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Hi there, my name is Ms. Lambo.
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 proportionality for directly proportional relationships and that's within the evening, direct and inverse proportion.
By the end of this lesson, you'll be able to use the general form for the directly proportional relationship.
Y equals KX to the power of N and you'll be able to find K.
Quick reminder, I know you won't need it direct proportion.
Two variables are in direct proportion if they have a constant multiplicative relationship, but of course I knew you knew that, today's lesson is split into two learning cycles.
In the first one we will look at direct proportion involving powers and in the second one we will look at direct proportion involving roots.
Let's get going with the first one.
Direct proportion involving powers.
Which of the following equations represent Y being directly proportional to X or a power of X in some way? Also, how do you know Y equals x plus one Y equals two X, Y equals two plus X, Y equals two root X, Y equals x over two and Y equals root two X, Y equals x squared, y equals x squared plus one, y equals two to the power of x and Y equals two cubed x.
Pause the video, decide which ones show Y being proportional to X or a power of X in some way.
What have you decided? Y equals two x plus one, no does not show a directly proportional relationship.
Y equals two x does, Y equals two plus x doesn't, Y equals two root X does, Y equals x over two does, Y equals root two X does, y equals x squared does, y equals x squared plus one doesn't.
Y equals two to the power of X doesn't and Y equals two x cubed does.
How can we tell if variables are directly proportional to each other? How can we tell that? Yeah, they have a constant multiplicative relationship.
Here are the non examples, just four of them.
Ones that do not show that there is a directly proportional relationship to X in some way and here are our examples of ones that did.
Y equals two X, Y equals x over two, Y equals root two X, Y equals two root X, Y equals x squared and y equals two x cube.
I'd like you please to find for me the multiplicative relationship in each of those.
Pause the video, find the multiplicative relationship in each of those and then come back when you're done.
Let's take a look.
First one is X multiplied by two.
Second one is x divided by two, but remember we can write that as X multiplied by half.
The third one is x multiplied by root two.
The fourth one is root X multiplied by two, y equals x squared is x squared multiplied by one and y equals two x cubed is x cubed multiplied by two.
In each of these, the numbers I've highlighted are K, they're the constant of proportionality, they are that multiplicative relationship.
The top three are probably ones that you've been used to seeing before.
In these three equations, Y is directly proportional to X.
I'd like you to have a go at complete in the following.
For Y equals two root X, Y is directly proportional to what? Y equals X squared Y is directly proportional to what and Y equals two x cubed, Y is directly proportional to what? And what did you say? Y is directly proportional to root X, Y is directly proportional to X squared and then X cubed.
Now I'm sure if you didn't quite get those right, you can see now Y we are saying Y is directly proportional to root X but Y equals two root X and so on.
When dealing with equations of direct proportion until now, we've used the equation Y equals KX.
The more general equation is Y equals KX to the power of N.
Since any root can be written as a fractional power, this equation also applies to roots.
Your turn to have a go at this, you're gonna match the following and I've done the first one for you.
M is directly proportional to T cubed, so that is M and then that proportionality symbol T cubed and that equals M equals KT cubed.
I'd like you please to match up the other four, pause the video and come back when you're done.
Let's check those then.
T is directly proportional to the cube root of M, that's the bottom one, and then the equation is the middle one.
M is directly proportional to the square root of T, is the middle one, and then the equation is the bottom one, M equals K root T.
T is directly proportional to the square of M is the second one in the middle column and the second one in the final column and M is directly proportional to the cubed root of T is the top one and the fourth one in the final column.
If you need to pause the video and just take a look and make sure you've matched those up, then obviously you can do that.
A is directly proportional to the cube of B.
Write a statement using the proportionality symbol.
Here are Jun, Izzy, Sam, and Alex's responses.
Who do you agree with? Sam is correct.
If we read the statement, it says A is directly proportional to the cube of B.
If we read Sam's statement A is proportional to B cubed, cube of B is the same as B cubed, I'd like you to write the correct statement for Jun, Izzy and Alex, please, pause the video, write those statements and come back when you're done.
And you wrote for Jun, B is directly proportional to the cube of A, for Alex, A is directly proportional to the cube root of B and Izzy, A is directly proportional to the square root of B.
Did you get those right? Well done.
Here is the graph of Y equals three x squared.
Alex says, surely X and Y are directly proportional because as X increases, so does Y, Jun says yes Alex, but they do not increase at the same rate.
There is not a constant multiplicative relationship between them.
Here's our table of values.
Negative two squared multiplied by three is 12 negative one squared multiplied by three is three and so on.
Let's look for the relationship between X and Y.
You can see negative two multiplied by something is 12, that's negative six.
What do I multiply negative one by to get three? That's negative three.
I don't need to go any further.
I can clearly see that there is not a constant multiplicative relationship between X and Y.
Y is directly proportional to x squared.
