Lesson video

Hi everyone, my name is Miss Ku, and I'm really happy that you're joining me today.

Today, we're going to look at some beautiful representations of direct and inverse proportion.

So let's make a start.

Hi, everyone, and welcome to this lesson on proportion modelled algebraically, under the unit Direct and Inverse Proportion.

By the end of the lesson, you'll be able to identify and solve proportion questions involving algebra.

Let's have a look at some keywords, starting with inversely proportional.

Now, two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.

And also, remember, two variables are in direct proportion if they have a constant multiplicative relationship.

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

Firstly, we'll be identifying different graphs, and then we'll be identifying from context.

So let's make a start identifying different graphs.

Now, understanding key features of graphs is important so you can distinguish between these very different types.

For example, for any value of X, can you pair up the correct graph with the correct equation? Here, we have our equations on the left, and we have our graphs on the right.

See if you can pair them up.

Well done.

Well, hopefully you spotted y equals 3x is a linear graph.

We have y equals four over x is our reciprocal graph, and y is equal to 5x squared shows a quadratic graph.

Well done if you got this.

Now let's investigate how changing a major part of each graph impacts how it looks on the graph.

Let's start with directly proportional graphs.

Here, we know y is directly proportional to x and we're going to change the value of k.

In other words, for any value of x, can you link the graphs with the equations? And I want you to explain how you know, given the scale for each is the same.

See if you can give it a go.

Press pause if you need more time.

Well, hopefully you've spotted y equals 4x is our first graph.

y equals 0.

4x is our second graph, and y equals 40x is the third graph.

So with direct proportion, the constant of proportionality shows the gradient.

Therefore, the bigger the value of x, the steeper the gradient.

Well done if you spotted this.

Now let's change the value for k, for which y is proportional to x squared.

And let's have a look at these proportional graphs.

Here are our equations.

See if you can link the graphs with the equations and explain how you know.

See if you can give it a go.

Press pause if you need more time.

Well done.

Well, hopefully you've spotted we have the first graph is y equals x squared.

The second graph is t equals 10x squared.

And the third graph is y equals 0.

1x squared.

So what happens when we change the value of k? Well, as k increases, the graph of the x squared looks more narrow.

Now let's do a quick check, looking at our quadratic graphs and our linear graphs.

Here's a sketch of y equals 5x squared, and what I'd like you to do is I'd also like you to sketch y equals 10x squared and y equals 0.

5x squared on the same axis.

And then for the question on the right, here is a sketch of y equals 2x, and I want you to sketch y equals pi x and y equals root 2x on the same axis.

See if you can give it a go.

Press pause for more time.

Well done, let's see how you got on.

Well, looking at our quadratic, we have y equals 10x squared.

So what does that look like in comparison to y equals 5x squared? Well, a parabola with a gradient which increases more quickly than y equals 5x squared, and it still must pass through the origin and must show a parabola which is narrower than 5x squared.

Let's have a look at B.

A parabola with a gradient which increases more slowly compared to y equals 5x squared, it still must pass through the origin.

Now it looks wider in comparison to y equals 5x squared.

Really well done if you got this.

Let's have a look at our linear graph here.

We have a sketch of y equals 2x.

So what does y equals pi x looks like? Well, it's a straight line, and obviously it has a steeper gradient than y equals 2x, as pi is a greater number than two.

And obviously it must pass through the origin.

Let's have a look at y equals root 2x.

Well, root two is less than two, so we know it's a straight line, but the gradient is less steep compared to y equals 2x.

And clearly, again, it passes through the origin.

Really well done if you got this.

Now let's change the value for k for inversely proportional graphs.

Here, we know y is inversely proportional to x, and we're going to change the value for k.

And for any value of x, can you link the graphs with the equations? And I want you to explain how you know.

See if you can give it a go.

Press pause if you need more time.

Well done.

Well, let's see how you got on.

Well, hopefully you've made these pairings.

The constant is the dividend, therefore the larger the dividend, the further the reciprocal graph is from the origin when plotted on the same axis.

So comparing our graphs, the first graph shows the constant of proportionality be 1/4.

The second graph shows the constant of proportionality to be one.

And the third graph shows the constant of proportionality to be three.

And when the constant of proportionality gets larger, the further the distance the reciprocal graph is from the origin.

Therefore, the one over 4x shows the reciprocal graph being closest to the origin.

And y equals three over x shows the reciprocal graph which is furthest from the origin.

Well done if you got this.

Now what I'd like you to do is a quick check.

Use each number only once, and insert the following into the sentences below.

The numbers we have are one, two, three and 12.

We know curve A is y equals x squared.

Curve B is y equals 3x squared.

You need to find out what would the constant of proportionality be for C, the constant of proportionality be for D.

And you might notice, we have a point intersection on the curve B and the line E.

