Lesson video

Hiya, my name's Ms. Lambell.

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, expressing one number as a fraction of another number, and that falls in our unit of understanding multiplicative relationships with fractions and ratios.

By the end of this lesson, you will be able to express one number as a fraction of another.

Some key words that we will be using in today's lesson.

So it's worth a quick recap as to their definitions.

And those are, reciprocal and terminating decimal.

The reciprocal is the multiplicative inverse of any non-zero number.

Any non-zero number multiplied by its reciprocal is equal to 1.

A terminating decimal is one that has a finite number of digits after the decimal point.

We've got some examples there, and non-examples.

If we look at the non-examples, we can see that those decimals contain the recurring dot to show that those digits recur.

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

In the first one, we will look at finding the multiplier efficiently, and in the second one we'll concentrate at writing one number as a fraction of another.

Let's get going on that very first learning cycle, finding the multiplier efficiently.

Here we go.

Lucas and Sam are trying to find the missing numbers in the following.

3 multiplied by something is 6.

3 multiplied by something is 39.

3 multiplied by something is 81.

3 multiplied by something is 4.

5.

Would you be able to fill in the missing boxes? I'm gonna pause a moment, and let you have a think.

Lucas says, "I can do the first two, but then I get a little bit stuck." So 2 in the first one, and 13 in the next one.

But he gets a little bit stuck when he gets to 3 multiplied by something is 81.

Sam reminds Lucas that you can use the inverse of multiplication.

Ah right, Lucas has remembered, "Yeah, of course.

I can do 81 divided by 3 for the third one." Yeah, well remembered Lucas.

What about this one? 3 multiplied by something equals 10.

So Lucas and Sam are having a go at working out this one.

So Lucas has now remembered.

We can do the inverse of multiply by 3, which is divide by 3.

10 divided by 3 is 3.

3.

That's what Lucas says.

He says, "If 10 divided by 3 is 3.

3, then doesn't that mean that 3.

3 multiplied by 3 has to equal 10?" And Sam says, "It should, but 3.

3 multiplied by 3 is 9.

9." Yeah, that's right, isn't it? 3.

3, multiplied by 3.

That definitely is 9.

9.

Here's the calculator display.

This is what Lucas has on his calculator display.

"Ah," he says, "I should have written 3." And I'm not gonna try and read all of those threes out, not 3.

3.

So he spotted that actually there were more threes on the screen.

Sam says he'll check that to make sure that it does give 10.

So 3.

33333, however many threes there are, multiplied by 3 is 9.

99, and a few more nines.

Lucas says, "So that's still not right." Well it isn't is it? Because we wanted to find what we multiply 3 by to make 10, not almost 10.

Let's take a look at what they typed into the calculator.

Here we can see what they've typed into the calculator.

Now the screen at the top, they've decided to change their answer into a decimal.

But here we can see that actually the calculator gives the answer firstly as a fraction, and Lucas should have used the fraction as that is the exact value of 10 divided by 3.

It's really important that if we end up with a recurring decimal that we use its fractional equivalent, because this is the most exact value of it.

So like I just said, we should use the exact value in our calculations where appropriate, and that's the majority of the time to be honest with you.

So let's just take a look at these.

3 multiplied by something is 16.

So I would do 16 divided by 3, and my fraction would be 16 over 3.

3 multiplied by something is 4/7.

So I do 4/7 divided by 3, and that's 4 over 21.

And the final one, I'd do 7/12 divided by 3, and that would give me 7 over 36.

So we are using the exact fractional value, not a rounded decimal equivalent.

Now we're ready for you to have a go at this quick check for understanding.

So you'll need your calculator.

You need to decide what should be in each of the boxes.

Answers are on the right hand side, you just need to match 'em up.

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

Well done.

Let's have a look and see if you managed to match those up properly.

I'm confident you did.

First one is 3/4.

The second one is 5/3.

The third one, 1 3/4.

