Lesson video

Hiya, my name's Ms. Lambel.

I'm really pleased that you've decided to join me today to do some maths.

I'm really looking forward to working alongside you.

Welcome to today's lesson.

The title of today's lesson is Deepening Understanding with Fractions of Amounts.

This is within the unit Understanding Multiplicative Relationships with Fractions and Ratios.

By the end of this lesson, you'll be able to find the original amount when given a fraction and the result.

A keyword that we'll be using in today's lesson is reciprocal.

Here's a quick reminder of what the definition of reciprocal is.

Reciprocal is the multiplicative inverse of any non-zero number, any non-zero number multiplied by its reciprocal is equal to one.

Today's lesson I've split into three separate learning cycles for us.

In the first one, we'll use a bar model to help us solve these problems. Remember, we are going to be finding the original amount.

We'll then move on to looking at using double number lines to do the same thing, and then we'll look at using division.

So lots of different ways of tackling very, very similar problems today.

Let's get started on that first one, bar models to find the original amount.

Here we go.

Alex gives three eighths of his sweets to his brother.

That's very kind of Alex, isn't it? Would you give your sweets to your brother? His brother gets 15 sweets.

How many sweets did Alex have to begin with? We don't know how many sweets were in the bag to begin with.

You may be tempted here to draw a bar model out and find three eighths of 15, but this time we don't know how many sweets were in the bag, we don't know the whole, but we can still represent this problem with a bar model.

Here's my whole bag, that's the whole entire bag of sweets.

We got three eighths, Alex gives three eighths to his brother.

So those are the ones that he's given to his brother, and we know that he's given his brother 15 sweets, so we know that that part of our bar model must be equal to 15.

We need to take the 15 and we need to divide into three equal parts, which gives us five.

Remember, the bar has been split into equal parts.

That means that each part is the same size, so each part must have the same value, so all of the parts must be worth five.

Now we can work out, remember I said the whole bar, was how many sweets were in the bag to begin with? Eight multiplied by five is 40.

So originally Alex had 40 sweets.

And here we could check if we find three eighths of 40, we should find that we get 15 and if we checked it, we'd find out that we did.

Let's look at this one now.

The special offer box contains 600 grammes.

How heavy is the box usually? So we've got a special offer here, one fifth extra free.

You often see that on the boxes of things, don't you? Or on packets, you're getting extra free.

And here we're getting one fifth extra free.

So here's my box of cereal and here's my extra fifth.

So there's my normal size and there's my extra fifth.

This represents the special offer box, 600 grammes.

So we know that 600 grammes is equal to one and one fifth of the usual weight.

I'm just going to spin that bar model round just to make it a little bit easier to see what's happening.

We know that this new box is 600 grammes.

We need to look at how many parts 600 grammes is, which is six.

So we take the 600 and we divide it by six, meaning that each part is worth 100 grammes.

We want to know how heavy the box is usually.

So that was just the first box.

If we think about that image on the previous one, it was the original box, which is five multiplied by 100, which is 500.

The box is usually 500 grammes.

Izzy is saving for a new computer game.

When Izzy has saved enough money to buy the game, she notices that the price has increased by two fifths to 49 pounds.

Her mum gives her the extra money to buy the game.

How much had Izzy saved? What would a bar model look like to represent that situation? Let's have a look.

The original cost, and her mum gave her two fifths.

So the original cost of the computer game was our whole, and then her mum gives her an extra two fifths because that's how much the price is increased by.

Because the second bar is split into fifths, we must make sure that the first bar is also split into fifths.

This is how much the computer game cost in total.

That was the new price, which was 49 pounds.

How many parts are equal to the 49 pounds? Count them up, one, two, three, four, five, six, seven.

So we need to take the 49 divide by seven.

We're gonna put seven in each part.

We've got the original price, remember it was just that hole, and so therefore it's seven multiplied by five, which is 35.

So the original price was 35 pounds and again, you could check, you could find two fifths of 35 and then add it on.

Two fifths of 35 is 14, 35 add 14 is 49.

So we know our answer is definitely right.

Izzy saved 35 pounds, well done, Izzy.

