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 are 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 explaining with and without an image how to add and subtract related non-unit fractions that bridge a whole.

So we're going to be thinking about mixed numbers here and mixed numbers that go beyond one and then numbers that start as mixed numbers and end up with less than one.

But we're going to be thinking about those related fractions.

So let's have a look at the keywords.

We've just got one key phrase today, which is common denominator.

So I'll say it, and then it'll be your turn.

Are you ready? My turn, common denominator, your turn.

Excellent.

I hope you're quite familiar with that term.

You may have be using common denominators already to make adding and subtracting fractions easier.

So let's just remind ourself what it means.

So when two or more fractions share the same denominator, you can say that they have a common denominator, and you might hear that referred to as a sort of common unit of counting.

It means that we can add and subtract fractions easily.

Without a common denominator, it's really quite difficult because we're not counting in the same units.

There are two parts to our lesson today.

In part one, we're going to be bridging across a whole.

And in part two, we're going to be working with multiple parts.

So let's make a start on part one.

And we've got Jun and Sam helping us in our lesson today.

Jun is working with his granddad to create a model train track.

Ooh, that sounds exciting.

"Granddad," he says, "uses feet and inches when he is talking about measuring length." You may have heard people talking about feet and inches.

Sometimes we refer to height in feet, don't we? If somebody's tall, we might say, "Oh yes, they're over six foot tall." I wonder if we know what that looks like in metres.

So feet were created because they were about the length of an adult foot, so it made it easy to measure.

And an inch is about the distance of that top joint of your thumb is roughly an inch, again for an adult, maybe not quite yet for you, but maybe for some of you.

And a foot is around 30 centimetres as well.

Your 30-centimeter ruler is that length because that is around one foot, which is 12 inches.

We're gonna find out more about that as we go on.

So there we go, one foot is about 30 centimetres, so the length of your ruler, and there are 12 inches in a foot.

And if you've got an old ruler, you might see that it has 30 centimetres on one side and 12 inches on the other.

So this obviously is not to scale, but this is our representation of one foot, around the same as 30 centimetres.

It's an approximation, they're not exactly the same, but one foot is exactly 12 inches.

So our bar model shows that 30 centimetres is roughly the same as a foot, but one foot is exactly the same as 12 inches.

So Jun and his granddad found two planks of wood that they need to glue together.

One plank is 3/4 of a foot long, and the other is seven inches long.

So let's represent that on our bars here.

About 3/4 of a foot and seven inches, but we need to glue them together.

So how long will the piece of wood be when it is glued together? Jun says, "We can represent this as an equation." We have 3/4 of a foot, so there's 3/4.

Our one whole is one foot on this occasion, and we also have seven inches, which is 7/12 of a foot.

There are 12 inches in a foot.

So we've got one plank that's 3/4 of a foot and one plank that's 7/12 of a foot long.

We needed to say 7/12 because 7/12 is of a foot, so our wholes are the same.

3/4 of one foot and 7/12 of one foot.

And we can add these together to find the total length of the wood.

Can we add them straight away though? No.

"The fractions need a common denominator before I can add them," says Jun.

He says, "We can see that 3/4 of a foot is equivalent to nine inches." Can you see that 3/4 comes up to nine inches along our second plank, and that can be represented as 9/12 of a foot.

There we can see it.

He says, "We can also scale the numerator and denominator by the same factor.

3/4 of a foot, we want it to be in a number of inches effectively, so an out of 12, a denominator of 12.

We can scale the denominator by three, and we scale the numerator by a factor of three, and we create an equivalent fraction of 9/12.

So 3/4 of a foot is 9/12 of a foot, nine inches.

So now we can rewrite our equation as 9/12 plus 7/12.

9/12 is equivalent to 3/4.

What do you notice? Nine plus seven is going to be greater than 12, so we're going to make an improper fraction, so this gives us 16/12.

And we can convert the improper fraction of 16/12 into a mixed number.

How many twelfths make one whole? Well, that's 12/12, and we've got four extra twelves.

So our mixed number is 1 4/12.

And did you see what happened there with our two bars? So we had nine inches effectively and seven inches.

And so three of those inches complete one whole, and then we've got four extra left over.

So our mixed number is 1 4/12.

