Lesson video

Hello, I am Mr. Gratton, and thank you all so much for joining me for this lesson on Similarity, where we will look at a range of different contexts that involve direct proportion.

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

First of all, let's have a look at some general contexts for direct proportion.

Jacob here is absolutely correct.

We can find examples of direct proportion wherever we look.

For example, in a supermarket, three lemons cost 75 p.

Lemons and cost are directly proportional.

So we can use this information to calculate the price of 10 lemons.

And Sam prefers to use ratio tables to show proportional thinking like so.

Three lemons cost 75 pence, and so the 10 lemons goes in the same column as the three lemons.

When two things or variables are directly proportional to each other, there is a multiplicative relationship in two different ways.

Way one is between the two variables.

The multiplicative relationship between lemons and cost is 75 over three, which simplifies to a multiply by 25.

This means that the cost of no matter how many lemons in pence is 25 times the price of the number of lemons that you are buying.

So for 10 lemons the cost is 10 times 25, or 250 pence in total.

But there is also a relationship between two different quantities of one variable.

In this case, the price of 10 lemons is 10 over three times the price of three lemons.

So for 10 lemons the price is 75 times by 10 over three, which gives us the same total price of 250 pence.

I'm convinced that since calculations from both directions give the exact same answer, that my calculations are correct and my conclusion that 10 lemons costs 2.

50 pounds is also correct.

Calculating your answer using only one of the two forms of multiplicative relationship is absolutely fine, and sometimes using one direction is more straightforward to calculate or spot than the other direction.

It is down to you to spot the most efficient way yourself for any given situation.

So we can also show direct proportion between two variables on a straight line graph.

Properties of a straight line graph that shows direct proportion include; the line must either start at or pass through the origin, the coordinates 0,0.

This means that zero lemons costs zero pence.

That makes sense.

We can then use the graph to calculate the price of any quantity of lemons, in this case, eight lemons costs 200 pence, or two pounds.

And vice versa, how many lemons can be bought for a given price? 2.

75 pounds will pay for 11 lemons.

Right, for this first in a series of checks, pause here to find the multiplier between two different quantities of British pounds.

From four to 52 is a multiplier of 52 divided by four, or 13.

Onto the next check.

With this currency conversion, knowing that the multiplier from four pounds to 52 pounds is a multiply by 13, pause here to calculate the equivalent amount to 52 pounds in US dollars.

We take the $5 and we multiply it by the multiplier of 13 to give us $65.

Here's another currency example, but this time with Malaysian Ringgits.

One pound is equal to six Ringgits.

By placing the 52 pounds 50 pence into the correct part of the ratio table.

Pause here to calculate how many Ringgits, 52 pounds and 50 pence is worth.

52 pounds and 50 pence goes in the pounds column because the money is currently in pounds.

The multiplier from pounds to Ringgits is a multiply by six as we can see in the top row.

This means our pounds value, no matter how much we are looking at, is a multiplier by six to convert it into its Ringgits equivalent.

So six times 52.

5 equals 315 Malaysian Ringgits.

Sticking with the one pound equals six Ringgits, how many pounds can be exchanged for 531 Ringgits? Pause here to consider how we can modify the use of the multiply by six multiplier for this question.

If the multiplier from pounds to Ringgits is a multiplier by six, then the multiplier from Ringgits back into pounds is divided by six or times by 1/6.

531 divided by six is 88 pounds and 50 pence.

For this next currency example, this graph shows the conversion from British pounds to Philippine Pesos.

Pause here to identify 35 pounds on the x-axis and use this information to calculate how many Pesos this is equivalent to.

Going vertically up then reading to the left gives us approximately 2,500 Pesos.

Onto the same again, but this time in the opposite direction.

Pause here to identify 1,600 pesos on the y-axis, and then use this information to identify how many pounds this is equivalent to.

Read right onto the line, and then down.

We get a value somewhere near the middle of 20 pounds and 25 pounds.

So any answer between 22 pounds and 23 pounds is absolutely sensible.

And lastly, pause here to identify the sentence that explains why this graph does not show direct proportion.

All direct proportion graphs must pass through the origin, the coordinates 0,0.

Right, let's have a look at some misconceptions with direct proportion.

Jacob claims that these two ratio tables both show examples of direct proportion because they both show a relationship that's the same across two variables, the two columns in each ratio table.

