Lesson video

Hi everyone, welcome to our lesson.

Today we'll be looking at multiplicative relationships.

It's one of my favourites because it appears so much in everyday life.

Really excited that you're learning with me today.

So let's make a start.

Hi everyone and welcome to this lesson on multiplicative relationships represented graphically and it's under the unit understanding of multiplicative relationships, percentages, and proportionality.

By the end of the lesson, you'll be able to use a graph to represent a multiplicative relationship and connect to other known representations.

We'll also be looking at these keywords.

Let's have a look at variable.

Well variables are in proportion if they have a constant multiplicative relationship.

We'll also be looking at the word ratio.

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

In today's lesson, it'll be broken into two parts.

The first part will be looking at percentages of amounts graphically.

And the second part will be looking at other representations of proportional relationships.

So let's make a start looking at percentages of amounts graphically.

Now, fractions, percentages, and ratio all show proportion, and there are lots of different ways to show this proportion.

For example, double number lines, bar models, and ratio tables.

Using a bar bottle, let's have a look at the number 240 and find some percentages of our 240.

Well, first of all, here is our 240 and it represents 100%.

All I've done here is divided it into 10 equal parts and I've found 10%.

Each 10% is 24.

Now showing this on a double number line, we have our percentage is on the top and our amount is on the bottom.

Let's identify our zero as 0% and our 240 is 100%.

Here you can see we're showing exactly the same information.

All I need to do now is show 10%, which is 24.

And you can see those 10 equal intervals, which represent those 10 equal parts of our bar model.

We also know we can show relationships graphically.

For example, here is the same double number line showing 240 as a percentage.

All I'm going to do is change it into a graph by simply rotating that top line to make a y-axis.

Here you can see the y-axis now shows the percentage and the x axes shows the amount of money we have.

So let's plot what 240 as 100% percent looks like.

Well, it'll be plotted here.

So you can see that we've drawn another representation to show 240 as 100% percent.

But what's really important, just like our double number lines, is graphs must be correctly scaled.

So let's do a check question.

Here you can see we have a graph showing that proportional relationship.

We have our percentage on the y-axis and our amount on the x-axis.

Now we're asked to pair it with the correct number line represented here.

Do you think it's A or do you think it's B? Press pause if you need more time.

Well done.

Let's see how you got on.

Well, hopefully you can spot it's B.

And the reason why it's B is because you can see 100% on our number line represents 200 as an amount and you can see it plotted right here.

Well done if you got this one right.

Now, using the graph, how do you think we can find 50% of 200? Same again, you can see our y-axis has our percentage and our x-axis has our amount.

What do you think? Well, first things first, it's important to read the scale carefully.

What you can spot is the y-axis has 10 equal intervals.

So if we divide our 100% percent by 10, that means each interval represents 10%.

All I'm going to do is identify where 50% would lie.

Now from here, what are the intervals for the x-axis? Because reading across, I've got my 50%, but I need to find out what amount that is.

Well, let's find out those intervals.

Well, we know there are 10 intervals between the zero and the 200.

So I'm gonna simply divide my 200 by 10, thus identifying each interval to be 20.

All I'm going to do is label it 20, 40, 60, 80, and 100, which tells me what my amount is as a percentage.

It's always important to interpret and use the scale correctly.

Now let's have a look at another check question.

Here you are using the graph to find 40% of 64.

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, first things first.

Hopefully, you can spot on that y-axis, which goes from zero to 100% and there are five equal intervals.

So all I'm going to do is divide a hundred by five, thus telling me each interval represents 20.

Now remember, the question wants us to find 40%.

So identifying 40% here, I need to find out what this value is.

Well, to do this, I need to work out what my scale is on my x-axis.

Well, to do this, if you notice the scale on the x-axis goes from zero to 64, and we have five equal intervals.

So 64 divide by five means each is 12.

8.

So I've labelled it here.

Now we can extract this information quite clearly.

40% of 64 is 25.

6.

Really well done if you got this one right.

So now what I'm going to do is ask why do you think it might be difficult to state what 59% of 64 is using this graph? I need to have a little think.

Well, how easy and accurate it is to work out the percentage of an amount.

Using the graph really does depend upon the accuracy of the graphs and the scale of its axes.

So very well done if you've recognise this.

