Lesson video

Hi everyone, and welcome to today's lesson.

We'll be looking at multiplicative relationships.

It's one of my favourites, and I'm really happy that you're joining with me today.

The reason why it's one of my favourites, well, it's because it appears so much in everyday life.

Really happy that you've joined me today.

So let's make a start.

Hi everyone and welcome to this lesson on scaling diagrams for multiplicative relationships.

And it's under the unit understanding multiplicative relationships, percentages and proportionality.

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

In this lesson we'll be looking at the keyword proportion, and proportion is a part to whole, sometimes part to part, comparison.

And if two things are in proportion, then the ratio of part to whole is maintained, and the multiplicative relationship between the parts is also maintained.

Our lesson today will be broken into two parts.

We'll be looking at scaling diagrams first and then comparing representations second.

So let's make a start on scaling diagrams. Well, proportion can be represented in lots of different ways.

I want you to look at these diagrams and what do you think the question is to all of these answers? See if you can give it a go and press pause for more time.

Well done.

Well, hopefully you spotted all of these diagrams are showing the answer to 25% of 500.

Looking at the double number line and the graph, these show how a pair of numbers align when scaled proportionally.

And when we explore where this comes from, we're looking at a scaling diagram as this emphasises multiplication as scaling.

For example, let's have a look at this double number line.

What do you think it's showing? Well, hopefully you can spot it's showing that 100% is £200.

Now, a double number line is a condensed version of a scaling diagram.

So what I'm gonna do is I'm going to separate the double number line to make two number lines.

So you can see here.

Now what I'm going to do is I'm going to un-squash or stretch the bottom number line.

So you can see, I'm just stretching it out here.

Now, a completed scale diagram now shows the mapping of our numbers.

In other words, we know 100% is £200.

And to ensure we map the other numbers correctly, we can find the intervals.

So what do you think the intervals are for each scaled line? Have a little think.

Well, hopefully you've worked out that for the percentage, we have 10 intervals, so I'm going to divide 100% by 10.

This means each interval represents 10%.

Now for the amount of money, you can see there are 10 intervals too.

So I'm going to divide the £200 by 10.

That means each interval represents £20.

Now it's clear to see which percentage maps to which amount of money.

100% represents £200.

90% represents £180.

80% is £160.

70% is £140.

So on and so forth.

So what you can see is we've mapped the percentage to the correct amount.

Let's have a look at a quick check.

Here's a scaling diagram.

And what I want you to do, I want you to fill in the missing intervals and write down what is 60% of 350.

See if you can give it a go, and you can use gridded or squared paper to help you if you need.

Press pause for more time.

Well done.

So let's see how you got on.

Well, first things first, let's have a look at that percentage scale.

Well, you know it goes from 0 to 100% and there are five equal intervals.

So dividing 100 by 5 means each interval represents 20%.

So I've put it here.

Now let's have a look at our amount.

Well, there are 10 equal intervals.

So 350 divided by 10 is simply £35 for each interval.

So I've identified it here.

So now let's map our numbers.

Now remember, the percentage had five intervals and the amount had 10 intervals.

So what does that mean? Well, if you divide 10 by 5, it's 2.

So that means each percentage interval will map to the second interval of our amount.

So 20% will map to £70.

40% will map to £140.

60% will map to £210.

80% will map to £280.

And 100% will map to £350.

So from here, we can identify the answer of 60% of £350 is £210.

Really well done if you got this one.

Now it's time for your task.

What I want you to do is fill in the missing information, including those arrows, using the scaling diagram.

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

Well done.

So let's move on to question two.

Question two wants you to fill in the missing information again, including those arrows, using the scaling diagrams. See if you can give it a go, and press pause for more time.

Well done.

So let's move on to question three.

Question three, same as before.

Fill in that missing information with those arrows using the scaling diagram.

Please give it a go, and press pause for more time.

Great work.

Let's move on to the answers.

Well, first things first, let's identify those intervals.

Well, hopefully you've spotted for 1A, it should be going up in twenties, because 100 divided by those five intervals means 20% is each interval.

Now for the second part, we have 10 equal intervals.

So 150 divided by 10 means each interval is 15.

From here, we can identify 60% to be mapped onto that number 90.

