are red.

Okay? So 1/4 of the blue bar is represented by the red piece.

Okay.

Let's have a look at this next example.

What addition calculations do these two models represent? So if we look at the model on the left-hand side, we have five equal segments and two equal segments.

And if we look at the model on the right hand side, we have two equal segments and five equal segments.

So if we were adding these together, how could we write this calculation? Okay, have a think, pause the video and press Play when you're ready to continue.

Welcome back.

So if we have a look at this first example here on the left-hand side, you can see that we have 5/9 and we can add 2/9.

Okay? So this represents 5/9 plus 2/9.

How many ninths have we got altogether? Well, we can count them up.

One, two, three, four, five, six, seven.

So we have 7/9 all together.

Okay? If we look at the model on the right-hand side, we have 2/9 to start off with, and then three, four, five, six, seven.

So again, we have 7/9.

Okay? So both of these additions have the same sum.

The one on the left-hand side starts with 5/9 and we add 2/9 to it.

And this one on the right hand side starts with 2/9 and we add 5/9 to it, but we get to the same answer, and the answer is 7/9.

Now, using the same example, we can notice here that we have some fact families.

So pause the video for a moment and just have a read through of those fact families and see if you can understand where they're coming from, and then we'll explain them in a moment.

Pause the video, and then press Play when you're ready to continue.

Okay, so if we have a look then at these examples on the right hand side, then you can see that the first example says, 5/9 plus 2/9 is 7/9.

And if we're looking at our bar model, what we're saying here is this much here plus this much here equals this total.

Now the second fact is essentially working the other way around.

So we're adding the 2/9 on the right-hand side.

We're starting with the red, then we're adding the yellow, and we're still getting that same total, 7/9.

So even though the addition is the other way around, the answer remains the same.

Now, a little bit trickier, there's this third one, it says 7/9 subtract 2/9 is 5/9.

So here from the diagram, we're starting here, and we're saying if the total is 7/9 to start off with, and then I take away these two red ones, then I'm left with 5/9.

And then finally, we're just doing the same idea as that, but working the other way around, starting with my 7/9.

This time I'm going to subtract the 5/9 and I'm left with these two at the end.

So 7/9 subtract 5/9 is 2/9.

That's my final fact family.

Now, your turn to have a think about it.

So this time we have a different bar model, and I want you to have a think about what this could represent as an addition, as an addition in a different way, so working right to left; and then as a subtraction, and see if you can find a fact family, just as we did in the previous example.

Pause the video and press Play when you're ready to continue.

Welcome back.

So in this example, this is our whole here and here we have one, two, three, four, five, six, equal pieces.

So our way of doing this is saying, okay, this is 1/6 and this is 5/6, okay? Because we've got six equal pieces altogether, the red one represents one of them and there are five of the yellow ones.

So our fact family could be 1/6 plus 5/6 equals one whole.

That could be fact number one.

Similarly, we could work the other way around and we could say 5/6 plus 1/6 is also a whole.

And if we wanted to do a subtraction, one subtract 5/6 is 1/6, or one subtract 1/6 is 5/6.

Okay, Those are our facts.

1/6 Plus 5/6 is one.

5/6 plus 1/6 is also one.

One subtract 5/6 is 1/6, and one subtract 1/6 is 5/6.

Now we're going to develop that learning with some tasks.

Here we go.

So here we have a pizza that's equally divided into eight slices.

Bella eats three slices and Lara eats two slices.

That's important information for this question.

What fraction of the pizza has been eaten and what fraction is left? Now take note it says, has been eaten.

So it's a total of how much Bella and Lara has eaten, how much they have both eaten together.

Okay, pause the video when you're ready and try and work out this calculation and then press Play when you are ready to continue.

Okay, so let's have a look then.

We can see here that this pizza has been divided into one, two, three, four, five, six, seven, eight equal slices.

Bella ate three of these slices.

So we could say that's these three here.

Those have been eaten, they're gone.

And Lara eats two slices.

That could be these two here.

Okay, so all of this has been eaten.

So what's left? Well, we still have.

We finished there, started there.

