Lesson video

Hello there, my name is Mr. Tilstone.

I hope you're having a great day, filled with smiles and success.

Let's see if we can make it even better with a great maths lesson.

So if you're ready for the challenge, let's begin.

The outcome of today's is this.

I can subtract a mixed number from a mixed number, explaining which strategy is the most efficient.

You might've had some recent experience of subtracting a proper fraction from a mixed number.

Our keywords today are, my turn, mixed number, you turn.

And my turn, improper fraction.

Your turn.

What do those words mean? Let's have a reminder.

A mixed number is a whole number and a fraction combined.

So for example, this is one and a half.

That's how we say it, that's how we write it.

Can you think of a different example of a mixed number? And an improper fraction is a fraction with a numerator, that top number is greater than or equal to the denominator, that's the bottom number.

So let's give an example.

This is five thirds and this is nine eighths.

They're both improper fractions.

Can you think of a different improper fraction? Our lesson today is split into two parts, two cycles.

The first will be strategies for subtracting mixed numbers and the second will be problem solving.

So if you're ready, let's begin by looking at strategies for subtracting mixed numbers.

In this lesson you're going to meet Andeep and Izzy.

Have you met them before? They're here today to give us a helping hand with the maths.

Andeep is trying to solve this calculation.

Two and one thirds subtract one and two thirds.

Now, what do you notice about those numbers? What kinds of numbers are there? They're both fractions, they're a particular kind of fraction.

They're mixed numbers.

So mixed numbers have got a whole number part and a fractional part, and that's true of both of those.

Now, which strategy would you use to calculate this? How would you do that? What ideas have you got? Well, one strategy is to use a number line and count back.

And they're nice and easy to draw.

So our blank number line, we're going to put two and one thirds on the right-hand side of it and then count back.

Now, first we subtract the whole number, because that mixed number has a whole number part and a fractional part.

So let's take away or subtract the whole number part.

So subtract one and that leaves us with one and one third.

What could we do now? Then we subtract the fractional part, partitioning as needed.

So we're going to take away, subtract those two thirds and we can do that by subtracting one third, which would take us back to the whole number, that's one and then another one third, which would take us to two thirds.

So the answer is two thirds.

"There is another strategy that we could use," says Andeep.

Okay, well let's see.

We could convert the mixed numbers to improper fractions.

Oh yes, you might've had some very recent experience of doing that.

So first we multiply the whole number by the denominator and then add the numerator.

So in this case, two and one third, if we multiply two by three, that tells us how many thirds we've got in the whole number, the two.

So that's six thirds and then the extra one third is equal to seven thirds.

So that's seven thirds subtract something.

And then one multiplied by three.

That's the whole number multiplied by the denominator, plus two, that's the numerator, is equal to five.

So that's five thirds.

So that becomes two and one thirds subtract one and two thirds, that's seven thirds subtract five thirds.

That seems easier, doesn't it? The denominators are the same, so we can subtract the numerators.

Keep the denominators the same.

And that gives us two thirds.

Izzy says, "I think there's a third strategy, which is more efficient for these numbers." Mm, what do you notice about those numbers? Can you think of a third strategy? Well, what do you notice about the whole and the part that we are subtracting? The numbers are quite close to each other, yes.

Let's represent this on a number line.

So we can see two and one third and one and two thirds on the number line.

We're finding the difference and when finding the difference, we can count forwards or backwards.

So in this case, we can also count forwards as well.

So to find the difference between those two numbers, adding one third will take us to two and adding another one third will take us to two and one third.

So that means that's two thirds.

That's the difference between those numbers.

So if the numbers are close together, we can find the difference by counting on.

Okay, well let's have a little check point.

Calculate by finding the difference.

I'll do one and then it's your turn.

So four and two 10ths subtract three and seven 10ths, the denominators are the same.

Well, we could partition it.

We could use a number line.

And we can count on, because those numbers are quite close together.

So three and seven 10ths, if we add three 10ths, so if we count on another three 10ths, that takes us to four and then if we add on another two 10ths, that takes us to four and two 10ths, so the difference between those numbers is five 10ths.

Okay, let's see if you can do that with three and one fifth subtract two and three fifths.

And again, those numbers are fairly close together, so see if you can use counting on.

Pause the video.

How did you get on? Let's have a look.

Well, did you partition them using a part-part-whole model like that? Did you then make a number line with two and three fifths on the left-hand side and three and one fifth on the right-hand side? They're pretty close together.

So we're going to count on from two and three fifths.

Another two fifths will take us to three and another one fifth will take us to three and one fifth.

Add those two fractions together and we've got three fifths.