So now it's time for your task.

All I want you to do in question one is to match the double number line with the corresponding graph.

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 shows a graph and you can see from the graph the y-axis represents 100%, and the x-axis represents the mark.

Now, question two states, a class of students took a test and the grades available were either A, B, C, or D, and the graph helps the teacher convert a score out of 65 into a grade.

Now Jun scored 35 marks on his assessment.

Question 2a asks, what grade did Jun get? B asks, estimate his percentage score.

C asks, estimate the minimum mark you need to get a grade.

What calculation could you carry out to be more accurate? See if you can give it a go and press pause if you need more time.

Well done, let's move on to question three.

Question three says, we have to fill in the missing information given the graph and our double number line.

So you can see we're missing some information on our double number line and we have to extract it from our graph.

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

Great work, so let's move on.

For question 3b, the same again.

What we have to do is find that missing information, give them the graph, and/or double number line.

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

Great work.

Let's see how you got on.

You should have paired up the first graph with that middle double number line.

Hopefully you can spot the 80 pounds corresponds with the 100%.

For the middle graph, hopefully you can spot it's the first double number line and that's because the 80% corresponds with that a hundred pounds.

And for the last one, hopefully you can spot it's the very last graph, 80% corresponds with that 50 pounds.

Well done if you got this one right.

For question two, we're asked, when Jun got 35 marks, what grade did he get in his assessment? Well, 35 marks is a grade C.

For part B, we're asked to estimate what his percentage score could be.

Well, Jun scored 54%.

An estimate between 50 and 60% is absolutely reasonable.

So well done if you got this one right.

Question 2c wants us to estimate the minimum mark that we need to get a grade and also it asks what calculation would we carry out in order to be more accurate.

Well, the minimum mark that's needed is 13 marks.

You can see on the graph that would be 13 marks to get a grade.

And the calculation is equivalent to 65 multiply by not 0.

2 as this gives us 13.

Great work if you got this one right.

For question 3a, let's see how you extracted the results from the graph and popped it into our double number line.

Hopefully you can spot on the first graph, 180 pounds is 100%, and that means 40% represents 72 pounds.

Well done if you got this one right.

The second graph shows, well, 100% has to be 60 pounds, so that means 20% would have to be 12 pounds.

Well done if you got this one right.

For 3b, extracting information from our graph, we can't see what 100% represents, but what we can see is what 180 pounds represents and that represents 40%.

So that means we can work out 100% to be 450 pounds.

Well done.

Next.

Well, hopefully you've spot we have 100% is equal to 240 pounds, but our graph does not go all the way to 240 pounds.

So let's see if we know what a third or 33.

3 recurring percentage of our amount.

Well, that would be 80 pounds.

So we can plot that for sure.

So 80 pounds is 1/3 of our 100%.

So we can draw our graph from here.

Really good work if you've got this one right.

It was tough.

Great work, everybody.

So let's move on to the second part of our lesson.

We'll be looking at other representations of proportional relationships.

You can use a ratio table to work with proportional relationships.

For example, let's have a look at this ratio table.

Here we have two pound is 10% and 10 pound is 50%.

Let's find out why.

Well, you could multiply by five here and multiply by five here, thus giving us are 10 pound is 50%.

Now we can do this because the multiplicative relationship is constant, therefore the amount and percentage are directly proportional to each other.

Let's have a look at the same table.

You could also multiply the amount by five.

You can spot we still get the same values.

Same again, this is true horizontally because they are proportional and there is a constant multiplicative relationship.

We can form these quantities in our ratio table either looking at it horizontally or vertically.

And that's what's fantastic about ratio tables.

So let's have a look at our question.

How would you show 40% of 720 on a double number line? Well, using our percentages are top line and the amount is our bottom line.

You can see we're lining up that zero amount and that 0%, and we're lining up that 100% with 720.

Just like we did at the start, let's work out 10%.

Well, 10% is simply done by dividing by 10.

So 10% would be 72.

Now from here, we can work out 40% by simply multiplying by four.

Multiplying our 10% by four gives me 40%, thus multiplying our 72 by four gives us 288.

So that means we've identified 40% to be 288 using our double number line.

Let's see how we can show this in a ratio table.

Well, we know 720 represents 100%.