That means that we've got these three slices left, slice six, slice seven, and slice eight.

So if we write that as a fraction, that's three different pieces of pizza that are left.

So we need to make sure that our denominator is eight.

We could represent this as a bar model like this.

Here you can see that we have a total here, a total here representing the full pizza.

And we have Bella's representation of 3/8 and Lara's representation of 2/8 of the pizza.

And we can see here, using the squares in the background of the diagram, that we have three left.

Okay, so there's 3/8 of the pizza left.

That's just another way of representing this problem.

Okay, let's try a slightly trickier one.

Bella eats five slices of a pizza and Lara eats seven slices of a pizza.

In this example, we're using two full pizzas, but again, they're still divided by the same amount.

So each pizza is eight slices.

Bella eats five, Lara eats seven slices.

How much pizza have they eaten in total and how much is left? So pause the video, try and work these out and see if you can do it in a couple of different ways.

So there's a couple of prompts here.

You can use a number line if you want to, a bar model if you want to, or you can just use writing fractions and doing the calculation that way.

Pause the video and press Play when you're ready to continue.

Welcome back.

So here we have a couple of different representations or the beginnings of representations to help us calculate.

So if we take that first one, the number line, okay.

The number line is going to go from zero all the way to 16.

Now, where did 16 come from? Well, there's eight slices here and there's eight slices here, so we have a total of 16 slices of pizza.

Bella eats five slices, so she gets to here, okay? Lara is seven, so we need to count on seven.

One, two, three, four, five, six, seven.

So the seven slices between these two red lines.

Okay, so how many slices are left? Well, it's this much.

Okay? One, two, three, four slices left.

Okay? How many did they eat in total? Well, we started with the five slices, then the seven slices.

And where did we finish? We finished at 12.

So 12 have been eaten in total, four are left.

If you want to represent that as a bar model, this is a pizza one, and this is pizza two.

Each one has eight slices.

So you can see that there's eight segments there and another eight segments there.

And we know Bella ate five, so there's five yellow.

I've just chosen yellow.

And Lara at seven, I've chosen red to represent Lara's slices.

And again, if I want to work out how much they've eaten, I count the coloured segments.

And how many left? I count the white segments.

One, two, three, four.

Okay, same answer, just a slightly different way of calculating it.

If you want to do this as fraction work, just using abstract fractions by just writing numbers and numerators, vinculums, and denominators, then you could do it like this example at the bottom, I'm starting with 16/8.

That represents 16 slices of pizza.

That's going to give me two wholes because I've got two whole pizzas, and then I'm going to take away Lara's pizza.

Oh, sorry, that's a Bella's pizza.

I'm going to take away Lara's pizza, and then that's going to leave me with 16 take away five, take away seven for my numerator, which is four.

And my denominator stays the same, which is eight; 4/8.

So all three methods give me the same answer, just different ways of representing it.

Okay.

Next tasks.

One, two, three, four written questions this time.

I want you to read through them carefully and see if you can work them out with your pencil and paper.

Take your time, and then pause this video.

Do your calculations, and then press Play when you are ready to continue.

Okay, so we have four written questions here.

Let's work through them together now.

If I take this first example, if I ate 1/3 of a bag of chips, what do I have left? Well, if we just keep this in the.

We'll just write it as fractions.

So I start with one whole bag and I'm subtracting 1/3 of the bag, okay? Now you can use different models if you want to, but just to save time, I'm going to work through it this way.

That's going to give me 2/3 as the answer.

Now, if you want to work that through a different way, you could see one as 3/3.

That might make it simpler to see where that 2/3, as an answer, has come from.

Second one.

What is 1/4 of a bag of sweets add 3/4 of a bag of sweets? Well, we'll assume that they're the same size bag.

We have 1/ plus 3/4, and that's going to give us 4/4, which is one whole bag.

The third one.

What is 3/4 of an hour add another 3/4 of an hour? Well, 3/4 plus 3/4 is going to give me the same denominator and six as the numerator.

So this is 6/4.

Now we can simplify this.

We can turn this into one whole and 2/4, which is the same as 1/2.

