Lesson video

Hello, my name's Mrs. Hopper, and I'm really looking forward to working with you in this lesson from our unit on addition and subtraction of fractions.

Now, fractions are one of those things that some people absolutely love, and I hope you're in that camp.

I love fractions.

I think they're really clever ways of expressing numbers and parts of wholes.

So if you're ready to make a start, let's do some work on adding and subtracting fractions.

So in this lesson, we're going to be solving problems involving adding and simplifying fractions with different denominators.

You may have added fractions with the same denominators in the past, and that's quite simple because we're adding the same unit, but we're going to look at how we can add fractions when they're not the same denominator.

What do we have to think about? We've got one key phrase I think we'd have to call this, two words together today is our keywords today, and that's common denominator.

So I'll take my turn and then you can take your turn.

Are you ready? My turn.

Common denominator.

Your turn.

I'm sure you know what those two words mean on their own, but I wonder if you've ever used them together.

Let's have a look at what that means.

So when two or more fractions share the same denominator, you can say that they have a common denominator and we know that when the denominators are the same, we can add and subtract fractions quite easily.

So we're going to start to think about how we can create fractions with common denominators.

So there are two parts to our lesson today.

In the first part, we're going to be representing contexts as equations, and then in the second part we're going to be finding a fraction of a whole.

So let's make a start with part one.

And we've got Aisha and Alex helping us with our learning in the lesson today.

We're going to think about gardens and allotments.

So you might have things growing in your garden, and you might know of a place nearby where you go past and you see what looks like lots of gardens together, and they're called allotments and some people have an allotment and they can grow vegetables or flowers.

So we are going to think about a vegetable patch.

So when people grow vegetables, they often section their patch into different parts so they know what's growing where.

So we can have a look here at a vegetable patch that's got carrots and onions growing in it.

And that's all at the moment in this patch, carrots and onions.

So our question is, what fraction of the vegetable patch is being used to grow carrots and onions? Aisha says, "It's hard to tell because the whole isn't divided into equal parts." Can you see that? We've got one bigger part with the carrots in, and then six smaller parts and one of those parts has got onions in.

Alex says, "I think I can sort that!" Can you visualise what Alex is going to do to divide the vegetable patch into equal parts? Ah, there we go.

Is that what you thought? So our questions changed slightly.

What fraction of the vegetable patch is being used to grow carrots? Can you think? Alex says, "What do you notice now?" Aisha says, "You've divided the larger part into the same size parts as the others." So now each carrot effectively is in the same size part as the rest of vegetable patch was divided into.

Aisha says, "There are now three equal parts of vegetables which have carrots in." And how many in the whole? It's nine, isn't it? "So," Aisha says, "three-ninths of the vegetable patch has carrots growing in it." What about the onions? What fraction is being used for onions? And Alex says, "It's also easier to see what fraction of the patch is use to grow onions," because we've now got equal parts across the whole patch.

"Yes," says Aisha, "the whole is divided into nine equal parts and one part is being used to grow onions." "So one-ninth of the vegetable patch has onions in it." So how much of the vegetable patch is being used altogether? Well, Alex says, "Three-ninths is being used to grow carrots," so we can record that as a fraction.

"And one-ninth is being used to grow onions." So we're going to add on the one-ninth.

So to find how much of the vegetable patch is being used altogether, we need to add those two fractions together.

And we know that because they are both ninths, three-ninths plus one-ninth is equal to four-ninths.

Three of something plus one of something is equal to four of them.

So four-ninths of the vegetable patch is being used.

And you could see that in the picture as well.

Alex says, "I think we can write this equation like this as well." Ah, 1/3 + 1/9 = 4/9.

Why does he think that? And Aisha says, "Why do you think that?" Can you think why Alex thinks this is a good way to record the equation? He says, "Well, we know that the onions take up one-ninth of the whole patch," so there's our one-ninth.

What about the carrots? What did we do to the bit with the carrots in it? Ah, he says, "If we divide the whole differently, we can see that the carrots take up one-third of the patch." Can you see there are three columns of equal size and in one column we've got carrots? So one-third of the vegetable patch is used for carrots.

There we go.

"If we divide it differently, we can see easily that the carrots are one-third." Now, we can't see so easily that the onions are one-ninth.

