Lesson video

Hi there.

My name's Ms. Lambell.

You've made such a fantastic choice deciding to join me today to do some maths.

Come on, let's get going.

Welcome to today's lesson.

The title of today's lesson is Finding the Constant of Proportionality for Direct Proportion, and this is within the unit Direct and Inverse Proportion.

By the end of this lesson, you'll be able to use the general form for a directly proportional relationship to find k.

Two variables are indirect proportion if they have a constant multiplicative relationship, and that's going to be key to what we are doing today's lesson.

I've split the lesson into two learning cycles.

In the first one, we will look at expressing proportionality algebraically, and in the second one we will look at solving problems algebraically.

Let's get going with that first one, expressing proportionality algebraically.

Which of the following equations represent y being directly proportional to some form of x? And I'd also like you to think about how you know, how do you know that? We've got y = 2x + 1, y = 2x, y = 2 + x, y = 2 root x, y = x/2, y = root 2x, y = x squared, y = x squared + 1, y = 2 to the power of x, and y = 2x cubed.

Which of those do you think show that y is directly proportional to some form of x? What did you decide? Well, let's take a look.

The first one doesn't, second one does, y = 2 + x doesn't, y = 2 root x doesn't, y = x/2 does, y = root 2x does, y = x squared doesn't, y = x squared + 1 doesn't, y = 2 to the power of x doesn't, and y = 2x cubed also doesn't.

How can we tell if variables are directly proportional to each other? And I mentioned that when I went through that keyword slide.

Yeah, that's right.

They share a constant multiplicative relationship.

y is directly proportional to x.

Here were our non-examples.

These were the examples of one that were not directly proportional.

And here are our examples of the ones that were.

And they were y = 2x, y = x/2, and y = root 2x.

What is the multiplicative relationship in each of those examples? The first one is x multiplied by 2, the second one is x divided by 2 or x multiplied by 1/2, and the final one is x multiplied by root 2.

We can see here we've got 2, a 1/2, and root 2.

These are k, the constant of proportionality.

These are showing the multiplicative relationship between x and y.

I'd like you to have a go at this check now please.

I'd like you to sort the following into examples and non-examples of equations that represent y being directly proportional to x.

For the examples, write down the value of k.

Pause the video, decide which box each goes into, and if they are examples, then I'd like you to also state the value of k for each.

Pause the video.

Good luck with this.

And then I'll be ready and we'll check your answers when you get back.

Super.

Let's check those answers.

y = 3x/4 is an example.

Did you put it in the example box? Yeah, of course you did.

y = 4x cubed was a non-example, y = 3 to the power of x was a non-example, y = root 3x was an example, y = 3 + 4x was a non-example, y = 3.

4x was an example, y = 3 root x was a non-example, and then y = 4/x was also a non-example.

I then asked you to write down the value of k for the examples.

k for y = 3x/4 is 3/4.

What do we multiply x by to find y? That's 3/4.

k in the second one was root 3, and in the third one was 3.

4.

y is directly proportional to x.

This symbol is the symbol for proportionality and we will use that from now on.

If the variables x and y are directly proportional, we can write the relationship using the proportionality symbol, and this reads, y is proportional to x.

Any two variables can be directly proportional to each other.

a is proportional to b.

y is proportional to x.

We know that any variables that are directly proportional produce a linear graph.

Therefore, it must be of the form y = mx + c where c is zero as the y-intercept of a direct proportion graph is the origin.

When dealing with equations of direct proportion, we use the equation y = kx and you'll be familiar with that from previous learning.

If we know the values of x and y we are able to find the value of k which is the constant proportionality.

Task A.

I'd like you please to pause the video and then to fill in the missing words or letters or statements on the following.

Pause the video and then when you are ready you can come back.

Question number two: You are gonna sort the following into examples and non-examples of equation that represent y being directly proportional to x.

And just as you did in the check for the examples, I'd like you please to write down the value of k.

Again, pause the video and come back when you're ready.

Great work.

Let's check those answers then.

