Lesson video

In progress...

Hi everyone, my name is Ms Khoo.

And I'm really happy and excited to be learning with you today.

It's going to be a fun lesson full of some words maybe that you may or may not know, and we'll build on that previous knowledge too.

Super excited to be learning with you, so let's make a start! Under the unit "Understanding Multiplicative Relationships: Fractions and Ratio," we'll be representing a multiplicative relationship.

And by the end of the lesson, you'll be able to use a double number line to represent a multiplicative relationship and connect to other known representations.

We'll look at the word proportionality again.

Proportionality means when variables are in proportion if they have a constant multiplicative relationship.

And a ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

Today's lesson will be broken into two parts.

Firstly, we'll be looking at double number lines, and then we'll be looking at graphical representations.

So let's have a look at double number lines first.

Now there are lots of different ways to represent a ratio.

For example, I want you to have a think.

How many different ways can you represent this ratio? "For every one star, there are three hearts." I'm going to show you with a ratio table.

Here, you can see one star and three hearts.

I'm also gonna show you with a bar model.

One star and three hearts.

I'm gonna show you with a fraction.

A quarter are stars.

Here are just a few, all illustrating the same ratio.

Well done if you got this one right.

Now another way to show a ratio use is using a double number line.

For example, using the same ratio for every one star, there are three hearts.

What does this look like on a double number line? Well, we draw our two parallel lines, ensuring that we start from zero for both line.

I'm gonna identify my axes.

Stars are at the top and hearts are at the bottom.

Now for every one star, we have three hearts.

That means for two stars, how many hearts do you think we have? Well, it's six.

And for three stars, how many hearts do you think we have? Well it's nine.

This is a nice simple example of a double number line showing the ratio of, for every one star there are three hearts.

When drawing the scale number line, it's important to remember both lines must start from zero to illustrate the proportionality.

So what I'm gonna do, I'm gonna show you a bar model.

And here the bar model shows one rabbit and two carrots.

Which double number line do you think shows the above ratio? And what ratio do you think that other double number line show? See if you can give it a go.

Press pause if you need more time.

Well hopefully you've spotted it's B.

For every one rabbit, there are two carrots.

What do the other number lines show? Well this one shows for every one rabbit, there are three carrots.

We know this is one rabbit and there are two carrots.

This is two rabbits to one carrot.

Well done if you got this one right.

Now what I'm gonna do, I'm going to show you a ratio, but we have some things blanked out.

For every two reds there are how many green? In a ratio table, it shows you these amounts.

And our double number line is partially completed.

See if you can fill in the blanks and press pause for more time.

Great work.

So let's see how you got on.

Well you can use the ratio table as a reference.

For every two red, there are five green.

So we can put that on our double number line.

Two for five green.

That means four reds is 10 green.

Also means six reds are 15 green.

Notice how our axes, whether it's the top number line, or bottom number line starts from zero, and all of the ratios are equivalent.

Two red to five green is equivalent to four red to 10 green, which is equivalent to six red to 15 green.

Well done if you got this one right.

Now I'm gonna show you another check question.

Lucas is given this ratio table and he draws this number line.

Where did Lucas make his mistake? And can you draw the correct double number line given the ratio? See if you can give it a go, and press pause if you need more time.

Well done.

So let's see how you got on.

First of all, each line must start from zero.

And the scale for each line must be consistent.

He has added one to each part on the number line, A.

Ratio is multiplicative, not additive.

So let's draw the correct number line.

Firstly, we know for every two A, there are 11 B.

That means let's increase A by two each time.

Two, four, and six.

For B, we know there are two A's for 11 B's.

So that means we're increasing 11 each time, ensuring that we start from zero.

11, 22 and 33.

Well done if you got this one right.

I'm going to do one question, and I'd like you to do the second one.

Here is a double number line converting kilogrammes into pounds.

Let's fill in the blanks for part A, B and C.

Well it says the amount in kilogrammes multiplied by a what, gives us the amount in pounds? To find that multiplier, you have different options, really.

You could do 3.

3 pounds divided by three, or 4.

4 pounds divided by two, or 2.

2 divided by one.

All giving us the multiplier of 2.

I've chosen 2.

2 divided by one equals 2.

What does this mean? Well it means for every kilogramme, there are 2.

2 pounds.

And if we're asked to work out what 40 kilogrammes is, well we simply multiply the kilogrammes by 2.

2, giving us the pounds.

So that means it's 88 pounds.

If you spotted this, huge well done.

Now I'm going to give you this question.

Here's a double number line converting pounds into dollars.