Actually only the first of the two tables shows direct proportion.

This is because two variables must share the same multiplicative relationship.

It cannot be an additive relationship that the second table implies.

For example, if this variable is multiplied by three, then this other variable is also multiplied by three.

This is an example of direct proportion because both variables follow this same multiplicative relationship.

However, if this variable is multiplied by three, then this variable is not multiplied by three.

6.

5 multiplied by three is not 12.

5.

Therefore there isn't a directly proportional relationship between the two variables.

Just because we have a consistent plus six relationship does not mean that the relationship leads to direct proportion.

Okay, let's see if you can put the idea of multiplicative relationship into action.

Pause here to identify which of these ratio tables shows direct proportion through a multiplicative relationship.

Only A, C, and E do.

Sometimes a context might look like there is a directly proportional relationship when actually the variables involved are not directly proportional to each other.

For example, a taxi charges 1.

50 pounds for every one minute in the taxi, plus a fixed fare of five pounds at the very beginning of the journey.

Let's see this context in action.

For one minute in the taxi, we have one lot of 1.

50 pounds plus that five pound fixed fare for a total of 6.

50 pounds.

For two minutes in the taxi, we have two lots of 1.

50 pounds, plus that fixed five pound fare for a total of eight pounds.

The time in the taxi has doubled, so we expect the total fare to also double if there was a directly proportional relationship.

However, eight pounds is definitely not double 6.

50 pounds so, as the amount of time doubles, the total fare does not double.

Therefore, the time in the taxi and the total fare are not directly proportional to one another.

Well why is this? Because of that fixed five pounds at the beginning of the journey, that does not change regardless of how long or short a time you spend in the taxi.

That fixed charge makes it a non-directly proportional relationship.

If there was no five pounds fixed fare, then there would be a directly proportional relationship between the time and the total fare.

We can represent this context on a graph.

Whilst this is a straight line graph, it does not pass through the origin, and so it is definitely not a directly proportional relationship.

A taxi journey of zero minutes would still cost five pounds rather than the zero pounds that a directly proportional relationship would have.

For this check, both Jacob and Sam buy bananas and a carrier bag at the supermarket.

The results for bananas and the total cost are represented in this table.

Pause here to identify whether the number of bananas and the total cost that Jacob and Sam spent are directly proportional or not.

They are definitely not directly proportional to each other.

And pause here to analyse why the price of the bananas and the total amount they spent are not directly proportional to each other.

Double five bananas is 10 bananas, but double one pound is not 1.

70 pounds.

And even though Sam bought more bananas, they only bought one carrier bag.

Like Jacob, this means the price of the bag was fixed, and not directly dependent on the number of bananas bought.

A different non-example of direct proportion is this.

A population of 100 ants increases each week by 20%, compared to the previous week.

Representing this information on a table, we have 100 ants to begin with, which increases by 20% one week later.

This is the same as multiplying the previous week's population by 1.

20, giving 120 ants.

Similarly for week two, which will be 20% higher than 120 gives us 144 ants.

As the number of weeks increases by one, there is a multiplicative relationship of 1.

2 for the number of ants.

Whilst per one week that passes, the number of ants increases by that constant multiplier of a multiply by 1.

2, we can see that the time taken doesn't have a multiplicative relationship but rather an additive relationship of plus one.

Therefore, these two variables are not directly proportional.

One variable has a multiplicative relationship whilst the other has an additive one.

We can also see that there is not a directly proportional relationship straight away by looking at this zero week.

If there was a directly proportional relationship at zero weeks, there would be zero ants, which there's not.

The graph shows the lack of direct proportionality for two reasons.

One, the graph is curved, and the graph does not pass through the origin.

Pause here to explain why the relationship between time and the population of fish is not directly proportional.

The graph is not a straight line, nor does it pass through the origin.

And there is not a constant multiplicative relationship.

Great stuff.

Onto the practise task; For questions one and two, complete the ratio tables for these two contexts that have variables that are definitely directly proportional to each other.

Pause now for questions one and two.

Next up, question three shows a context of theme park tickets and total price spent on those tickets.

For parts A and B, assume that these variables are directly proportional to each other, but for part C, come up with a possible explanation for why ticket number and price may not be directly proportional.

And for question four, by first calculating the cost of a shed, explain whether this context is directly proportional or not.