1, a, cost, c, is proportional to boxes, b, c is proportional to b with that proportionality symbol.

b, you are missing e is proportional to f, p is proportional to m, d is proportional to s, and T is proportional to f, we would write as T and then the proportionality symbol f.

Notice, we must maintain the exact letter.

So if it was capital P, it must stay as capital P.

Lowercase f stays as lowercase f.

Question two; examples.

y = x/5, that gives us k is 1/5.

y = 5.

2x, k is 5.

2.

y = root 5x, k is root 5.

And y = 5x/2, k is 5/2.

And our non-examples were: y = 5x + 2, y = 5/x, y = 5x squared, y = 2 + 5x cubed, and y = 5 to the power of x.

How did you get on? Brilliant.

Well done.

Now let's move on then to that second learning cycle, solving problems algebraically.

We know variables other than x and y can be directly proportional to each other.

T is directly proportional to m.

Write a statement using the proportionality symbol.

Aisha says, it is T is proportional to m, the symbol, and Alex says, it is m is proportional to T.

Who do you agree with? They're actually both right.

Both of those expressions are correct, and we will check this with our first example.

T is directly proportional to m when T = 840, m = 350.

We're going to use Aisha's proportionality expression, which was T is proportional to m.

We need to write an equation connecting T and m.

We know that it's directly proportional, so therefore, has the form y = kx, but here we're using the variables T and m instead of y and x.

Therefore, my equation is going to be of the form T = km.

We substitute in our values of T and m.

We know when T is 840, m is 350, so I've substituted those values into my equation.

Now, we can solve to find the value of k.

k is 840 divided by 350, so k is 2.

4.

We now substitute k into our original equation.

This gives us T = 2.

4m.

Now we can find any value of T or m if we know the other variable.

So if I know T, I can find m.

And if I know m, I can find T.

Find T when m is 230.

We've got our equation.

Substitute in m is 230, T is 2.

4 multiplied by 230, so T is 552.

We can also find the value of m when T is given.

T = 2.

4m.

That's our original equation.

Substitute in the value of T, which was for 144, m is gonna be 144 divided by 2.

4, and I've decided this time to write that as a fraction and then I work out that value and that gives me m is 60.

When T is 144, m is 60.

There are Aisha's answers.

Now let's have a look if we used Alex's proportionality expression.

m is proportional to T.

We're gonna do the same thing.

We write our equation.

This time we've got m = kT.

Substitute in the values of T and m.

That gives us 350 = 840k.

We then solve to find the value of k.

k is equal to 350 divided by 840, k is equal to 5/12.

Then substitute k into our original equation, giving us the m is 5/12 T.

Now again, we can find any value of T or m if we know the other variable.

Find T when m is 230.

There's our equation.

Substitute m is 230, get 230 = 5/12 T, T then is 230 divided by 5/12, so T is 552.

We can see we get the same answer as Aisha did.

Now, same equation and we're going to this time substitute in T is 144.

So we end up with m is 5/12 multiplied by 144, which is 60.

And again, we can see that it matches.

It doesn't matter which way round we write that proportional statement.

Now let's compare the two equations.

When m was proportional to T, the equation of proportionality is m = 5/12 T.

When T was proportional to m, the equation of proportionality is T = 2.

4m.

Aisha says, there must be a link between the values of k, and Alex says, I think writing 2.

4 as a fraction will help us find the link.

2.

4 is the same as 24 over 10, which is the same as 12/5.

Aisha says, they are reciprocals of each other.

Yes, you are right Aisha.

And that makes sense, doesn't it? 12/5 is a reciprocal of 5/12.

True or false? If y = 0.

75x, then x = 4/3y.

Make your decision.

Is that true or false? And don't forget, I want that justification.

And you decided that the correct answer was true, and b was the correct justification.

The product of the two values of k is 1.

0.

75, I could write as 3/4, 3/4 multiplied by 4/3, because they're reciprocals of each other, their product is 1.

y is directly proportional to x.

When x = 6, y = 7.