Alex's cat goes to the vet once a year.

The vet tells Alex that his cat has lost two sevenths of its mass.

His cat now has a mass of 4.

5 kilogrammes.

What was the mass of Alex's cat last year? Let's look at the bar model to solve this problem.

Here's our bar and that's the original mass.

That's how much Alex's cat weighed originally, so how much he weighed last year.

We're splitting it into sevenths because he's lost two sevenths of his mass.

There's the two sevenths.

So this is how much Alex's cat now weighs, the new mass, and we know that the new mass is 4.

5 kilogrammes.

4.

5 kilogrammes, we can see there, is over five parts.

So we're going to take the 4.

5 divided by five, which gives us 0.

9, but 0.

9 in all of the boxes, remember, because each of the boxes represents an equal part.

Now we can find the original mass.

We can see that we've got seven lots of 0.

9, so 0.

9 multiplied by seven is 6.

3 kilogrammes.

Alex's cat had a mass of 6.

3 kilogrammes last year.

A sunflower increases its height in one month by one sixth of its original height.

It now measures 1.

4 metres.

How tall was it at the beginning of the month? Who has drawn the correct bar model? Pause the video, have a think and come back when you're ready.

What did you decide? Was it Alex's bar model or Izzy's bar model? Alex has drawn the correct bar model.

We can see that it's grown by one sixth, so we need to add on the extra sixth and Izzy's shows a decrease of one sixth.

Now we're ready to have a go at one final question together and then you'll be ready to have a go at one independently.

You buy a sweatshirt in this sale, you pay 30 pounds.

How much was it originally? Now the sale is one third off.

We didn't pay for the purple part, so we only paid 30 pounds, which was the two parts.

So 30 divided by two is 15.

We now know that each part is equal to 15 pounds and originally we would've paid the full price, so the entire bar, that would be three multiplied by 15 pounds, which is 45 pounds.

Here's your one to have a go at.

You buy a sweatshirt in the sale, so still the same sale, one third off, and you pay 50 pounds.

How much was it originally? Good luck with this and I'll see you when you come back.

You can pause the video now.

Well done, let's take a look and see how you got on.

Here's my bar, remember, we are not paying for that final third.

So the 50 pounds is above the two boxes here or two parts, 50 divided by two is 25.

If we'd paid full price, we would be doing the whole entire bar, which is three multiplied by 25 pounds, which is 75 pounds.

Did you get that right? Yes, you did, well done.

Alex and Izzy are trying to decide if the following is true without working out the answers.

Two fifths of 150 equals one fifth of 75.

What do you think? It says, "I think it's true.

"The fraction of the amount have both halved." And Izzy says, "It must be false.

"Both the fraction and the amount are smaller, "so the answer must be smaller." Who do you agree with? Two fifths of 150, and one fifth of 75.

Notice my bottom bar I've drawn half the size of the 150 because it needs to represent 75 rather than 150.

So we can see from the bar models that that is false.

Let's take a look at this question.

Four fifths of 80 equals two fifths of something.

So we've got four fifths of 80.

Here's my bar representing four fifths of 80.

Using Alex's logic from the previous slide, we've half the number of fifths we're finding, so we half the amount, that's what that bar model looks like.

So my bar is half the size and I've got two fifths.

We can clearly see that those two purple sections are not equal in size.

What about if I double the length of the bar? We can now see that they are the same size.

So Alex says, "Because we halved the fraction, "we then had to double the amount." And Alex is right, isn't he? Four fifths of 250 equals one fifth of something.

Alex says, "I think that here we have to divide the fraction "by four, so we must have to multiply the amount by four." And Izzy says, "Yes, I think you are right, Alex." 250, four fifths, 250, there's my bar to represent that.

So Alex is suggesting that we multiply the amount by four and we find one fifth of that.

Let's just check then that that bar is four lots of 250, so two lots, three lots, four lots.

We can see that the bar is exactly the same length.

So yes, Alex was right, because we had divided the fraction we were finding by four, we needed to multiply the amount by four, which was four multiplied by 250, which was 1,000.

Let's give this one a go.

One six of 240 equals five sixths of something.