The fourth one, 1 1/3.

The fifth one was 3 over 2, or you may have written there 1.

5, or you may have written 1 1/2.

And then the final one is 2/3.

Just quickly have a think.

Which of those could have been written as terminating decimals? Remember terminating means it's got a finite number of digits after the decimal point.

And those would be 3/4, 3 over 2, and 1 3/4.

It's absolutely fine if it is a terminating decimal to use the decimal value rather than the fractional, but I tend to sit with a fractional in all cases so that I don't forget at any point when I need to definitely be using the fractional value.

Lucas has bought a piece of material for £10.

50.

Here's his piece of material, and we can see that it's three metres in length, and that cost him £10.

50.

Sam says, "I really like that material Lucas.

I would like to get some, but I would need six metres.

How much will that cost?" Let's take a look at that on a double number line.

So I've got my length on the top line, and my costs on the bottom.

We know that the cost, and the length are proportional to each other.

Let's fill in what we know.

We know that three metres costs us £10.

50.

We want to buy, or Sam wants to buy, six metres.

We need to find the cost for six metres.

We need to look for the multiplicative relationship between 3 and 6, and that's a nice easy one, isn't it? 3 multiplied by 2 is 6.

Because the length and the cost are proportional to each other, because we've multiplied the length by 2, we need to multiply the cost also by 2.

£10.

50 multiplied by 2 is 21.

It would cost Sam £21 to buy six metres of the same material.

How could we use the double number line to find the cost of 10 metres? So notice here, same double number line, 'cause I know three metres cost £10.

50.

I've replaced the 6 with the 10, because I'm now finding the cost of 10 metres.

What is the multiplicative relationship between 10 and 3? So that's just what we were doing on the previous slides.

3 multiplied by something is 10.

So we do the inverse and we do 10 divided by 3.

Remember, fraction is another way of representing a division, so we can literally just write that as multiplied by 10 over 3.

Therefore I'm going to need to multiply my cost also by 10 over 3, giving us 35.

It will cost £35 to buy 10 metres of the same material.

Different example here, we've got a recipe, and it requires 195 grammes of chocolate chips to bake 30 cookies.

What weight of chocolate chips would you need to make 250 cookies? So I've got cookies on the top, and I've got chocolate chips on the bottom.

30 cookies requires 195 grammes, and we want to make 250 cookies.

What is the multiplicative relationship between 30, and 250? 30 multiplied by what is 250? So we do the inverse, and we do 250 divided by 30, and that simplifies to 25 over 3.

Our multiplier to take us from 30 cookies to 250 is 25 over 3.

So we need to multiply the chocolate chips also by 25 over 3, giving us 1,625.

You'll need 1,625 grammes to make 250 cookies.

That is a lot of chocolate chips.

Gonna do one more together, and then you'll be ready to have a go at one of these independently.

So here we can see the double number line has already been set up for us.

We know that 30 packets of something, doesn't matter what it is, costs us £25.

80, and we want to find the cost of 115 packets.

So we're looking for the multiplicative relationship between 30 and 115, and that's multiply by 23 over 6.

If you put into your calculator 115 divided by 30, it will give you its simplified fraction of 23 over 6.

So we need to do the same on the bottom now to the cost.

That gives us £98.

90.

115 packets will cost £98.

90.

Now you are ready to have a go at one of these independently.

Here's your question.

I'd like you to work out what the cost of 70 packets is? What number should be where the question mark is? You can pause the video now, remember you can use your calculator, remember to use that exact value, and then when you're ready come back.

You can pause that video now.

Great work.

Let's check that answer.

Multiplier was 7 over 3, so we do the same on the bottom to the cost, which is £19.

60.

70 packets will cost you £19.

60.

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

I knew you would.

Now we're gonna take a look at this function machine.

We know the input is 27, and the output is 24.

We don't know what the multiplier is.