2.

Write an equation connecting x and y.

y is proportional to x.

This means that y is equal to kx.

Substitute in the values of x and y given in the question.

7.

2 = 6k.

Solve to find the value of k.

k is going to be 7.

2 divided by 6.

k is 1.

2.

And then substitute k into the equation y = kx, giving us the equation that links x and y is y = 1.

2x.

Now, we can find any value of x and y if we know the other.

Find y when x is 18.

Start with our equation.

Substitute in x is 18.

We get y = 1.

2 multiplied by 18, y = 21.

6.

Y = 1.

2x.

The equation isn't changed, so it's the same equation linking them.

Substitute in y is 9.

6.

We end up with 9.

6 = 1.

2x.

And then to solve this we get x = 9.

6/1.

2, x = 8.

a is directly proportional to b.

When a = -14.

8, b = -4.

Write an equation connecting a and b.

a is proportional to b.

This means that a = kb.

Substitute in our values of a and b.

We end up with -14.

8 = -4k.

Solve to find the value of k.

k is -14.

8 divided by -4.

k is 3.

7.

Substitute k into a = kb.

Our equation of proportionality is a = 3.

7b.

Now we can find any other variables.

Find a when b is 62.

There's our equation of proportionality.

Substitute in b is 62.

We end up with a is 3.

7 multiplied by 62.

A is 229.

4.

And then find b when a is 3108.

Equation of proportionality, substitute in our known value, which this time is a, we end up with 3,108 = 3.

7b.

So b is 3108/3.

7 'cause remember, we're dividing, and that gives us b is 840.

Let's take a look at another example.

We've still got the a is directly proportional to b.

So therefore our equation of proportionality is going to be a = kb.

Substitute in our values of a and b, which are 10.

4 and 5.

6.

Solve to find the value of k.

k is 10.

4 divided by 15.

6.

k is 2/3.

Remember to write your answer as a fraction if you end up with a recurring decimal or a decimal that is just a bit too long to write.

And then substitute k into the original equation of proportionality a = 2/3 b.

Now we can find other values of a and b.

Find a when b is 1.

41.

Equation of proportionality, substitute in b is 1.

41, a is 2/3 multiplied by 1.

4, a is 0.

94.

And find b when a is 2.

6.

There's our equation of proportionality.

Substitute in a is 2.

6.

b is going to be 2.

6 divided by 2/3 giving us that b is 3.

9.

Let's do one more together and then you can have a go at the one on the right hand side independently.

So I'm sticking with a and b.

We're writing an equation to connect a and b.

a is proportional to b, so a = kb.

Substitute in our values.

We end up with 12.

6 = 37.

8k.

Solve to find K.

k is 12.

6 divided by 37.

8.

This gives us a value k of 1/3.

Substitute that back into the equation.

We end up with a equals, instead of writing k, I write is now is known value, which is 1/3 b.

a = 1/3 b.

Find a when b is 1.

86.

Start with your equation of proportionality.

Substitute in the value for b.

a is 1/3 multiplied by 1.

86, so a is 0.

62.

And the same with the second one.

Start with your equation of proportionality and then we end up with substituting in that a is 41, we end up with 41 = 1/3 multiplied by b.

Therefore b is 41 multiplied by 3, b is 123.

Now you can have a go at this one please.

Pause the video, come up with your equation and then find the value of a when b is 8, and b when a is 4.

02.

Good luck with these.

I'll be waiting for you when you get back.

Great work.

Let's check.

a is proportional to b.

So a = kb.

Substitute in a is 18, b is 12.

Solve to find k.

k is 18 divided by 2, k is 1.

5.

Our equation then linking a and b is 8 = 1.

5b.

Now we can substitute in b is 8, a is equal to 1.

5 multiplied by 8, a = 12.

So when b is 8, a is 12.

And then again using that equation of proportionality, and we know this time we want to find b when a is 4.

02.

Substitute a is 4.

02.

That gives us 4.

02 equals 1.

5 multiplied by b.

Solve that to find b, we end up with 4.