Now, what are our units? Well, we've been working in hours, so this is 1 1/2 hours.

Final question.

What is 6/8 of a pizza subtract 4/8 of a pizza? 6/8 of a pizza subtract 4/8 of a pizza.

Well, the denominator made stays the same.

The numerator is going to change.

Six takeaway four is two.

I can simplify that to 1/4.

What are my units? Pizzas; so I have 1/4 of a pizza left.

Now, here we have some different representations of fraction operations.

We have some sort of pies to add together.

We have fractions written in abstract form, so with a numerator and denominator; and we have bar models to help us.

So there are one, two, three, four, five, six, seven questions here working the left column first and then the right column.

Have a go at these.

Pause the video and press Play when you are ready to continue.

Okay, so if we look at this first example, we have two different pies.

So each pie is worth a whole.

And so the first pie is one, two, three, four, five out of eight.

And the second pie is three out of eight.

So our answer is eight out of eight, so it's going to be one whole.

So that just equals one.

Okay? Second example, we have one whole already and we have 3/8, three out of eight; one, two, three, four, five, six, seven, eight, if you're wondering where that came from.

So our answer is one whole and 3/8.

That'd be a mixed number.

The third example, 4/5 takeaway 3/5 is 1/5.

Next example: 2 1/3 take away 2/3 is going to be 1 2/3.

Okay, if you count on from that number in thirds, you'll get back to 2 1/3.

Let's look at our bar models, 5/8 plus 1/8.

Well, this is already divided into one, two, three, four, five, six, seven, eight.

So I've got one, two, three, four, five plus one.

It means I've got 6/8 Again, the second model is already divided into eight: one, two, three, four, five, six, seven, eight.

I've got seven here to start off with, by taking away this time.

So I'm going to go back three spaces and that's going to give me these ones.

So that's 4/8, which is 1/2.

And the last one, I started with one whole.

This is already divided into six.

So these are sixths.

I'm going to take away; this is my whole here.

I'm going to take away 5/6, which leaves me here, 'cause I've taken away these five, and that leaves me with 1/6.

Okay.

Now, I'd like you to try your independent task.

So here's your independent task.

We've got a few ideas here for you based on a fraction wall, which is the picture at the bottom.

So using that picture, what calculations can you show? For example, can you show a fraction, a calculation that gives you an answer of a fraction above one? A calculation that gives you an addition of three fractions and the answer is less than one? A fraction equivalent to a half as your answer? A calculation that gives you two fractions with a difference of 3/8? That'll be a subtraction problem.

And finally, a calculation with an answer above two? Have a go at these ideas and pause the video and we'll run through them when you press Play, when you are ready to continue.

Okay, let's have a look at them.

So if we write on here, you can see that to get a fraction above one, my answer needs to be in this area here or beyond.

So as an example, I could do one, two, three, four, five; I could start with maybe 6/8 and I could add 5/8, okay? And that's going to take me one, two, three, four, five, six; one, two, three, four, five.

That's going to take me here, which is above one.

This is 1 3/8.

Add three fractions with an answer less than one.

So this time I want to stay this side.

So it could be that I add 2/8 plus 1/8 plus 3/8.

That would, as an example, that keeps me below one.

Something that gives me a fraction equivalent to 1/2.

Well, 1/2 is 4/8.

So I could have 3/8 plus 1/8.

That would give me a 1/2.

There are other ways; you could do 2/8 plus 2/8 or 1/8 plus 3/8; it's okay.

If you want to do this subtraction, you could have 7/8 take away 3/8, and so on.

Two fractions with a difference of 3/8.

So as an example, you could have 1/8 and 4/8.

And if you subtract one from the other, you'll get 3/8 as your answer.

Anything where you subtract one from the other and get 3/8, your answer should work.

And finally, a calculation with an answer above two.

Well, we could start here and say, okay, we've got 1 7/8.

That's where we are at this point.

And we could just add 1/8.

That will take us past two.

That would be 2 1/8 as an answer.

There are lots of different ways to do these.

So don't worry if your example wasn't the same as mine.

That brings us to the end of today's lesson.

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