Alex says, but here it states they are directly proportional.
How can this be true? We know that graphs showing direct proportion are linear with a positive gradient and a Y in intercept of zero.
Totally agree with everything Alex has said there.
Positive gradient, linear y intercept of zero.
Jun says yes, you are right Alex, and this is clearly not linear.
That graph's not linear is it? No, it's a curve, a very definite curve.
Alex says, I think I know why our graph shows relationship between X and Y, but the statement states Y is directly proportional to x squared.
Alex is right, isn't he? Here, we've plotted X against Y.
Jun says, so if we plotted Y against X squared, do you think it would show a linear relationship? What do you think? Let's plot the graph of y equals three x squared, but instead of plotting X against Y, we'll plot x squared against Y, X squared against Y.
So if x squared is zero, Y is three multiplied by zero, that's zero.
If x squared is one, we end up with three multiplied by one and that's three, x squared is two.
We end up with three multiplied by two, which is six.
Next one would be three multiplied by three, which is nine and then three multiplied by four, which is 12.
If we plot those coordinate points, we can see we end up with a linear graph.
Alex says, now that shows Y is directly proportional to x squared and Jun says, and the constant of proportionality must be three.
Do you agree with Jun? And Jun is correct? The constant of proportionality is three.
It's the value of K, which is the coefficient of x squared, which we can clearly see is three, direct proportion equations always take this form.
Y equals KX to power of N.
This means that we can apply the same process as when one variable is directly proportional to the other.
If we know the values of X and Y, we're able to find the constant proportionality.
Remember that's what K is.
Using the variables X and Y, generally we will consider problems where Y is directly proportional to X, X squared, X cubed.
Let's take a look at some examples.
Y is directly proportional to x squared when X equals five Y equals 75.
Write an equation connecting X and Y.
So we've done this previously or you should have done this previously when we were doing things like Y is proportional to X.
It's just now we are saying it's proportional to x squared.
The rest of the process is exactly the same, therefore Y must be equal to KX squared.
Substitute in the values that we know for X and Y.
Y is 75, X is five, so we got five squared equals K, solved.
Find the value of K.
K is 75 divided by 25.
Where did that 25 come from? Yeah, that's right, it's five squared, so K is three.
And then remember we substitute K into our original equation of proportionality and we end up with Y equals three x squared.
Now we can find any value of X and Y, if we know the other variable, we're going to find Y when X is 12.
Equation of proportionality substituting in the X is 12, Y equals three multiplied by 12 squared, Y equals 432.
Find X when Y is 3.
63.
Given that X is positive, equation of proportionality, 3.
63 we substitute in for Y, which is three multiplied by x squared.
X squared is 3.
63 divided by three.
What do I do if I know what X squared is? If I want to find X, if I'm solving for X, yeah, I find the square root and that is the square root of 1.
21.
X is 1.
1 M is directly proportional to P cubed.
When P is four M equals 96.
Write an equation connecting M and P.
M is proportional to P cubed, therefore M equals KP cubed substitute in our values we end up with 96 equals four cubed multiplied by K solve to find the value of K, K is 96 divided by 64.
The 64 came from.
Yep, you're right 'cause four cubed is 64.
K is 1.
5.
Substitute back into our original equation.
The equation of proportionality is M equals 1.
5 P cubed and again, we can find any value of M or P if we know a value of M or P, find M, when P is 0.
5, we start with our equation of proportionality.
Substitute in P is 0.
5.
We end up with M is three sixteenths.
Find P when M is 12,000.
Given that P is positive, start with our equation.
Substitute in M is 12,000.
Solve to find P cubed.
P cubed equals 12,000 over 1.
5.
Then what we gonna do now? Yeah, you're right, we're gonna find the cubed root of that.
So P equals 20, you could check, couldn't you? It's always worth thinking here you could check if I do 20 cubed multiplied by 1.
5, check that I do get 12,000.
A ball is dropped from the viewing platform on the Eiffel Tower.
After T seconds, the ball has travelled a distance of D metres.
D is directly proportional to T squared.
After two seconds the ball has travelled 19.
2 metres A, how far has the ball travelled after five seconds? B, the height of the viewing platform is 276 metres.
How long will it take for the ball to reach the ground? Give your answer to two decimal places.
We're told that D is directly proportional to T squared, therefore D equals KT squared.
We're also told that after two seconds the ball has travelled 19.
2 metres.
So we're gonna substitute in our values for D and T.
19.
2 equals two squared multiplied by K solved to find the value of K, K is 19.
2 divided by four because two squared is 4, K is 4.
8.
Substitute K into the equation, we end up with D equals 4.
8 T squared.
Now we can use that equation to find other values of D and T.
Here's our equation.
We want to know how far the ball is travelled after five seconds, so that is when T is five.
So starting with our equation of proportionality, substitute in T is five.
We end it with D equals 4.