Can you find out what those coordinates are? That's a great extension question.

See if you can give it a go.

Press pause if you need more time.

Great work, let's see how you've got on.

Well, for C, you should've had the constant of proportionality is two.

For D, the constant of proportionality is 12.

You can see D is the furthest reciprocal graph from the origin.

And that intersection has the coordinate one, three.

And that coordinate can be worked out by using the equation y equals 2x plus one, and equating it to the curve y equals 3x squared.

Really well done if you got this.

Now let's look at slightly changing the variable.

For any value of x, we know the reciprocal graph generally looks like this.

And an inversely proportional graph can take the form of y equals k over x to the power of n.

Now, when n is even, what do you think the graph would look like? And I want you to explain your answer.

See if you can have a little think.

Well, the graph would look like this.

And the reason why is because when we have an even exponent, the y values can only be positive.

Therefore, in summary, if you know an inversely proportional reciprocal graph can take the form of y equals k over x to the n, and when n is an odd number, the reciprocal graph generally takes this form at the top right of our screen, given the y values can be positive or negative.

And when an inversely proportional reciprocal graph takes the form of y equals k over x to the n, and n is even, that means the y values can only be positive, and is generally in this form at the bottom right of our screen.

So let's see how you get on with a quick check.

On the left hand side, here we have a sketch, y equals two over x to the five.

Using this sketch, I want you to sketch y equals 10 over x to the five and y equals one over 2x to the five on the same axis.

And the question on the right says here's a sketch of two over x to the four, and I want you to sketch y equals the square root of eight over x to the four, and y equals pi over x to the four on the same axis.

See if you can give it a go.

Press pause if you need more time.

Well done, let's see how you got on.

Well, let's have a look at the first one.

Well, hopefully, you've drawn a sketch of y equals 10 over x to the five, giving a reciprocal graph which is further from the origin than two over x to the five.

Now, for y equals one over 2x to the five, hopefully you've spotted that reciprocal graph has to be closer to the origin compared to two over x to the five.

Really well done if you got this.

And for the next one, well, we have two over x to the four is plotted.

y equals root eight over x to the four must be further from the origin than two over x to the four.

And this is simply because root eight is greater than two.

And y equals pi over x to the four must be the furthest from our two over x to the four and our root eight over x to the four.

This is because pi is greater than two and pi is greater than square root of eight.

Really well done if you've got this.

Fantastic work, everybody.

So now let's have a look at your task.

Here, we have lots of beautiful equations and lots of beautiful graphs.

So what I want you to do is match the equation with the graph.

See if you can give it a go.

Press pause if you need more time.

Well done, let's have a look at question two, match the equation with the correct graph, and some cannot be matched.

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

Well done, let's have a look at question three.

When a is greater than c, which is greater than d, I want you to sketch the following on the same axis, given that y is equal to c to the x squared has already been plotted.

I want you to sketch y equals axe squared and y equals dx squared.

See if you can give it a go.

Press pause if you need more time.

Great work, so let's look at question four, which of the following graphs shows the directly proportional relationship of y and the square root of x.

And I want you to explain how you know.

See if you can give it a go.

Press pause if you need more time.

Well done, let's go through these answers.

Well, for question one, here are our equations matched to our graph.

Press pause if you need more time to mark.

Great work, let's look at question two.

Here are all our answers, press pause if you need more time to mark.

And for question three, your graph should look something a little like this.

Hopefully, you've spotted y equals dx squared is the narrowest to them all, and y equals axe squared should be the widest parabola.

Great work, so let's have a look at question four.

Which of the following graphs shows the directly proportional relationship of y and the square root of x, and we're asked to explain how we know.

Well, it has to be B because the x values can only be positive.

There cannot be any negative x values.

Really well done if you got this one.

Great work, everybody.

So let's have a look at identifying from context.

Knowing the difference between directly proportional questions in context and inversely proportional questions in context allows you to have a general idea of the graphical representations of these relationships.

Can you fill in the sentences below to help you spot the differences in context questions? Direct proportion, as one variable increases, the other something proportionally.

Next one, inverse proportion, as one variable increases, the other variable something proportionally.

See if you can give it a go.

Press pause if you need.

Well, hopefully you spotted for direct proportion, as one variable increases, the other increases proportionally.

For inverse proportion, as one variable increases, the other variable decreases proportionally.

Well done if you got that.

So for example, let's match the statement with the graph.

State the equation of the graph, and then we're gonna answer the question.

So here we have a car company, and it has a number of robots are working at the same rate to build a car in n days.

Now, it takes 12 robots to build a car in a day.

How many days will it take three robots to build the car.

Which graph do you think this matches to? As the number of robots increase, it will take fewer days.

Therefore, we know n and r are inversely proportional.