Pause now for questions three and four.

And finally question five, pause here to analyse this straight line graph.

Amazing effort on understanding these contexts.

Onto the answers for question one.

The price of 12 bottles is six pounds and 12 pence, and the price of two bottles is just over one pound at one pound and two pence.

For question two, 63 euros equals 54 pounds, and 93 pounds and 24 pence equals 180 euros and 78 cents.

And one pound is equivalent to one euro and 17 cents.

For question three, four tickets costs 166 and 50 pence, and for three B, 286 people visited the theme park.

For part C, it is possible that adults and children's tickets cost different amounts, or there may be family discounts.

There are also other valid answers.

For question four, pause here to check the answers and explanations on screen.

For question five A, in eight minutes, approximately 270 to 280 litres of water is displaced.

For Part B, 200 litres of water would take approximately 5.

7 to 5.

9 minutes to displace.

For part C, this straight line graph also passes through the origin, meaning there is a directly proportional relationship.

In 28 minutes, approximately 960 litres of water is displaced.

To show this, I chose seven minutes, which was a value on the graph that was also a factor of 28.

Now that we've looked at some general contexts for direct proportion, let's look at some context that involve shape.

Converting a length from one unit of measurement to another is an example of direct proportion.

The same is also true if we wanted to convert from one unit of area to a different unit of area, or unit of volume to volume.

We can convert the side lengths of this rectangle into metres using a ratio table.

Centi metre means 1/100th of a metre, so one centimetre means one lots of 1/100th of a metre.

Therefore 420 centimetres means 420 lots of 1/100th of a metre.

The multiplier from centimetres to metres is 100 times smaller, or a divide by 100, giving a conversion into metres of 4.

2 metres.

Here we have a congruent rectangle with its width written in metres, not the centimetres written above.

The same conversion can be applied to the height of 150 centimetres.

Remember that directly proportional relationships can use the multiplicative relationships in any direction.

So 150 divided by 100 is the calculation, either going left to right, or top to bottom in that ratio table giving a conversion of 1.

5 metres as the height of that congruent rectangle.

Notice how we can convert each length from centimetres to metres by dividing each length by 100.

This is an example of direct proportion.

Right, but will it be the same with area? Well, the area in centimetres squared is the width times by the height of 63,000 centimetres squared.

Whilst the area in metre squared is oh, 6.

3 metres squared.

The multiplier is in fact 1/10,000, or a divide by 10,000.

This is not the same multiplier as when converting the lengths, centimetres to metres.

Converting from centimetres to metres and centimetres squared to metres squared are two separate examples of direct proportion.

However, lengths and areas have different multiplicative relationships from each other, and so the two sets of conversions are not directly proportional to one another.

For this check, pause here to complete the ratio table and find the length of the side labelled A in centimetres.

A has a length of 4,000 centimetres.

And now using the information gained from that ratio table, find the length of the side labelled B also in centimetres.

Sticking with these rectangles, the perimeter in centimetres is 14,400 centimetres.

Pause here to identify the calculation for the perimeter in metres.

From centimetres to metres is a divide by 100.

And now onto the area in metres squared, the area is 1,280 metres squared.

Pause here to write down a multiplication and answer for the area in centimetres squared.

1,280 times by 10,000 is 12,800,000 centimetres squared.

Notice how we multiplied by 10,000, not 100, because areas follow a different relationship than lengths.

Right, let's bring this all together.

Pause here to find the area of this triangle in centimetres squared.

That's 300 centimetres squared.

Now using the relationship between areas, write down a calculation and answer for the area of that same triangle, but now in metres squared.

From centimetres to metres is a divide by 100, and from centimetres squared to metres squared is a divide by 10,000.

So 300 divided by 10,000 is 0.

03 metres squared.

The relationship between the angle of a circular sector and its arc length is an example of direct proportion, assuming the radius remains the same.

Let's show this on a ratio table.

An angle of 40 degrees results in an arc length of four pi.

Triple that angle to make 120 degrees results in an arc length of 12 pi.

Both variables have been multiplied by three, therefore the angle and arc length are directly proportional.

On the other hand, if we look at the relationship between the angle and the perimeter of a circular sector, not its arc length, we have a non-example of direct proportion, let's have a look.

The perimeter of the sector of 40 degrees is an arc length of four pi, plus two radii at 18 centimetres each for a total of four pi plus 36.