Remember we're going to use the inverse multiplication, which is division, and we're going to divide 24 by 27.

I can write my division as a fraction.

Then what have I done? That's right.

I've written each of those integers as a product of its prime factors to help me with the simplification, and I can see that this then simplifies to 8/9.

My multiplier here is 8/9.

Your turn.

Pause the video, come back when you've got an answer.

Great work.

Let's check in.

42 over 30.

So we do the output divided by the input.

Write them as product to prime factors.

We can then see we've got 2 multiplied by 3, and 2 multiplied by 3 as a numerator denominator, which is equivalent to 1.

So it simplifies to 7/5.

The multiplier is 7/5.

Now I'd like you to have a go at this task.

You need to complete the table.

So I've given you an input, and an output, and you need to find the multiplier.

Or sometimes I've given you the multiplier, and you need to find the missing input, or output.

Pause the video.

Remember here it's absolutely fine to be using a calculator.

I suggest you write down the working out that you've done though so that if you've made an error you might be able to spot what error you've made, but I'm sure you won't make any errors.

I'll be here when you get back.

You can pause the video now.

Great work.

And question number 2, I'd like you please to complete the double number lines just as we did in that check for understanding.

Pause the video, and I'll be here when you get back.

Great work.

Let's check in now with those answers.

So question number 1.

The missing multiplier was 2.

5, or you may have 5 over 2.

The missing multiplier on the second one is 2/3, and you must have 2/3 there.

You may have an equivalent fraction, which is okay, but if you've used a calculator it would give it to you in its simplest form anyway.

The missing output in the next one was 60.

The next one, the missing input was 72, the missing output was 69, then we were missing a multiplier of 5.

Then an input was 16x and then the output 5x.

How did you get on with those last three? Put those in as a bit of a challenge.

I'm sure you got them right.

It's just implying exactly the same thing as you've done previously, but we've just added in a little bit of algebra for an extra challenge.

Now we can look at question number 2.

Here we were missing on A, a multiplier of 17 over 7.

So we end up with 102 packets are going to cost us £24.

48.

And then my multiplier on B, was 5 over 12.

And so 35 packets is going to cost us £18.

55.

Notice in A, we were moving to the right of our double number line and in B, we were moving to the left.

So our double number line can work either way around.

We can move right, or we can move left.

That's learning cycle number one done, and dusted.

Now let's take a look at our second learning cycle.

Writing one number as a fraction of another.

What fraction of 30 is 20? Now I find it quite difficult to get my head around that.

What fraction of 30 is 20? Let's take a look at some representations that might help us with this.

Here's a bar that represents 30, and here's a bar that represents 20.

We want to know what fraction of the 30 bar is the 20? I can split my 30 bar here.

I've got my 20, and I've got my 10.

So does that mean that 20 is 1/2? Of course it doesn't because we know that we have to have the parts have to be equal size.

I'm going to make my parts equal size now.

Now I can see that my 20 is equivalent to 2/3.

Hopefully the visualisation of those bars will help you to understand why the answer is 2/3.

20 is 2/3 of 30.

Notice this is just 20 over 30 in its simplest terms. So we have the 20 over 30, and that simplifies to 2/3.

Let's take a look at another one then.

What fraction of 250 is 150? Here's my bar representing 250, and here's my bar representing 150.

Here's my 250 bar with the 150 shown, but my parts are not equal size.

I need to make sure that this is split into equal parts, and here I'd have to split it into fifths.

And now I can see that this is 3/5.

150 is 3/5 of 250.

Using what we spotted on the previous slide, that it would be 150 over 250, and then we can simplify that to 15 over 25 by dividing the numerator, and the denominator by 10.

And then we can simplify that again, by dividing the numerator and the denominator by 5.

Remember, we're dividing both the numerator, and the denominator, so therefore we're dividing by 1, and not changing the value.

We could also represent this type of problem on a double number line.