That's the difference between those two numbers.

Very well done if you got that.

You're on track and ready for the next part of the learning.

And the next part of the learning is a practise.

Number one, calculate these.

You should check your answer using a different strategy.

Number two, Andeep says that he cannot calculate this, because he cannot subtract three quarters from one quarter.

Do you agree? Give reasons for your answer.

And number three, complete the missing numbers in each box.

Have a good think about that, that's a bit trickier.

If your teacher's happy for you to do so, I always recommend working with a partner or in a small group and that way you can bounce ideas off of each other.

Okay, pause the video and away you go.

Welcome back, how did you get on? Let's give you some answers.

So number one, calculate these.

So four and three sevenths subtract three and four sevenths.

Both mixed numbers.

Did you notice they're quite close together? Well, we can use counting on.

So let's do that, let's use a number line.

Let's find that difference.

So if we add three sevenths onto the four sevenths, that takes us to four and then there's a difference of three sevenths between four and four and three sevenths.

Add those two three sevenths together and it gives us six sevenths.

That's definitely the most efficient way when they're close together like that.

You might have checked your answer there using a different method, but we're looking for six sevenths.

You may have checked it by converting to an improper fraction.

So instead of having four and three sevenths, maybe we could have four multiplied by seven plus three, which is 31, so that's 31 sevenths and instead of three and four sevenths, three multiplied by seven plus four which is equal to 25, so that's 25 sevenths.

31 sevenths subtract 25 sevenths gives us six sevenths.

That's probably not as efficient, but it was a good way to check.

And you may have noticed that the other fractions were quite close together too, so that find the difference strategy really is useful, that counting on strategy.

So B is one and four eighths, C is five sevenths and D is one and five ninths.

And Andeep says that he cannot calculate this, because he cannot subtract three quarters from one quarter.

Do you agree? Hopefully you didn't.

Hopefully you gave a different strategy.

The numbers are quite close together, aren't they? So we could use a number line and find the difference between them.

You could use counting on.

So we could write a number line just like this.

A quick one, five and three quarters and six and one quarter and then the difference could be used by counting on.

So one more quarter will take us to six and one quarter takes from six to the six and one quarter.

Add those two one quarters together and we got two quarters.

And complete the missing numbers in each box.

Nice challenge here, you might've started by reasoning that the missing denominator must be eight, because the unit that we are working with is eighths.

You can see that eight appear twice.

So it must be eight there.

You might then have reasoned that the missing whole number part must be three, because the difference is not a mixed number and does not have a whole part.

So that's got to be three and that leaves one final missing part.

You might then have solved the equation using the find the difference strategy and determined that the missing number is in fact five.

So adding on three eighths takes us to three and another two eights takes us from three, add those together and we've got five eighths.

Well done if you said that.

You're doing very, very well and you're ready for the next part of the lesson which is problem solving.

Izzy has three and seventh 10ths metres of rope.

She cuts a one and nine 10ths metres of length off to use when den building.

That sounds like fun.

What might the question be? Mm, so we've got three and seven 10ths, what kind of number is that? It's a mixed number.

And one and nine 10ths, what kind of number is that? That's a mixed number.

The question is, what length of rope is left? Well done if you predicted that.

Let's represent it as a bar model.

So our whole length, the top part of the bar model is three and seven 10ths of a metre.

And the known is one and nine 10ths, so a metre, so we can add a part in one and nine 10ths.

We're looking to find the difference.

What could you say about those numbers? To find the unknown part, we subtract the known part from the whole.

So this is three and seven 10ths subtract one and nine 10ths, that's the equation.

We now need to use our fraction sense superpower to decide which strategy to use.

And I noticed those numbers are quite close together.

Did you notice that too? So we can use the find the difference strategy, that counting on strategy.

So let's sketch our number line.

One and nine 10ths on the left, three and seven 10ths on the right and we're finding the difference.

Well, one more 10th will take us to the whole number, that's two.

Now, we can't go straight to three and seven 10ths, because there's another whole number in between, so we can jump to that.

So between two and three, that's one.

And then we need to add seven more 10ths on to get from three to seven 10ths.

What happens when we add those parts together? We get one and eight 10ths.

We should check our answer using a different strategy.

How about we turn those mixed numbers into improper fractions? Let's do it.

So we multiply the whole number by the denominator and then add the numerator.

Three multiplied by 10 plus seven is equal to 37.

That's 37 10ths and then one multiplied by 10 plus nine is equal to 19, so that's 19 10ths.

The denominators are the same, so we can subtract the numerators and that leaves us with 18 10ths.