Aisha says, "So we can record the problem in two different ways." 3/9 + 1/9 = 4/9 and 1/3 + 1/9 = 4/9.

And you can represent this on a number line.

Our number line is from 0 to 1, or 9/9, divided into ninths.

So we had three-ninths for the carrots and one-ninth for the onions recorded with that equation.

And Alex says, "Here you can see the three-ninths and the one-ninths being added." And we know that three-ninths is equivalent to one-third.

There's a common factor of three there.

We can divide the numerator and denominator by three.

Three-ninths is equal to one-third.

So we can also record this as 1/3 + 1/9 = 4/9.

And you can see that one-third and three-ninths are equivalent jumps on the number line.

Aisha says, "I recorded it like this on the number line." 1/9 + 3/9.

Well, that is equal to 4/9 as well, isn't it? And Alex says, "Whilst mathematically correct," there's nothing wrong with that, "I don't think it is as easy to see the one-third now on the number line." I think he's right.

There is one-third there between one-ninth and four-ninths.

We've got three-ninths which is equal to one-third, but it's not as easy to see.

It's much easier to see the one-third as that first jump of three-ninths, isn't it? So we can see that it is the same, but we sort of miss out the easy way of looking at three-ninths being equal to one-third.

Time to check your understanding.

The neighbours next door also have a vegetable patch.

How much of the patch is being used for potatoes, and how much for lettuces? Have a look at what you can see in the way the patch is being divided up.

Can you continue that to make it easier to see? Pause the video, have a go, and when you're ready for the answer and some feedback, press play.

What did you think? Ah, did you add those extra lines so that we can now see that the whole is divided into 12 equal parts? And we can see that four-twelfths of the patch was used for potatoes and two-twelfths of the patch was used for lettuces.

Time to continue the check.

How much of the patch was used altogether? Pause the video, have a go.

When you're ready for the answer and some feedback, press play.

How did you get on? So we know that four-twelfths of the patch is used for potatoes and two-twelfths are cabbages, and 4/12 + 2/12 = 6/12 4 + 2 = 6.

But you could also have thought about what it looked like before we put those extra divisions in.

The potatoes took up one out of three equal rows, so one-third.

So we can also say that 1/3 + 2/12 = 6/12.

But the way Aisha has recorded it are fractions have common denominators.

They're both twelfths, so that makes it easier to add.

The way Alex has recorded it, we need to think of one-third as an equivalent fraction in twelfths in order for us to be able to add them easily.

Time for you to do some practise.

What fraction of the whole is used altogether? And can you write two equations to represent each image for the animals at the zoo, the electrical items in a shop, and the objects in the park.

Remember you can add extra lines to your drawing to create those equal parts that might make it easier to write one of the two equations.

Pause the video, have a go, and when you are ready for some feedback, press play.

How did you get on? So what fraction of the whole is used altogether? Let's look at the animals in the zoo.

Well, we can see that there are four equal columns, aren't there? And the giraffe is in one of them.

So we can represent the giraffe as one-quarter, but if we imagine that quarter divided up, we'd have 12 equal parts in the whole and the snake is in one of those parts.

So we can represent this as 1/4 + 1/12.

Or if we put those extra lines in, the giraffe would be in three-twelfth of the zoo.

So 3/12 + 1/12.

But in both of those we can see that 4/12 of the image is being used by animals at the zoo.

What about the electrical items in the shop? Well, again, if we look, we've got one row that is being used by laptops and that's one outta four rows, that's one-quarter.

But then we've got five divisions in each of those quarters.

So we've got five lots of four, we've got 20 parts in the whole if we divided it up.

And so the mobile phones are just in one-twentieth.

So where's our six-twentieths come from? Well, if we imagined our laptop zone being broken up, we'd have five-twentieths there.

So we'd have 5/20 + 1/20 = 6/20.

And what about the park? What can we see? Gosh, we've got a lot of extra divisions to do there.

The squirrels bit might help us.

We've got two divisions there.

So our mushroomy toadstooly bit would be two more divisions and our tree would be four, so two, four, six, so that will be eight parts altogether.

So without making those extra divisions, we can see that one-half of the park has got trees in it, one-quarter has got mushrooms in it, and one-eighth has got squirrels in it.

But we could convert all of those into eighths.