I want you to fill in those missing numbers.

See if you give it a go, and press pause if you need more time.

Well done.

Let's see how you did.

Well using our double number line, did you find that multiplier? I'm going to find the multiplier by doing three divided by two, which is 1.

So that means our pounds multiplied by 1.

5 gives us our dollars.

This means for every one pound, we have $1.

50.

And if we're asked to find 80 pounds, well, 80 multiplied by a 1.

5 is $120.

Great work if you got that one right.

Let's have a look at another check.

Izzy and Laura are going on holiday.

And Izzy converts her pounds using shop A and Laura converts her pounds using shop B.

Who has the better deal? And explain why.

See if you can give it a go.

Press pause if you need more time.

Well there's lots of different ways in which you could work up the conversion.

I spot that both shops identify what two pounds is in rupees.

So on both double number lines, two pounds in shop A is 210 rupees.

And two pounds in shop B is 213 rupees.

So that means in shop B, you get more rupees for two pounds.

Well done if you spotted this.

Now it's time for your task.

I want you to use the double number line and/or the ratio table to fill in the equivalent ratios.

See if you can give it a go, and press pause if you need more time.

Well done.

Let's move on to C and D.

C and D gives you more blanks.

See if you can fill these in, and press pause if you need more time.

Great work.

Let's move on to question two.

So for question two, Sofia and her dad walk side by side along a straight path.

The number of steps are represented by the double number line.

Which line do you think represents Sofia's steps? And I want to explain how you know.

And when Sofia has taken 24 steps, how many steps has her dad taken? Part C wants you to write the ratio of Sofia's steps to her dad's steps.

See if you can give it a go.

Press pause if you need more time.

Let's have a look at question three.

Question three says, here's a double number line.

Aisha says to find A, she's going to do five multiply by 1.

Andeep says to find A, he's going to find one quarter of 30.

Do you agree with Aisha or Andeep? And I'd like you to explain why.

See if you can give it a go and press pause for more time.

Well done.

Let's move on to question four.

See if you can fill in the blanks.

See if you can give it a go and press pause for more time.

Well done.

Let's move on to these answers.

For A, we should have these answers.

For every one A, there are four B's.

Great work if you got that one right.

For B, you should have these answers.

For every one A, there are three B's.

Great work if you got that one right.

For part C, you should have these answers.

For every two A's, there are seven B's.

And for D you should have these answers.

For every two A's, there are five B's.

Great work if you got that one right.

For question two, which line represents Sofia's? Well it would be the top line.

And how do we know? Well, Sofia must be the top line, as she takes more steps compared to her father.

Now when Sofia's taken 24 steps, how many steps has her dad taken? Well if you double 12 steps of Sofia, you get 24 steps.

So that means doubling the eight give 16 steps.

What's the ratio of Sofia's steps to her dad's steps? Well if you do 12 divided by eight, it's 1.

That means for every one step Sofia's dad makes, Sofia has to do 1.

5 steps.

Well done.

For question three, Aisha says to find A, she's going to multiply five by 1.

And Andeep says he's going to find a quarter of 30.

Both are correct.

I'm gonna show this in a ratio table.

Well if you know A, 20 A's make five B's.

That means 30 A's can be found by multiplying 20 by 1.

So five multiplied by 1.

5, which will give us the answer.

Or, you can find that multiplier from A to B, which is multiplying by a quarter.

Both would give you an answer of 7.

So both Andeep and Aisha are correct.

Question four, you should have got these answers.

And for question 4b, you should have got these answers.

Great work, well done everybody.

Now let's have a look at these graphical representations.

You can represent a ratio on a graph.

For example, the following multiplicative relationship, I'm going to show it through a bar model.

One X is two Y's.

You can see this in our ratio table as well.

Now I'm gonna show it on a double number line.

Here you can see one X is two Y's, or two X's is four Y's, or three X's is six Y's.

Now I'm going to change it to an axis.

As we change it to an axis, everything still remains in proportion.

Drawing a straight line, we can identify the ratio is exactly the same.

And the line represents all the points where the ratio is equivalent.

One X is two Y.

Or two X's is four Y's.

Or even 2.

5 X's is five Y's.

There are an infinite number of equivalent ratios.

Now what I'm going to do is show another graphical representation.

Jun and Sofia have each represented a ratio graphically.

Jun thinks they both represented the same ratio.

Do you agree with him? Well hopefully you can spot that they're different ratios.

Because you need to pay attention to the scale on the axis.

For Jun, for every one X, it's one Y.