What have we done with the fraction? We've multiplied the fraction by five, therefore we need to divide, we need to do the inverse.

We're gonna divide the amount by five, which is 48.

You give this one a go.

This time let's look at what's happened to the fraction.

We've divided by three, therefore we need to do the inverse.

We need to multiply the amount by three, so it was 45.

Well done if you've got that.

Now you can have a go at this question.

So I'd like you please to match each problem to the correct bar model and when you've done that, you're then going to solve the problem, pause the video, good luck, come back when you are ready.

Well done, now we can check those answers.

You can pause the video, mark your answers, and then come back when you're ready.

How did you get on, all of it right? Fantastic, now we can move on to our second learning cycle.

So we're going to be looking at using double number lines.

I think of a number, four ninths of my number is 28.

What is my number? Here's double number line.

We know that four ninths is equal to 28.

So on the top line I'm putting my fraction, on the bottom line, I'm putting the number.

So four ninths is equal to 28.

We want to find the original number.

So that's the whole, so this is what we're trying to find.

What is the multiplicative relationship between four ninths and one? We know that this is the reciprocal of four ninths, which is nine quarters, so we're gonna multiply by nine over four, nine quarters.

We know that our double number line shows a multiplicative relationship, so if we multiply the fraction by nine quarters, we need to multiply the number also by nine quarters, giving us 63, I was thinking of 63.

Remember you could check your answer here.

You could find four ninths of 63 and check that we get 28.

We can also use double number lines to find the original amount.

We're going to use Alex's cat example.

Here is the double number line.

So on the top we have the fraction, and on the bottom we have the mass in kilogramme.

What fraction of the cat's original mass is it now? The whole bar represents the mass of Alex's cat last year.

The two boxes that are shaded represent how much Alex's cat has lost.

We can see from the bar model that the cat is now five sevenths of the original mass.

We know that five sevenths is equal to how much Alex's cat weighs now, which is 4.

5 kilogrammes.

Remember we're trying to find the original amount, which is the whole, I'm looking for that multiplicative relationship between five sevenths and one, which is multiplying by the reciprocal of five sevenths.

So that's multiplied by seven over five.

I do the same to the mass and I get 6.

3, Alex's cat was 6.

3 kilogrammes.

And if you remember from earlier, that agrees with it.

So we can use a double number line in that situation as well as a bar model.

Alex has been challenging himself each week to complete as many times table questions as he can in two minutes.

Over the last year he's improved his score by three quarters.

Wow, that's a huge improvement, well done, Alex.

His score is now 140.

What was his score at the beginning of the year? My double number line, I've got my fraction on the top this time, we've got the score.

What was his score? We know that his score has improved by three quarters.

So we've got his original score and it's improved by three quarters, so that's one in three quarters and we know that that's equal to seven quarters.

We also know that seven quarters is equal to his current score of 140.

We need to find out his original, which is one, What is my multiplicative relationship between seven quarters and one? We multiply by the reciprocal of seven quarters, which is four sevenths.

We do the same to the score, we get a score of 80.

So Alex, at the beginning of the year, was doing 80 in two minutes.

He's now doing 140.

And again, you could check, find three quarters of 80, which is 60, 80 add 60 is 140.

So all of these questions you could do that double check yourself and be confident you've got the answers right.

I'd like you to look at this check for understanding now.

There is a mistake.

I'd like you please to spot it, pause the video, find the mistake, and then come back when you're ready.

Where was that mistake, what did you find? Here, the arrow was pointing in the wrong direction.

We were working from 23 20ths to one.

Then you'll notice that actually the bottom arrow is incorrect as well.

Well, the arrow's not incorrect, it's what's written against the arrow.

It should say multiply by 20 over 23 and that gives us 400.

Just looking at the answer of 529, how could you tell that it was wrong? The price of the laptop has increased, therefore the original price must have been lower and 529 was not lower than 460.

You can always do a quick sensibility check to make sure whether your answer looks like it's right.

Are you ready to have a go at some questions independently? So just like the first task where I'd given you some bar models, here I've given you some double member lines and I'd like you please to match it to the problem and then solve the problem.