So that means we know it must be either one of these, as they show inverse proportion.

Looking at the axes, we know r cannot be negative, therefore we know this is the only graphical representation of our question.

Now, knowing it will be in the form of n is equal to k over r, as we know the number of robots is inversely proportional to the number of days, we can now substitute what we know.

We know it takes 12 robots one day to build the car, so therefore our equation looks like one equals k over 12 so far.

Working out k, we have k to be 12.

So this means we have our equation of the curve.

It is n is equal to 12 over r.

Now we can answer the question.

We now have our equation to be n is equal to 12 over r, and we know the question wants us to work out how many days will it take three robots to build the car? Well, that means we substitute r to be three, identifying that it'll take four days for our robots to build the car.

Well done, so let's move on to a check.

The mass of an object m given in kilogrammes is inversely proportional to the radius squared, r squared in centimetres of a solid.

Now, when r is equal to two centimetres and m is equal to 3.

25 kilogrammes, we're asked to express m in terms of r.

Then on the same axis, sketch m against r.

Then we're asked to work out the radius when m is 29.

25 kilogrammes.

See if you can give it a go.

Press pause if you need more time.

Well done, let's see how you got on.

Well, first of all, we know m is inversely proportional to r squared, so we can represent it as an equation.

m is equal to k over r squared.

Let's substitute what we know.

We know r is two when m is equal to 3.

25.

So we can work out k to be 13.

Now we've identified our equation.

In other words, we've expressed m in terms of r.

m is equal to 13 over r squared.

Let's have a look at B.

Well, we know it's inversely proportional, so the graph should look like this.

Same again, notice how we did not plot the negative values of r, as we can't get a negative radius of our solid.

Lastly, let's identify the radius when m is 29.

25.

Substituting this into our equation, we have 29.

25 is equal to 13 over r squared.

Making r squared the subject, we can simplify to give 4/9, so therefore r is equal to the square root of 4/9, which gives me the radius to be 2/3.

Once again, we're only using the positive value for r.

Really well done if you got this.

Great work, everybody.

So now it's time for your task.

I want you to match the statement with the graph, state the equation of the graph and answer the question.

The question states that the area a of a shape is inversely proportional to the length l squared.

When l is equal to two centimetres, the area is 144 pi centimetres squared.

And we're asked to work out the area when l is equal to five centimetres, and we're asked to also leave our answer in terms of pi.

See if you can give it a go.

Press pause if you need more time.

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

Some Oak pupils draw a graph of the proportions.

They've all made errors.

Describe their errors and draw the correct graph.

Izzy says, "This shows pounds is proportional to dollars." Jacob says, "This shows pressure p "is inversely proportional to r squared." And Jun says, "This shows the length is proportional "to the square root of the width." See if you can give it a go and press pause for more time.

Great work, let's move on to question three.

The pressure p is inversely proportional to the length squared in centimetres, l squared of a solid.

When l is equal to four centimetres, the value of p is 15 newtons per centimetre squared.

We're asked to express p in terms of l.

Then we're on the same axis, sketch p against l.

And for c, it wants us to work out what the length is when we know p is 0.

6.

See if you can give it a go.

Press pause if you need more time.

Great work, let's have a look at these answers.

Well, for question one, we should have had this graph.

Remember, they show an inversely proportional relationship, and the length can only be positive.

We should have also worked out the equation for a is 576 pi over l squared.

Using the value of l to be five centimetres, we can substitute this into our equation, getting a to be 23.

04 pi centimetres squared.

Really well done if you got this.

For question two, well, let's have a look at Izzy's graph.

For direct proportion, the graph must start from the origin.

So the graph should look like this.

Now, for Jacob, he's saying p is inversely proportional to r squared, but for inverse proportional graphs, the graph must not touch the axis.

So this would be the correct graph.

And for C, well, the width cannot be negative, therefore there will be no graph plotted in that negative x quadrant, so the correct graph should look like this.

Really well done if you got this.

And for question three, great question, we should have worked out the equation to be p is equal to 240 over elsewhere, and the graph shows an inversely proportional relationship between p and l squared, and substituting p to be 0.

6 gives us the length to be 20.

Notice how we can get positive or negative 20.

But given the context of the question, we cannot get a negative length.

Really well done if you got this one.

Great work, everybody.

So in summary, we've looked at different types of graphs of directly proportional relationships and the impact of changing k.

For example, y is equal to kx.

As k increases, the gradient gets bigger.

We've also looked at y is equal to x squared.

And as k increases, the graph looks more narrow.

For inversely proportional reciprocal graphs, it will take the form of y equals k over x to the n.

And the general shape when n is odd is given here, and the general shape when n is even is given here.

Furthermore, we've applied this to context questions.

Great work, everybody.

It was wonderful learning with you.