In total, the piece of wood was glued, and it was one foot and 4/12 of a foot long, or one foot and four inches.

And we can represent this on a number line.

So our number line shows quarters and twelfths, and we can see that equivalence, can't we? We can see that there are an exact number of quarters matching an exact number of twelfths.

So we started on 3/4, or 9/12, and we added on 7/12.

So our answer was 16/12, or one foot and 4/12.

Granddad's dad found a third plank of wood, which was too long.

"Granddad says we need to cut 1/4 of a foot off the plank.

How long will it be when the piece of wood has been cut?" Again, we can represent this as an equation.

We have one foot and 2/12 of a foot.

Remember one inch was 1/12, and we've got two inches.

So we have one foot and 2/12 of a foot, 1 2/12, and we need to subtract 1/4 of a foot, so we're subtracting a quarter.

"It would help if we had a common denominator to complete this calculation." You're absolutely right, Jun.

We know we can do that because quarters and twelfths are related fractions.

So we can scale the numerator and denominator by the same factor to find a common denominator.

1/4 we can write as a number of twelfths.

We can scale the denominator by three, so we do the same to the numerator.

One times three is three, so 1/4 is equivalent to 3/12.

Now we can rewrite our equation.

1 2/12 subtract 3/12, because 3/12 is equivalent to 1/4.

Or can you see the fractional part we're subtracting is greater than the fractional part of our mixed number.

So let's have a think.

Let's convert our mixed number to an improper fraction.

One foot and 2/12 of a foot is equivalent to 14 inches or 14/12 of a foot.

12/12 plus two more, 14/12.

And we're subtracting 3/12, so that gives us 11/12 of a foot altogether.

And we could have thought of that of partitioning our 3/12 that we're subtracting into 2/12 and 1/12.

We subtract 2/12 to get to one whole, and then subtracting one more twelfth will take us to 11/12.

Time to check your understanding.

Can you write an equation to represent adding these two pieces of wood? Have a look carefully at where the shaded area of the one foot comes up to on the 12 inches scale so that you can work out what fraction that is.

Pause the video, have a go, and when you've written your equation, press play for some feedback.

How did you get on? Did you spot that the shaded parts of one foot was 1/2? So we had 1/2 plus 8/12.

Well, we know that 1/2 is equivalent to 6/12, so that was our equation, 6/12 plus 8/12.

And you might have gone on to work out that that was 14/12 or one whole and 2/12, 1 2/12 of a foot time.

Time for you to do some practise.

Can you add the following lengths together? They're not all in twelfths here.

Some of them are, but some of them aren't.

But you've got two lengths shown here.

Can you add them together? So, for example, in the first one, we can see that the top bar represents 1/2 of the whole and the bottom bar represents 4/6.

So we're going to combine 1/2 and 4/6 to see what our total is.

And in question two, you're going to do some subtraction.

So this time we've represented one area as a purple area and one area as a green area.

The circles are there to show you what the fractions are.

So in the first example we've got 1 1/4, which is purple.

And we can see that we've got 5/8, which is the green area.

And we're going to imagine subtracting 5/8 from that purple area.

Pause the video, have a go at your tasks, and when you're ready for some feedback, press play.

How did you get on? So for question one, we were adding the following lengths together.

So we had 1/2 plus 4/6.

Well, they're related fractions, aren't they? 1/2 can be expressed as 3/6, and we can sort of see that in the bars, can't we? So we can rewrite 1/2 as 3/6, 3/6 plus 4/6 is equal to 7/6.

And there we are, our common denominator, so we can add the fractions.

You may have turned your answer into a mixed number of 1 1/6.

So for two and three, we can apply that same thinking.

We had 1/4 plus 10/12, and 1/4 is equivalent to 3/12.

So 3/12 plus 10/12 is equal to 13/12.

And you might have converted your answer into a mix number of 1 1/12.

What's about the other one? What was slightly strange here? Well, that's right, that one of our fractions we hadn't put at the beginning of the bar, but that doesn't matter, does it? We also haven't filled in all the parts on the bar.

We could see that it would be 1/3, but we've got four equal parts there.

So four, eight, 12, so that will be 4/12.

So we've got 22/24 plus 4/12.

Well, we could convert that to 20/4, we could have converted it into twelfths.