Four-eighths would have trees, two-eighths would have mushrooms, and the one-eighths with squirrels.

So seven-eighths of the park is covered by those objects at the moment.

Well done, and I hope you added in those extra lines to help you to see those fractions more clearly.

And on into the second part of our lesson, we're going to be finding a fraction of a whole.

So what fraction of the whole is shaded? It looks a bit like the garden and the zoo, but this time we haven't got any things in it.

It's just shaded in.

So can you think about what fraction of the whole is shaded? What do you notice? Well, Aisha says, "We can represent this as one-quarter plus one-sixteenth." Can you see where the quarter is and where the sixteenth is? Well, the pink shaded area at the top right, the larger shaded area represents one-quarter of the whole square.

And if we imagine those divisions carried on across it, there would be sixteen parts in the whole and the blue would be one of those sixteenths, so 1/4 + 1/16.

Alex says, "But it would be difficult to add if this is the case because we do not have a common denominator." We've got one-quarter and one-sixteenth.

Aisha says, "So we will need to divide the whole into equal parts." Alex says, "I can do that!" Aha, so he's divided up our one-quarter, and we've now got four-sixteenths again.

So now the whole is divided into sixteen equal parts.

The red section is four parts of the whole, so this is four-sixteenths.

And the blue part is still that one-sixteenth.

So now we've got common denominators, and we can add the numerator, 4/16 + 1/16.

Four of something plus one of something is equal to five.

So 4/16 + 1/16 = 5/16.

What fraction of the whole is shaded this time do you think? Aisha says, "I don't think we have equal parts this time." Hmm, do you agree with Aisha? What do you think? Alex says, "I think it's easier than you think it is." He says, "We know the top right part can be divided into four equal parts." So there we are.

"And the bottom right section is also composed of four equal parts." They don't look the same as the four equal smaller squares, but it is four equal parts, and it is the same size as the square at the top that we've just divided up.

Aisha says, "I don't think we have equal parts this time, although the red section is still one quarter." But Alex says no, "The bottom right section is also composed of four equal parts.

And we know that equal parts can look different." The area of each of these triangles would be the same as the area of each of one of these squares.

So we can still say that this is divided into four equal parts.

Ah, Aisha says, "So the bottom right section is once again equivalent to four-sixteenths." Plus the additional one-sixteenth, which is equal to five-sixteenths.

So it was the same fraction of the shape that was shaded before.

We just had to recognise that those four red triangles were actually equivalent to the four red smaller squares.

And we can represent this on a number line as well.

We had four-sixteenths plus one-sixteenth, which is equal to five-sixteenths, and we can represent that within an equation.

But we also saw that the four-sixteenths was equivalent to one-quarter.

And one-quarter plus one-sixteenth is also equal to five-sixteenths.

And there's the equation to represent that way of thinking about our fractions.

Alex says, by identifying a common denominator, it makes our fractions easier to add.

We know that one-quarter is equivalent to four-sixteenths, so we can replace the one-quarter in our equation with four-sixteenths and then we have fractions with common denominators, which are easier to add.

Time to check your understanding.

How many equal parts could the whole be divided into? So just thinking about how we would add extra lines to divide that whole into equal parts.

Pause the video, have a think, and when you're ready for some feedback, press play.

How did you get on? What did you think? Well, Aisha says, "The red section is equivalent to six-twentieths of the whole." So six of those parts would fit into the red section.

There we go, we can add those extra lines.

And what about the blue section? How many parts? Oh, and the blue section is four parts, four-twentieths.

And when we add all those, it's easy to see that there are 20 parts in our whole, and the red part is six-twentieths and the blue part is four-twentieths.

Another check.

Can you shade in the correct amount of the shape to match the equation and write the equation with a common denominator? So at the moment our equation says 1/3 + 2/12 = 6/12.

We've only got one part shaded.

So what's the other part that needs to be shaded? And how could we rewrite the equation with a common denominator? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Did you remember that even though we've cut the shape into triangles, each of those triangles has the same area as one of the little squares? So we've still divided into four of those same equal parts.

Oh, but it's a bit more difficult to think about it now.

We've got three in each column, but how many of those columns would there be? I think there'd be four of them, wouldn't there? So maybe we are looking at twelfths here.

Let's have a think.

So now we've divided the shape into equal parts that look the same.