02/1.

5.

You may have written divided by 1.

5 there.

Remember, it doesn't matter.

So we end up with b is 2.

68.

When a is 4.

02, b is 2.

68.

Now we're gonna have a look at this question.

x is directly proportional to y.

y is directly proportional to z.

When x = 10, y = 30.

When y = 12, z = 2.

4.

Find a formula for x in terms of z.

Aisha says, I will work out the equation for x and y.

Alex says, okay, and I will work out the equation for y and z.

Maybe you'd like to do that.

Pause the video and work out the equation connecting x and y and then another equation connecting y and z.

And when you've got that, come back and see if you've got the same as Aisha and Alex.

You should have x = 1/3 y, and y = 5z.

Aisha says, what do we do now? Can you see what Aisha and Alex should do next? Alex says, we know y equal 5z.

Yeah, he's just work that out.

Therefore, we can replace y in the first equation.

Then we will have an equation of x in terms of z.

If y = 5z, which we know from that second equation, then x = 1/3 of y, but we know that y is 5z, so it's 1/3 of 5z, which we can tidy up and write as 5z/3.

We now have an equation connecting x and z.

If x = 5y and y = 2z/3, which of the following are true? Pause the video, work out which of you think are true, making sure you show all of your workings.

Good luck and I'll be here waiting as always for you when you get back so that we can check those answers for you.

What did you decide? Was A true? It was.

B? Yeah.

C? Yeah.

D? Yeah.

Hopefully you didn't get to C and think, oh, all those three are true, so D must have been false.

Hopefully, you continue to work out why they were true.

And they were all true, every single one of them was true.

So well done if you identified that.

And now for Task B, the final task for today's lesson.

Well done for getting this far, just a little bit further to go.

Gonna pause the video and then you're gonna answer these questions and then come back when you're ready.

Well done.

And question three.

You know me, you know I like a decode.

You are gonna decode the message to find out the answer to: Do you know what is odd? For Word 1, you are going to use, y is proportional to x, when x is 6, y is 7.

2.

For Words 2, you're gonna be using a as proportional to b.

And for Word 3, you're going to be using T is proportional to m.

You need to work out the answers, find the letters, rearrange them to answer the question.

Do you know what is odd? Pause the video and then come back when you're ready.

Great work.

Wonder if you're laughing? Probably not.

Question four.

Pause the video, have a go at this one.

Little bit more challenging.

Don't let the fact that there are some surds in there confuse you.

You've got all of the skills you need to be successful at that question.

Good luck.

Super work.

Let's have a look at our answers then.

Question number one.

The equation was y = 3.

5x.

A, y = 22.

5b; B, x was 8.

Question two.

The equation was a = 0.

6b or you may have 3/5 b.

A, the answer was a = 9.

6, and B, the answer was b = 4.

8.

The answers to Word 1 were V, R, E, E, Y, rearranging to "every".

The answers to Word 2 were O, R, T, E, and H, rearranging to make "other".

And Word 3, U, B, N, E, M, R, rearranging to make "number".

Do you know what is odd? Every other number.

Ha, ha, ha.

Funny, funny.

And then finally, question number four.

Final answer is x equals root 3 over 2 z.

Now you may need to pause the video and take a look at each of those steps.

I'm not gonna read through them, that would be a bit confusing.

So if you need to pause the video and then come back because we need to do that summary.

Summarising our learning from today then.

If two variables are directly proportional to each other they share a multiplicative relationship.

When dealing with equations of direct proportion we use the equation y = kx.

Remember that x and y can change.

We might have a and b or f and T.

It really doesn't matter.

If we know the values of x and y we are able to find the value of k, k being the constant of proportionality.

The equation of direct proportion can be used then to find any other values of x and y if we know either x or y.

And there's an example there of one of the questions that we went through during today's lesson.

Great work today, well done.

Hopefully you're still not laughing from my or groaning from my very, very corny joke.

Look forward to working with you again really, really soon.

Take care of yourself.

Goodbye.