We could have had 11/12 plus 4/12, but we've converted to 20/4, so we got 22/24 plus 8/24, which is equal to 30/24.

And again, we could have converted that into a mixed number as well.

In question two, we had some subtractions.

So in our first one, we can see that we've got 1 1/4 subtract 5/8.

Well, quarters and eighths are related fractions.

We can't convert the eighths into quarters this time 'cause we got an odd number of them, so we need to say 1 2/8 subtract 5/8.

And 1 2/8 is equivalent to 10/8.

10/8 subtract 5/8 is equal to 5/8.

So you might have converted your mixed number into an improper fraction to help you there, or you might have bridged through.

And for the second one, we can see we've got 1 3/5 subtract 9/10.

Well, 1 3/5 is equivalent to 1 6/10, which is like 16/10, and 16/10 subtract 9/10 is equal to 7/10.

What about the last one? Again, you can see we've shaded things in slightly different ways.

Things don't have to be together to be the same fraction.

So we've got 1 3/6, which is a bit like 1 1/12, isn't it? 1 3/6 subtract 2/3.

So we can change our 2/3 into 4/6.

And again, this time we're going to bridge back, aren't we? So we might want to think about that as 9/6 subtract 4/6, which is equal to 5/6.

Well done if you've got those right.

And on into the second part of our lesson, we're going to be working with multiple parts this time.

So how would you calculate this equation? Jun says, "I can think of a strategy to do this.

Firstly, we have different denominators, so we need to make sure they all have a common denominator." Are these related fractions? They are, aren't they? Six and three are factors of 12, 12 is a multiple of six and three.

We can use 12 as a common denominator.

What would 5/6 be as twelfths? Well, we're going to scale up the numerator and denominator by a factor of two, which creates a equivalent fraction of 10/12.

So we can replace 1 5/6 with 1 10/12.

So what about 1/3? We know we've got our whole, but we're just converting the fraction, aren't we? What's 1/3 expressed as a number of twelfths? Well, we've got to scale everything up by a factor of four.

So 1/3 is equivalent to 4/12, so we can replace 1 1/3 with 1 4/12.

And now we can complete our calculation.

He says, "I'm going to subtract 2/12 first from the 1 4/12." So 1 4/12 subtract 2/12 is 1 2/12, and now we can add them together.

So we've got 1 10/12 plus 1 2/12, which is 2 12/12, And that's equivalent to three, isn't it? But Sam says she had a different strategy for this.

Can you spot what Sam might have seen? She says, "I decided to use six as a common denominator." So she converted everything to sixths.

Well, hang on a minute.

Does that work? It does, doesn't it? 12 is a multiple of six, and we've got an even number of twelfths.

So 1/3 expressed as sixth is going to be 2/6, so we can replace 1 1/3 as 1 2/6.

What about these 2/12? Ah, she says, "I can scale 2/12 down by the same factor to make sixth." So 2/12, we can scale the numerator and denominator down by a factor of two to create 1/6, so 2/12 can be replaced by 1/6.

Again, we can now calculate.

She says she's going to add the wholes first, so there's one two wholes.

And then 5/6 plus 2/6 minus 1/6 is equal to 6/6.

And we know that 6/6 is equal to another whole, so we have our three wholes in total.

So we don't always need to go for the largest denominator.

Sometimes if we look carefully at the fractions, we can use a denominator that's smaller than the largest denominator and still create equivalent fractions.

Always worth checking out for.

So time to check your understanding.

True or false? The best common denominator would be quarters for this.

Is it true, is it false, and why? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, it's false, isn't it, in this case? Why is it false? We've got two, four, and eight as our denominators.

They're all related, so we could perhaps convert eighths into quarters, but what is stopping us? Ah, that's right.

You can sometimes convert eighths into quarters, but only if you've got an even numbered numerator.

We've got 3/8, and you would not have a whole number for the numerator here.

So we can't use quarters on this occasion, we'd have to use eighths.

So knowing that, can you now calculate 1 2/4 minus 1 1/2 plus 1 3/8? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? So we needed to use eighths, didn't we? 1 2/4 is the same as 1 4/8.

1/2 is equivalent to 4/8, 2/4 is equivalent to 1/2, isn't it? So they were both a half.

And then we're adding 1 3/8.