We found 10% by dividing by 10.

From here, we found 40% by simply multiplying by four.

This now shows 40% is 288.

So you can see how we can use a double number line or a ratio table to show the exact same information.

Now, let's have a look at a graph.

We can also graph this proportional relationship.

We know 720 is 100%.

So I'm going to plot it here.

We know 72 is 10%, so I'm going to plot it here.

And we also know 288 is 40%.

So I've plotted it there and I've drawn a lovely straight line.

This shows our proportional relationship.

But what's so important to remember is working out the percentage of an amount using the graph is dependent upon the accuracy of the graph and its axis.

So what we've done is we've looked at three representations showing the same proportion of relationship.

We've looked at a double number line, we've looked at a ratio table, and now we've shown this proportional relationship on a graph.

Now it's time for a quick check.

What I want you to do is fill in the ratio table using the graph and then use the ratio table to work out 65%.

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, hopefully, you can see from the graph, 100% represent 330.

So that means I'm going to use my ratio table and find out what 10% is.

10 percent's always a good starting point.

To do it, we divide by 10.

This giving me 33 as my 10%.

From here, I'm gonna find 50%.

There's lots of different ways you could do it.

So I multiply my 10% by five.

So that means I multiply my 33 by five to give me 50% is 165.

Now I'm gonna find 5% by simply dividing the 50% by 10.

So that means I'm dividing 165 by 10 to give me 16.

5.

Simply summing up the 50%, the 10%, and the 5% gives me my 65%, which is 214.

5.

Great work if you got this one right.

But why do you think I used a ratio table to get this amount rather than using the graph? Well, hopefully you spotted it's because the accuracy of the graph is really dependent upon the axis.

A ratio table isn't depending upon the axes.

The ratio table uses that multiplicative relationship as a row or column and can help you find the accurate answer.

So now it's time for your task.

Question one wants you to write what you think are the advantages and disadvantages of using a graph, double number line, and a ratio table when trying to find a percentage of an amount.

See if you can copy the table and fill it in and press pause if you need more time.

Well done.

Let's move on to question two.

Question two wants you to match the ratio table with the correct graph and work out that missing amount using the ratio table please.

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

Great work, everybody.

Let's go through our answers.

Well, for question one, here are some examples of some answers.

The advantage of using a graph is it's a really good visual representation.

An advantage of using a double number line is the same again.

It's a really good visual representation of those percentages.

And the advantage of using a ratio table is it's so concise, neat, and tidy.

Now, what are the disadvantages in using a graph? While it does take time to draw and the answers are dependent on the accuracy of the scale.

A disadvantage of the number line, well, it does take time to draw as it is scaled.

And a disadvantage of using the ratio table, well, it doesn't show the visual representation much like a graph or a double number line.

Well done if you've got any of those.

Moving on to question two, we're asked to match the ratio table with the correct graph and work out the missing amount using the ratio table.

Let's have a look at our ratio table first.

Well, if you know 100% represents 110 as an amount, I think sometimes finding 10%, 50%, 1% is really helpful.

So let's have a look at 50% first.

Well, 50% would simply be dividing 110 by two.

To work out 5%, I divide by 10.

Thus to work out 55%, I'm summing the 50% and the 5%, thus giving me 55%, which is 60.

5.

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

For the middle question, 100% represents 55.

So I'm going to work out 50%, which is 27.

5%, 5% which is 2.

75, and that same again, I can work out 55%, which is 30.

25.

Lastly, we were given 120%.

So I can divide by 10, which gives me 12%.

12% is 10.

Then to work out 36%, I simply multiply by three.

Massive well done if you got this one right.

Remember, there were lots of different ways in which you could have worked at the percentage, but as long as you get the right answer, that's really all that matters.

So now let's pair it with the right graph.

Here, you should have the first table paired up with the first graph, the middle table paired up with the last graph, and the last table paired up with the middle graph.

Really well done if you've got this.

Excellent work, everybody.

So let's have a look at what we've done.

Well, we know fractions, percentages, and ratio all show proportion, and we know proportion can be represented in a number of different ways.

And in this lesson, we really did focus on graphs, double number lines, and ratio tables.

And we've considered the advantages and disadvantages of each representation.

A huge well done, everybody.

It was great learning with you.