For the sector of 120 degrees, we have its arc length of 12 pi plus the same 36 for the two radii, which have not changed.

As we can see, the perimeters do not follow the same multiplied by three relationship of the angles, and therefore there is not a directly proportional relationship.

This is because of those two fixed radii that are part of the perimeter of that sector.

The radius's length does not change as the angle of the sector changes.

We've seen similar fixed parts of a relationship in the previous cycle, such as the fixed price of a carrier bag, or the fixed fee in a taxi fare and so on.

And so here we can see two straight line graphs, one for the arc length and one for the perimeter.

The arc length and perimeter of this 40 degree angle sector can be seen here.

And for the 120 degree sector, here.

The arc length graph is straight and passes through the origin, so there is direct proportionality.

However, the perimeter graph intersects at 0,36, not 0,0, that 36 being the sum of the two radii, 18 and 18.

For this check, a sector with fixed radius has an angle that increases from 70 degrees to 350 degrees.

Pause here to calculate the arc length of that sector.

The angle has been multiplied by five, and so will the arc length.

So 1,100 times by five is 5,500.

For a different sector with perimeter this time of 211 centimetres, the angle increases from 20 degrees to 80 degrees.

Pause here to choose the correct statements.

The angle and perimeter are not directly proportional, so if the angle is multiplied by four, the perimeter will not be multiplied by four.

Right, let's now have a look at the directly proportional relationship between angle and area of a sector with a fixed radius of 24 inches using this ratio table.

A sector of 45 degrees has an area of 72 pi.

Whilst a sector of 180 degrees has an area of 288 pi.

Both angle and area have a multiplicative relationship of a multiply by four, so there is direct proportionality.

On the other hand, the variables of radius and area of a sector are not directly proportional to each other if the angle stays the same.

Here's our calculation for our sector of radius 24 inches and an angle of 45 degrees.

Double the radius but keep the angle the same.

Ah, we also get an area of 288 pi.

The radius has doubled, but the area has been multiplied by four, not two.

There is a relationship between radius and area, but their relationship is non-linear.

A non-linear relationship between two variables means the variables cannot be directly proportional to each other.

Here we have the relationship between angle and area on a straight line graph.

It follows all of the properties of a directly proportional relationship.

So, a 45 degree angle gives a 72 pi area, whilst a 180 degree angle gives a 288 pi area.

But the graph for radius and area is clearly not directly proportional, because it is a curve, not a straight line.

A 24 inch radius gives a 72 pi area, but if we double the length along the horizontal axis to a 48 inch radius, then this more than doubles the height along the vertical axis from 72 pi to 288 pi.

Right, for this check we have some information about a sector whose angle increases from 25 degrees to 200 degrees.

Pause here to find its new area.

The angle is eight times larger, so its area is gonna be eight times larger.

60 times eight is 480 metres squared.

And last check, a sector has a radius of 12 centimetres, but that radius increases to 60 centimetres.

Pause here to identify the correct statements.

Yes, the area will be larger, but the area will not be exactly five times larger.

This nonlinear relationship will result in the area being greater than five times larger.

In fact, it will be 25 times larger.

Right, great effort in analysing all of those shapes.

For question one and two of this practise, pause here to analyse relationships between the lengths and areas of the rectangle and garden.

For question three, we've looked at 2D shapes, but the direct and non-direct proportional relationships extend in the exact same way into three dimensions.

Pause now to use all of the information given to complete the table of information about this cuboid.

And lastly, pause here to use direct proportion to complete the tables of information about sectors X and Z in this question four.

Great stuff, pause here to check your answers for questions one and two.

And pause here to check your answers in the table for question three.

And pause here once more to compare your answers to question four.

And a very well done if you manage to make all of the links between all of these features of these sectors.

Thank you all so much for all of your effort in this pretty diverse lesson where we have looked at direct proportion across a range of real world contexts using ratio tables and knowledge of the multiplicative relationships between variables.

We've also seen that two directly proportional variables can be plot on a straight line graph that passes through the origin.

And lastly, we've looked at direct proportion within properties and features of shapes, and also examples of non-linear relationships between different properties of those shapes.

That is all for this lesson.

I thank you all so much once again for your time and effort here today.

I've been Mr. Gratton, and so, take care and have an amazing rest of your day.