What fraction of 105 is 75? So we've got a fraction on the top, and our amount on the bottom.

1 is 105.

So the whole, if we think about how we're gonna draw the bar, the bar I would start with my whole was 105.

We are trying to find out what fraction 75 is.

So notice the 75 is to the left of 105.

The 75 is less than 105.

I now need to find my multiplicative relationship between 105 and 75.

What have I multiplied by? Well I'm gonna do 75 over 105.

I've written that as a product of prime factors so that I can see that simplification down to 5/7.

Moving to the left, I multiply by 5/7 as I move along my double number line.

So we can see that it's 5/7.

So you may prefer to use a double number line.

75 is 5/7 of 105.

Lucas and Sam have drawn a double number line to answer this question.

What fraction of 45 is 18? Let's have a look at Lucas's method.

Here's Lucas's method.

I'm gonna pause a moment and let you just have a look, and see what Lucas has done.

And here's Sam's method.

So again, I'm just gonna pause, and I'm gonna let you have a look at Sam's method.

Whose is correct? Whose number line is correct? Sam's method is correct.

We'll now take a look at each of the methods.

What mistake has Lucas made? He has marked 18 as the whole.

What question does Lucas's number line represent? What fraction of 18 is 45? 18 is 2/5 of 45.

45 is 5/2 of 18.

Lucas says, "I've noticed that the fractions are reciprocals of each other." "Oh yes, you're right Lucas," Sam says.

The integers in both are the same, but the fraction one is the reciprocal of the other.

Now I'd like you to have a go at this one.

I'd like you to decide what is missing from each box.

It will either be a number or a fraction.

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

How did you get on? Three answers? Brilliant, well done.

24 is 3/8 of 64, so 64 is 8/3 of 24.

6,804 is 9/4 of 3,024, so 3,024 is 4/9 of 6,804.

80.

5 is 1 3/4 of 46, so 46 is 4/7 of 80.

5.

That final one, you just needed to change 1 3/4 into an improper fraction first so that you could then find its reciprocal.

What fraction of 144 is 120? We'll do this one together.

Here's my double number line.

We know that the whole is 144 because we want to know what fraction of 144.

We want to find what fraction is 120.

My multiplier is 5 over 6.

That is 120 over 144 in its simplest form.

So I'm going to multiply my fraction also by 5 over 6.

120 is 5/6 of 144.

Now you're ready.

Please give this one a go.

What fraction of 84 is 35? Pause the video, and I'll be here when you come back.

This time the whole was 84, and we are trying to find 35.

So my multiplier to take me from 84 to 35 is 5/12.

So my multiplier on the fraction will also be 5/12.

So 35 is 5/12 of 84.

You're now ready to have a go at these questions.

What fraction of A is B? So what fraction of A is B? And you're going to write all of your answers in its simplest form.

You are gonna shade the answers in on the grid.

They may appear more than once.

What fraction of 123 is the revealed number? So when you've shaded in all of the correct answers, it will reveal a number, and then I'd like you to just finish off by telling me what fraction of 123 is the revealed number? Pause the video.

Good luck with this.

Hope you have fun trying to decide what the number is, and then I'll be here when you get back.

Pause the video now.

Great work.

Let's check what number did you get? You did? Right, okay.

205.

You should have got 205.

And well done if you worked out that it was 5/3 of 123.

Now we can summarise our learning from today's lesson.

To find a multiplier, you divide the two numbers.

You must always use the exact value.

If the multiplier is not an integer, or a terminating decimal, you must use the fraction.

So in the example, the one we looked at earlier is if we're trying to find the multiplicative relationship between 3 and 10, we do 10 divided by 3, and we must use our fraction answer of 10 over 3, and not a decimal form of that.

To write a number as a fraction of another number, we can use a double number line.

And there's an example there too.

Thank you so much for joining me today.

It's been a fantastic lesson, and I'm really looking forward to working with you again really soon.

Thanks.

Bye.