Lesson video

Hi everyone! My name is Ms. Ku.

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 top 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 2 or more values and allows you to compare a part with another part in a whole.

Today's lesson will be broken into 2 parts.

Firstly, you'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 1 star, there are 3 hearts"? I'm going to show you with a ratio table.

Here you can see 1 star and 3 hearts.

I'm also gonna show you with a bar model, 1 star and 3 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 1 star there are 3 hearts, what does this look like on a double number line? Well, we draw our 2 parallel lines ensuring that we start from zero for both line.

I'm gonna identify my axes.

Stars at the top and hearts are at the bottom.

Now for every 1 star, we have 3 hearts.

That means for 2 stars, how many hearts do you think we have? Well, it's 6.

And for 3 stars, how many hearts do you think we have? Well it's 9.

This is a nice simple example of a double number line showing the ratio of, for every 1 star, there are 3 hearts.

When drawing the scaled 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 1 rabbit and 2 carrots.

Which double number line do you think shows the above ratio and what ratio do you think that other double number lines 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 1 rabbit, there are 2 carrots.

What do the other number lines show? Well this one shows for every 1 rabbit, there are 3 carrots.

We know this is 1 rabbit and there are 2 carrots.

This is 2 rabbits to 1 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 2 red, 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 2 red, there are 5 green.

So we can put that on our double number line, 2 for 5 green.

That means 4 reds is 10 green, also means 6 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.

2 red to 5 green is equivalent to 4 red to 10 green, which is equivalent to 6 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 1 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 2 A, there are 11 B.

That means let's increase A by 2 each time, 2, 4 and 6.

For B, we know there are 2 A's for 11 B's, so that means we're increasing 11 each time ensuring that we start from 0, 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 multiply by what gives us the amount in pounds? To find that multiplier, you have different options really.

You could do 3.

3 pounds divided by 3 or 4.

4 pounds divided by 2 or 2.

2 divided by 1.

All giving us the multiplier of 2.

2.

I have chosen 2.

2 divided by 1 equals 2.

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, given 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 can 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 3 divided by 2, which is 1.

5.

So that means our pounds multiply by 1.

5 gives us our dollars.

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

50.

And if we're asked to find 80 pounds, well 80 multiply 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 convert 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 2 pounds is in rupees.

So on both double number lines, 2 pounds in shop A is 210 rupees and 2 pounds in shop B is 213 rupees.

So that means in shop B, you get more rupees for 2 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 2.

So for question 2, Sophia 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 Sophia steps? And I want to explain how you know.

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

See if you can give it a go.

Press pause if you need more time.

Let's have a look at question 3.

Questions 3 says, here's a double number line.

Aisha says to find a, she's going to do 5 multiply by 1.

5.

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 one more time.

Well done! Let's move on to question 4.

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 1 A, there are 4 B's.

Great work if you've got that one right.

For b, you should have these answers.

For every 1 A, there are 3 B's.

Great work if you got that one right.

For part c, you should have these answers.

For every 2 A's, there are 7 B's.

And for d, you should have these answers.

For every 2 A's, there are 5 B's.

Great work if you've got that one right! For question 2, which line represents Sophia's? Well, it'd be the top line.

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

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

So that means doubling the 8 give 16 steps.

What's the ratio of Sophia steps to her dad steps? Well if you do 12 divided by 8, it's 1.

5.

That means for every 1 step Sophia dad makes, Sophia has to do 1.

5 steps.

Well done! For question 3, Aisha says to find a, she's going to multiply 5 by 1.

5.

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 20 A's make 5 B's, that means 30 A's can be found by multiplying 20 by 1.

5.

So 5 multiply 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.

5.

So both Andeep and Aisha are correct.

Question 4, you should have got these answers.

And for question 4 b, 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, that following multiplicative relationship, I'm going to show it through a bar model.

1 x is 2 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 1 x is 2 y's or 2 x's is 4 y's or 3 x's is 6 y's.

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

As we change it to an axes, 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.

1 x is 2 y or 2 x's is 4 y's or even 2.

5 x's is 5 y's.

There are an infinite number of equivalent ratios.

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

Jun and Sophia 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 1 x it's 1 y, but for Sophia, for every 2 x's it's 7 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 1 x, it's 1 y.

For the second one, for every 2 x's, it's 7 y's and for the third one, for every 1 x, it's 7 y's.

You may have spotted for every 2 x's, it's 14 y's, which is equivalent to our ratio for every 1 x, it's 7 y's.

Really well done! Now 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, 1 pound is $1.

20, 2 pound is $2.

40, 3 pound is $3.

60.

Using a ratio table, 1 pound represents 1.

20.

And for every 1 pound, it's $1 20.

Well done! There are 4 different ways that we've represented the same ratio here: graphically, number line, new 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 you 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 3 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 2 centimetres on the map, it's 15 kilometres in real life.

And you can see that using our graph, 2 centimetres is 15 kilometres.

This is the same as for every 1 centimetre, it's 7.

5 kilometres.

So what's 12 centimetres? Putting this in a ratio table, I'm going to show 2 centimetres is 15 kilometres.

I'm gonna simply multiply by 6 to give me 12 centimetres, which is now 90 kilometres.

Well done if you got this one right! For how many centimetres is 3 kilometres? Well, I'm going to use our ratio table again and from our ratio table, we're going to look at 3 kilometres.

What do I need to do to 15 to make 3? I divide by 5 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 2, 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 3.

Question 3, Jacob has interpreted the graph incorrectly.

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

And he says for every 3 A's, there is 1 B.

And when there are 12 A's, there are 4 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 4.

Question 4 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 one more time.

Great work.

Let's go through these answers.

Well, for question 1, you should have had: a should have been A is 1, B is 2; for b, A is 1 and B is 8; and for c, A is 2 and B is 7.

5.

Well done! For question 2, here are our missing values and our graph.

Huge well done if you've got this one right! For 3, 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 3 A's there are 9 B's.

And this is because for every 1 A, there are 3 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 4, 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 3 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!.