We need to convert that back though into a mixed number.

One full group of 10 10ths can be made from 18 10ths and then there will be eight more 10ths.

That's one and eight 10ths.

So yes, we checked and it came up again.

Our answer is the same as when we used the find the difference strategy, so we must be correct.

Always worth checking if you got time.

There is one and eight 10th metres of rope left.

Let's do a little check.

Complete the missing boxes and answer the question.

Andeep has a box with some books in it which has a mass of three and four 10th kilogrammes.

A mixed number.

After removing a book, the box has a mass of two and seven 10th kilogrammes.

A mixed number.

What is the mass of the removed book? So that's three (blank) 10ths subtract two and seven (blank) is equal to (blank) 10ths.

And we've got a number line for you to help.

Okay, pause the video and away you go.

How did you get on? Well, we do have some known information.

We can see the four 10ths there.

That's three and four 10ths.

We know we're working with 10ths, so we can fill that 10 in there and we can start to work out the missing values.

So the difference between two and seven 10ths and three is three 10ths.

And the difference between three and three and four 10ths is four 10ths.

Add those together and we've got seven 10ths.

Well done if you got that.

The mass of the removed book is seven 10ths of a kilogramme.

Well, you're ready I think, for the final practise.

Let's see how you get on with these.

Number one, solve these problems. You should check your answers using a different strategy.

So A, Andeep collects three and one fifths litres of rainwater in his bucket.

He uses one and two fifths litres of water to water some plants.

How much water is left in the bucket? And B, Izzy runs four and two 10th kilometres on Saturday.

On Sunday, she runs one and six 10th metres less than this.

How far does she run on Sunday? You may wish to use bar models to represent those before you solve them.

You may also, if your teacher allows it, wish to work with a partner and compare strategies.

Righty-oh, pause the video and away you go.

Welcome back, how did you get on? Are you feeling confident? What strategies did you use there? Let's have a look.

So Andeep collects three and one fifth litres of rainwater in his bucket and he uses one and two fifth litres to water some plants.

How much water is left? Well, we could represent that using a bar model just like this.

You may notice, they are quite close together, so we could use a number line, and we could count on.

From one and two fifths, another three fifths will take us to two, another one will take us to three and another one fifth will take us to three and one fifths.

Add them together and we've got one and four fifth.

There is one and four fifth litres of water left in the bucket.

You might have checked your answer using a different method.

So how about we do this? Three and one fifths, subtract one, gives us two and one fifth, subtract one fifth, gives us two, subtract that other one fifth, we partitioned that two fifths into two one fifths, gives us one and four fifths.

So that's one and four fifths.

That's a different way to look at it.

The counting back method gave the same answer, so we are very likely to be correct.

If you get the same answer twice, you've probably got it.

There is one and four fifth litres of water left in the bucket.

And then B, Izzy runs four and two 10th kilometres on Saturday, on Sunday, she runs one and six 10th metres less than this.

How far does she run on Sunday? This is what that would look like in a bar model.

So we're finding the difference.

We could use a number line just like this.

And if we count on, another four 10ths will take us to two.

Another two will take us to four and another two 10ths will take us to four and two 10ths.

Add all of those together and it gives us two and six 10ths.

On Sunday, Izzy runs two and six 10ths metres.

And you might've checked your answer using a different method.

How about turning them all into improper fractions? So that's four multiplied by 10 plus two is equals to 42, that's 42 10ths, subtract, well one multiplied by 10 plus six equals 16.

So that's 16 10ths.

So 42 10ths subtract 16 10ths is equals to 26 10ths and then if we convert that back, 10, 20, that's two wholes and then six 10ths left over.

Two and six 10ths.

Both methods have the same answer, so we're very likely to be correct.

On Sunday, Izzy runs two and six 10th metres.

We've come to the end of the lesson.

You've been amazing.

Today, we've been subtracting a mixed number from a mixed number, explaining which strategy is the most efficient.

Mixed numbers can be subtracted from mixed numbers and to do so, we need to use our fraction sense to determine the most appropriate strategy to use.

Mixed numbers can be subtracted by counting back using a number line or by converting them to improper fractions.

If the mixed numbers are close to each other, it's often more efficient to find the difference by counting forwards on a number line from the part to the whole.

And number lines are very efficient, you can draw them really quickly.

Well done on your accomplishments and achievements today.

I think you've been amazing and you definitely deserve to give yourself a little pat on the back.

Well done, you.

I hope I get the chance to spend another maths lesson with you at some point in the near future, but until then, have a fantastic day.

Whatever you got in store, be the best version of you that you could possibly be.

Take care and goodbye.