Very good if you got this one.

For B, identifying those intervals again.

Well, we have five intervals for our 100%, so that means each interval had to be 20%.

Now for the £240, you might have spotted we have 20 intervals there.

So just focusing on those main 10, dividing 240 by 10 gives us 24.

So that means those 10 intervals, each interval is 24, or you could have broken it into those 20 intervals, with each interval representing 12.

Then let's map it.

Well, same again, you can spot we have five intervals for the percentage, but we have 20 intervals for our amount.

20 divided by 5 is 4.

So that means we map every interval of our percentage to the fourth interval of our amount.

20 maps to 48.

40 maps to 96.

60 maps to 144.

80 maps to 192.

And as we know, 100% maps to 240.

So we can identify 40% of 240 is 96.

Really good work if you got this one right.

Now let's have a look at question two.

For question two, we have 10 equal intervals.

So that means 100 divided by 10 is 10.

But notice how we're asked to find every other interval.

So knowing that each interval is 10, we can fill in our scale.

Now for £120 we have five equal intervals.

So we divide 120 by 5, thus identifying each interval to be 24.

So I can fill in my scale diagram like this.

Then we can work out 60% of 120 to be simply 72.

For B, well, we have four equal intervals for 100%.

So that means each interval had to be 25.

100 divided by 4 is 25.

Now for 140, we have 20 intervals here, but notice how we're only interested in every other interval.

So if you do 140 divided by 10, it's 14, or you could have done 140 divided by 20, which is 7.

Still would have give you the same scale.

Now let's map.

Remember, the percentage has four intervals, and the amount of money has 20 intervals.

20 divided by 4 is 5.

So we're mapping each percentage interval to the fifth interval on our amount.

So the 25% maps to 35.

The 50% maps to 70.

The 75% maps to 105.

And obviously 100% maps to 140.

Therefore, 75% is 105.

Well done if you got this one right.

For question three, look at that interval again.

We have 10 equal intervals, so that means we know each interval is 10.

But notice how we're asked to label every other interval, so we have these.

Now for the £80 we have five intervals.

So 80 divided by 5 is 16.

So each interval is 16.

Now we have 10 equal intervals for the percentage, and we have five equal interval for our pounds.

So how do we map the numbers? Well, we know 5 divided by 10 is 0.

5.

So that means every two percentage intervals maps on to one of our intervals for our amount of money.

So this gives us our mapping here.

So answering our question, 60% of 80 gives us 48.

For B, well, we have 100% and it's divided into 20 intervals, so that means each interval had to be five.

But because we're asked to label every other interval, that means we're labelling our scale like this.

Well done if you got this.

But notice how we have five intervals for our £70.

Well, 70 divided by 5 is simply 14, so we've labelled our scale from here.

So every four intervals of our percentage will map to one interval of our amount.

Really well done if you figured that one out, thus giving us a final answer of 40% of 70 to be 28.

Massive well done if you got this one right.

Well done, everybody.

So let's have a look at comparing representations.

Well, we know we can find percentages of amounts using scaling diagrams. So let's start by putting the information from our scaling diagram into a ratio table.

Here you can see in my scaling diagram 100% represents £200.

So that means I can work out 90% using my ratio table and/or the scaling diagram.

What do you think it is? Well, you can spot it's £180.

So let's work out 40% using our scaling diagram and put it into our ratio table.

Well, that would be £80.

10%, well, using our scaling diagram, you can see it will be £20.

So given an amount and percentage are proportional, that means we have the same constant multiplier.

Can you see it on our scaling diagram, or can you see it from our ratio table? Well, hopefully you can spot you are multiplying the percentage by two.

You can work out a constant multiplier to be multiplied by two from the percentage to give the amount.

So you can see that clearly using the mapping of our numbers on our scaling diagram or using the numbers in our ratio table.

Now, perhaps you saw a constant multiplier to be times by half from the amount of money to give the percentage.

So in other words, you half the amount of money and it gives the percentage.

That would've also been correct because you'll notice the direction of the arrows now changes.

And we're looking at the amount mapping onto the percentage.

Really well done if you spotted that difference.

So scaling diagrams show the multiplicative relationship as a stretch or a squash.