But for Sofia, for every two X's, it's seven Y's.

Now let's have a look at a check.

Fill in the gaps for the ratio.

See if you can give it a go.

Press pause if you need.

Well done.

Let's see how you got on.

Well for the first one, for every one X, it's one Y.

For the second one, for every two X's it's seven Y's.

And for the third one, for every one X, it's seven Y's.

You may have spotted for every two X's it's 14 Y's, which is equivalent to our ratio, for every one X it's seven Y's.

Really well done.

Now what I'm going to do is give you another check question.

Using the graph, complete the information.

See if you can give it a go.

Press pause if you need.

Well done.

Let's see.

Well, hopefully you've identified using our double number line, one pound is $1.

20.

Two pound is $2.

40.

Three pounds is $3.

60.

Using a ratio table, one pound represents one pound 20.

And for every one pound, it's $1.

20.

Well done.

There are four different ways that we've represented the same ratio here.

Graphically, number line, in a ratio, in a sentence, and a ratio table.

Amazing work.

Let's have a look at another check question.

Here, the graph shows a scale conversion on a map.

I want to use the graph to work out how many kilometres is 12 centimetres on our map.

And I want you to use the graph to identify how many centimetres in three kilometres.

See if you can give it a go, and press pause if you need more time.

Well done.

Let's see how you got on.

Well, for every two centimetres on the map, it's 15 kilometres in real life.

And you can see that using our graph.

Two centimetres is 15 kilometres.

This is the same as for every one centimetre, it's 7.

5 kilometres.

So what's 12 centimetres? Putting this in a ratio table, I'm going to show two centimetres is 15 kilometres.

I'm gonna simply multiply by six to give me 12 centimetres, which is now 90 kilometres.

Well done if you got this one right.

For how many centimetres is three kilometres? Well I'm going to use our ratio table again.

And from our ratio table, we're going to look at three kilometres.

What do I need to do to 15 to make three? I divide by five so I have to do the same to the centimetres to give me 0.

4 centimetres.

Well done if you got this one right.

Now it's time for your task.

Fill in the ratio table using the above graph.

See if you can give it a go.

Press pause if you need more time.

Well done.

Moving on to question two.

Fill in the missing information on the double number line and/or the graph.

Give it a go and press pause for more time.

Great work.

Moving on to question three.

Question three.

Jacob has interpreted the graph incorrectly.

He's written down that the double number line would be this.

And he says for every three A's, there is one B.

And when there are 12 A's, there are four B's.

I want you to explain where is his mistake, and correctly answer the question.

See if you can give it a go and press pause for more time.

Great work.

So let's move on to question four.

Question four gives us the answers, but we're missing some information.

So we know we have information about a scale drawing and a map.

And it states that for every something centimetres, it's something kilometres in real life.

Part A says, well, we don't know how many centimetres there are in kilometres, but we know the answer is 40 kilometres.

Part B says for 45 centimetres, it's 150 kilometres.

And part C says, well we don't know how many kilometres there are, but in centimetres, it's 18.

Can you work out those missing gaps and draw that graph correctly? See if you can give it a go.

Press pause for more time.

Great work.

Let's go through these answers.

Well, for question one, you should have had, A should have been been A is one, B is two.

For B, A is one and B is eight.

And for C, A is two and B is 7.

Well done.

For question two, here are our missing values and our graph.

Huge well done if you've got this one right.

For three, where did Jacob make his mistake? Well, Jacob's double number line has B as the X-axis, and A as the Y-axis.

It should have been the other way round.

So he made a mistake there.

So that means the correct number line should be this.

So for every three A's, there are nine B's.

And this is because for every one A, there are three B's.

It's really important to look at the X-axis and the Y-axis so you know how to write your multiplicative relationship.

For question four, let's see if we can use the answer to part B to work out our scale.

So if you find the highest common factor of 150 and 45, it's actually 15.

And the reason why we're doing this is because we're simplifying our ratio.

So dividing 45 by 15, and dividing 150 by 15, then gives us three centimetres for every 10 kilometres.

We've written our ratio and we've simplified it by using that highest common factor.

Now from here, we can work out our graph, and work out our missing values.

Massive well done if you got this one right.

Well done, everybody.

It was a great lesson.

Remember, proportionality means when variables are in proportion if they have a constant multiplicative relationship.

There are lots of different ways to represent a ratio.

Bar model, ratio tables, double number line, and graphs to name some examples.

But what is important is to understand how the same ratio can be represented in lots of different ways.

Great work, everybody.

Well done.

I've finished the video