Normally I say no calculators, you're probably fed up with me saying that, but here I'm gonna allow you to use a calculator.

Good luck, you can pause the video now and I'll be here when you get back.

Great, let's check those answers, pause the video and then come back when you've marked your answers.

And finally, let's move on to that final learning cycle, finding the original amount using division.

What is three fifths of 60? Remember the word of and the process of multiplying are the same in maths.

We could rewrite this as three fifths multiplied by 60.

And from this we can come up with some other facts.

36 divided by 60 is three fifths.

36 divided by three fifths is 60, three fifths multiplied by something is 96.

96 divided by something is three fifths and 96 divided by three fifths is something.

Which of those is most useful for finding the value of the cloud? And it's this one because we are finding the value of the cloud.

The cloud is the result of the calculation that we are doing.

Laura notices that some of her crayons are broken.

Two sevenths of the crayons are broken.

How many crayons are in the box? And Laura says that four of her crayons are broken.

So we know that two sevenths of the crayons are broken.

So two sevenths of the crayons.

We don't know how many crayons there are in the box in total, so I've just represented that with the letter C.

But we know that that is equal to the four crayons.

Rearranging that, we end up with C equals four divided by two sevenths.

Remember the inverse of multiplying by two sevenths is to divide by two sevenths.

And we know how to do this from previous lessons.

We multiply by the reciprocal of the divisor.

The reciprocal of two sevenths is seven over two.

Four multiplied by seven over two is 28 over two, which is 14.

Remember here, you could have simplified before you did that multiplication.

Jun did a sponsored walk.

He gave 150 pounds to his favourite charity and this was three fifths of the total that he raised.

How much money did Jun raise in total? We know that three fifths of the total amount of money, which I've just called t here, is equal to 150.

To solve this, we'll rearrange it into t equals 150 divided by three fifths because the inverse of multiply by three fifths is divide it by three fifths, which is 150 pounds multiplied by the reciprocal of three fifths, so five over three.

Here I've decided to simplify.

So I've done 150 divided by three first, which is 50 and then multiplied by five.

Jun raised 250 pounds in total.

And if you work out three fifths of 250, you can double check that that gives you 150 and be confident you've got your answer right.

The weight of a special offer box is 900 grammes.

What is the weight of the normal box? So the special offer is one fifth extra free.

This means we are getting the original box, which is one plus an extra fifth free.

So we've got one and one fifth multiplied by the weight of the normal box, and that is 900.

Rearrange that, we end up with w equals 900 divided by six fifths, which is 750 grammes.

The normal box weighs 750 grammes.

Here the special offer is one third extra free.

A special offer bag of sweets weighs 320 grammes.

How heavy is the bag usually? One and one third multiplied by w equals 320 grammes.

So the special offer gets an extra third free.

So we're adding it onto the original of one, one and one third.

We're going to do 320 divided by.

And all I've done there is written one and one third as it's improper equivalent, four thirds, and then we can work that out.

Now it's your turn, give this one a go.

A special offer bag of sweets weighs 200 grammes.

How heavy is the bag usually? So same special offer, one third extra free.

Pause a video, come back when you're ready and I'm gonna let you use a calculator for this one.

Well done, let's check your answer.

One and one third multiplied by w is 200 grammes.

200 divided by four over three equals 150 grammes.

Did you get that right? You did, fantastic, well done.

Now you are ready for the final task of today's lesson.

Well done for sticking with me, we are nearly there.

Pause the video, give these questions a go.

I'll allow you to use a calculator, but only if you show me all of your steps, you write all of your working out down, whatever you've typed into the calculator, get that down on your page.

So pause the video, good luck, come back when you're ready.

Well done, check your answers now.

Pause the video, check those answers and then come back when you're ready.

Great work, we can now summarise the learning that we've done for today's lesson.

So in all of the different learning cycles, we were finding the original amount and we just looked at three different methods for finding the original amount.

We started off looking at bar models, then we used the double number line and then we used that division method.

All of those methods are valid.

You may have a preference for one of them, but you may not.

Being able to interchange between them might be quite useful.

Thank you so much for joining me today.

I've had a fantastic time and I hope to see you again very, very soon, goodbye.