But how about trying to find more complicated percentages, such as 23%? What do you think the problem is there? So you can see 23% is quite hard to see using a scaling diagram.

And the accuracy of this is completely dependent on the accuracy of the scale.

So using a ratio table can allow for any amount or percentage to be calculated.

So let's have a look at that constant multiplicative relationship and see if we can work out 23% of 200.

Well, if you know that we're multiplying the percentage by 2 to give the amount, that means we simply have £46 as our amount.

And you can use our ratio table here to see.

You could have also used the scaling diagram once that multiplier has been identified.

Both are really good options.

Now let's have a look at a quick check.

Using the scaling diagrams, see if you can fill in the ratio table and work out those missing values.

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

Great work.

So let's see how you got on.

Well, looking at the scaling diagram, do you know what do you multiply the percentage by to give the amount? Well, hopefully you spotted you're multiplying by 0.

8.

So if you multiply that percentage by 0.

8, it maps onto the amount of money.

So that means we can multiply 40 by 0.

8 to give me 32, and 16 by 0.

8 to give me 12.

8.

Really well done if you got this one right.

Now let's have a look at the second question.

What's that multiplier to change the percentage to the amount? Well, you multiply it by 1.

5, so that means we can find out what 80% is.

18 multiplied by 1.

5 is 120.

And 34 multiplied by 1.

5 is 51.

Great work if you got this one right.

It's always important to pay attention to the direction of the arrows, as it tells you which multiplier you're working out.

Here we were working out the multiplier to map the percentage to the amount.

If the direction of the arrows were the other way, what do you think the multiplier would be for part A? Well, it would be five over four.

And what do you think the multiplier would be for part B? Well, it would be two over three.

So it's important to recognise the direction of the arrow as it tells you how to map one amount onto another one.

Great work.

So let's move on to your task.

Here, you're asked to fill in the missing information on the ratio table and the scaling diagram.

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

Well done.

Let's move on to question two.

Question two wants you to fill in the missing information using these different representations.

We have a scaling diagram, a ratio table, a double number line, and a graph.

Great question.

So you can fill in what you can, and press pause for more time.

Great work.

Let's move on to question B.

Question B shows again a graph, ratio table, double number line, and a scaling diagram.

And we're asked to fill in the information using the different representations.

They all show the same proportion.

See if you can give this one a go, and press pause for more time.

Fantastic work.

So let's have a look at these answers.

Well, hopefully you've spotted the multiplier would be 1.

6.

So working out our multiplier allows us to work out these amounts.

For the second part, our multiplier was 2.

1, so we're able to work out those missing amounts there.

Huge well done if you got this one right.

For question two, where did you start on these questions? Lots of different places that you could have started.

For me, I'm going to identify the multiplier, which is 1.

8, and then from here I'm going to identify the scale.

Given the fact that I have five equal intervals, 180 divided by 5 is 36.

Now I have my scale, and I think I can use my ratio table or my scaling diagram to identify 40% or 15%.

Really well done if you got this one right.

But what does that look like on a double number line? Well, we know 100% is 180, 40% is 72.

And this confirms our graphical representation as well.

Always a good check given these different representations.

Huge well done if you got this one right.

For B, well, I worked out the multiplier first, which was 0.

8.

And then I identified the scale.

Well, 80 divided by 5 is 16, so I now know my scale.

I can use my scaling diagram or my ratio table to fill in the amounts, and then from here, I can fill in the information on our double number line.

Massive well done if you got this one right.

Completing the graph, you can see all of these different representations show the same multiplicative relationship between 100% being £80.

Fantastic work.

Great work, everybody.

So in summary, a multiplicative relationship can be represented a number of ways, including double number lines, scaling diagrams, graphical representations, and ratio tables, each with their own advantage and disadvantage.

We know scaling diagrams are wonderful visual representations to show us how multiplication can be seen as scaling and how the double number line really does relate to it.

Yes, scaling diagrams do take a little longer to draw, but they do have that visual advantage.

The great thing is we can also use ratio tables and see that relationship between the scaling diagram and the ratio table.

I hope you've enjoyed this lesson.

I really enjoyed seeing how we can represent that multiplicative relationship in so many different and wonderful ways.

Well done.

It was great learning with you.