So 1 4/8 subtract 4/8 is equal to one, plus another 1 3/8 gives us an answer of 2 3/8.

Well done if you got that right.

And time for you to do some more practise.

So in question one, in each equation circle the denominators which could be used as a common denominator.

So we're not solving these at the moment, just circle the denominator that could be used as a common denominator.

And remember to think, do you always have to scale up or could you scale down? And in question two, you're then going to solve each equation using your chosen common denominator.

And in question three, you're going to write three different equations with a different common denominator for each equation.

And which conversion did you prefer? Pause the video, have a go at questions one, two, and three, and when you're ready for the answers and some feedback, press play.

How did you get on? So in question one, you were just circling the common denominator.

Well, for any of them, it could have been the largest denominator, but could we have been a bit smarter and gone for something different? So in question one, we had 4/5, subtract 2/3, add 2/15.

Well, in this case, fifths and thirds are not related, but they are both related to fifteenths, so we did need to have 15.

The only common multiple that would leave a numerator as a whole number each time would be fifteenths.

We couldn't scale the fifteenths down because we've got an even number this time and we'd have had to have divided by five and by three, so that wouldn't have given us a whole number of fifteenths.

What about the next one? 3/4 subtract 2/3 add 11/12.

Again, the only common multiple here is 12, and actually 11's a prime number, so we wouldn't have been able to scale down 11/12.

11/12 is in its simplest form.

What about the next one? Well, this time we could have converted each fraction into a number of ninths or eighteenths.

16/18, we could have simplified to 8/9, and 1 2/3, we could have created a fraction equivalent with ninths as the denominator.

So we could have used eighteenths or ninths.

What about the last one? We could have created fractions with 21 as the denominator, or three.

We'd have scaled down by a factor of seven if we'd created thirds from our 21st, and that would've worked.

We could have had 2/3 again.

2/3 and 14/21 are equivalent fractions, so it's always worth checking.

You don't always have to go for the largest denominator of your related fractions.

So now we're going to solve these equations.

So we had to use fifteenths for this first one.

So we've got 12/15 minus 10/15 plus 2/15 is equal to 4/15.

For the next one, again, we had to use twelfths, didn't we? Because of that prime number, the twelfths were in their simplest form.

So 9/12 plus 8/12 minus 11/12 is equal to 6/12.

Nine plus eight is 17, subtract 11 is equal to six.

This time though, we've gone for ninths.

So 16/18 is equal to 8/9, subtract 7/9, add 1 6/9.

So 8/9 subtract 7/9 is 1/9, so we're adding on one more ninth, so 1 7/9.

And for this final one, we could convert to thirds.

So one and 1/3 plus 2/3 minus 2/3, we've added 2/3 and then subtracted 2/3.

So our 1 1/3 has actually not been changed, so our equation is equal to 1 1/3.

So here we could actually use tenths, fifths, or twentieths as our common denominator.

So as tenths, we'd have 1 6/10 subtract 1 4/10 plus 6/10.

As fifths, we'd have 1 3/5, subtract 1 2/5 plus 3/5.

And as twentieths, we'd have 1 12/20 subtract 1 8/20 add 12/20.

So which conversion did you prefer? Well, I think I agree with Sam.

I preferred converting to fifths as it meant I calculated with smaller numbers.

And you might also have looked, our answers are all equivalent fractions, 8/10, 4/5, and 16/20.

4/5 is in its simplest form.

Four and five only share a common factor of one, so we can't write that fraction with any smaller values.

So by using fifths, the smallest of the common denominators, we actually end up with our answer in its simplest form as well.

I hope you enjoyed experimenting with different common denominators.

Always worth checking.

You don't always have to go for the largest value.

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

We've been explaining with and without images, how to add and subtract related non-unit fractions bridging a whole.

So what have we learned about? Well, we've learned again that fractions need to have a common denominator if you're to add and subtract them.

When adding and subtracting fractions with a common denominator, you can just add and subtract the numerators.

We know that we are now counting in the same unit.

And when the sum of the numerator is greater than the denominator, you can convert the improper fraction to a mixed number.

We've also explored a little bit about what we call imperial measures with feet and inches.

I hope you've enjoyed the lesson, and thank you for all your hard work and your mathematical thinking, and I hope I get to work again with you soon.

Bye bye.