So our red part still got four parts to it, but we've created the squares rather than the triangles.

So we can see that the red section is equivalent to one-third of the whole.

If you imagine those four squares as one of our rows, that would be one outta three equal parts, so one-third, and the blue section is two-twelfths of the whole.

So there were lots of different ways you could have shaded those two blue squares.

I wonder where you put them.

What about the common denominator though? Well now we can see that our one-third that was read is four of those twelfths.

So we can rewrite the equation as 4/12 + 2/12 = 6/12.

We know that one-third is equivalent to four-twelfths.

We can scale the numerator and denominator up by a factor of four and create four-twelfths.

And we can also see it in the diagram there.

And time for you to do some practise now.

In questions one and two, you've got the same shapes to look at.

Can you label in part one each part as a fraction of a whole and in two each part as a fraction with a common denominator? And then for question three, the same shapes are back again.

But this time we've shaded some of the areas.

So can you write two equations for each image to represent the fraction that has been shaded? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? So for questions one and two, you have the same shapes and you are labelling each part as a fraction of the whole.

And then with common denominators, so let's look at the shapes.

So in this first shape, if we imagine it divided up into equal, each of those smallest parts will be worth one-twentieth.

So then we can see that our two sort of horizontal bars at the top are worth three-twentieths each.

And our square in the bottom right is worth one-fifth of the shape.

We can imagine four of those fitting in.

And then a bar down the sort of left hand side I'm imagining it one column, which would also be four of those small squares, which is what that's worth.

It's one-fifth of the shape.

For the next one, each of our smallest divisions is one-twelfth, and we know that we can divide that square at the top into four parts, and we could represent that as four triangles or as four squares.

They're each one-twelfth.

And so we can see on the right hand side, that's one of four equal columns.

So that must be one-quarter of the shape.

For the final one, again, I can this time see four rows.

So I've got one-quarter and then I've got two rows where that quarter is divided into two equal parts.

So they must be an eighth and they must be an eighth.

So those triangles and rectangles, although they look very different, represent the same fraction of the whole.

And then I've got four equal parts in my bottom row, so those were sixteenths.

So now we're going to think about those common denominators.

So we've got lots of common denominators in the first shape.

So we can put in our one-twentieth and our three-twentieth but then we can see that one-fifth is equivalent to four-twentieths.

So I can relabel it as four-twentieths.

Again, we've got lots of twelfths in that middle shape, but I can see that that column on the right hand side is worth three of those twelfths.

And then on the final shape, our quarter will be worth four-sixteenths and each of our eighths will be worth two-sixteenths.

And then we've got our four-sixteenths at the bottom.

So we've got them labelled all with the same denominator in each shape.

And then we've got the shapes back again.

But this time we're asked to work out what the shaded area is.

And by using the work we've done in questions one and two, we can represent that in two ways.

So the first one could be 1/5 + 3/20 + 1/20, One-fifth was our square, three-twentieth was our rectangle, and then one-twentieth.

And that is eight-twentieths, let's think of it in twentieths to help us to see that.

3/20 + 1/20 + 4/20, which was our square, and that gives us our 8/20.

What about the middle one? Well, we've got one 1/12 + 1/12 + 1/4.

We could rewrite that as 1/12 + 1/12 + 3/12, which gives us our 5/12 in total.

And for the final shape, we had 1/8 + 1/8 + 1/16.

But we can rewrite those 1/8 as 2/16.

So 2/16 + 2/16 + 1/16 gives us our 5/16.

I hope you had fun thinking about the different ways that those shapes had been divided and reasoning about what your common denominator would be and how that related to your other fractions.

And we've come to the end of our lesson.

We've been explaining how to add related unit fractions with a representation or an image.

If you noticed all of those fractions we'd looked at when they weren't with a common denominator were unit fractions.

So what have we learned about today? Well, we've learned that in order to add fractions of a whole, context or shapes need to be divided into equal parts.

When you divide shapes into equal parts, you are finding a common denominator for the value of each part.

And once fractions are of a common denomination, so the fractions have common denominators, they can be easily added by adding the numerator together, because we know now that we're adding units of the same value, so we just need to think about the numerator.

Thank you for all your hard work and your mathematical thinking in this lesson, and I hope I get to work with you again soon.

Bye-Bye.