8 multiplied by five squared, which is 120 metres, after five seconds, the ball has travelled 120 metres and B, we want to know how long it's going to take to reach the ground and we know it's being dropped from a height which is D of 276 metres.
Start with your equation of proportionality.
Substitute in D is 276, T squared, then is 276 over 4.
8.
Then we're gonna take the square root of both sides to give us T.
T is 7.
58.
It will take 7.
58 seconds for the ball to reach the ground, we're gonna have a go at one more together and then you are gonna do one independently and then you'll be ready for task A.
A is directly proportional to B cubed when A equals 32 B equals four.
Write an equation connecting A and B.
So I've written my proportionality statement and then my equation, we substitute A, it's 32 B is four, we end up with 32 equals 64K.
The 64 has come from four cubed, K therefore is 32 divided by 64, K is 0.
5.
And then let's go back and substitute K into our equation.
A equals 0.
5 B cubed, starting with our equation of proportionality, now we can substitute in B is seven and that gives us D is 171.
5 and then starting again with the same equation of proportionality, but this time we know that A is one over 250, we need to solve then to find B cubed is one over 250 divided by five and then it's going to be the cubed root of that B equals 0.
2.
And now your turn, pause the video obviously here it's fine to use a calculator.
Come back when you're ready.
Let's check in then.
You should have K equals three.
And if you need to look back through to find out where that's come from, you can, you could pause the video.
So our equation is A equals three B squared, find A when B is 0.
5, A would be 0.
75 and then find B when A is 1.
08 B equals 0.
6.
And again you could pause the video if you need to look at any of those steps.
Here you go then question number one, pause the video and then come back when you're ready.
And question number two, question three.
Great work.
Let's check those answers question one, A.
The answer is E equals 2.
6 F squared and if you need to pause the video and look through how I got that, you can B is 509.
6 and again the the workings are there and C is 0.
5.
If you need to pause the video and look through 'cause you've made any errors, obviously you can do that.
Question two, A is Y equals four x cubed, B is 3.
2 multiplied by 10 to the power of five or 0.
000032 or one over 31,250 and C is three.
Question three, it would take 13 seconds to reach the ground if you dropped it from the Burj Khalifa, from the Gherkin and it would take six seconds and from the Eiffel Tower it would take eight seconds.
Now let's move on then to looking at direct proportion with roots.
Y is directly proportional to the square root of X, write an equation, we're going to do exactly the same thing as we've just done in that previous learning cycle, but this time we know that Y is proportional to the square root of X.
So we're right.
Our equation is going to be Y equals K root X substitute in the values of X and Y we end up with 17.
5 equals K multiplied by square root of 25.
We solve to find K square root of 25 is five, so K is 17.
5 divided by 5, K is 3.
5.
Then we substitute that back into our original equation and we end up with our equation of proportionality is Y equals 3.
5 root X.
Now we can find any other values of X and Y using that equation of proportionality.
Find Y when X is 2.
25.
Start with our equation of proportionality.
Substitute in X is 2.
25.
Solve to give Y equals 5.
25.
Find x when Y is seven.
root five start with your equation of proportionality.
Substitute in Y is seven root five we end up with root X is seven root five over 3.
5.
I need to square both sides that I got what X is equal to rather than a square root of x and then I get X is 20 and it might be worth at that point, putting it back into the equation and checking you get the right answer.
Y is directly proportional to the cube root of X.
Same thing.
Write your equation of proportionality, as long as you get that right, then you know everything else.
The steps are the same as we've been doing over the previous examples.
Substitute in your values for X and Y, solve to find the value of K.
We end up with K is six and now we can substitute K into Y equals K cube root of X given us our equation of proportionality as Y equals six cube root of X.
Then we can find any values that we might need to find Y when X is 0.
027.
Substitute in X is 0.
027.
We end up with Y equals 1.
8.
Again, start with that equation of proportionality with this time we're substituting in that Y is 1.
5, so the cubed root of X is equal to 1.
5 divided by six, X is equal to 0.
25 cubed.
1.
25 divided by six is 0.
25 given us one over 64.
You may have chosen to write your answer there as a decimal.
G is directly proportional to the square root of H.
Find the missing values in this table.
G is directly proportional to the square root of H, meaning G is equal to K root H substitute in the pair of values of G and H.
Here we know when G is 10, H is 400.
That's the only pair that we know.
That's the pair we need to use.
We're going to now solve K equals 10 over 20.
The square root of 400 is 20, so K is 0.
5.
You could obviously leave that as a half.
The equation linking G and H is G equals 0.
5 root H.
Now we can find what H is when G is one third substitute one third into the equation of proportionality.
We get one third equals 0.
5 root H rearranging this we get root H equals one third divided by 0.
5.
Then we need to square both sides of the equation.
One third divided by 0.
5 is two thirds, H is equal to two-third squared, H is four ninths.
You can fill that into our table.
Now we can find H when G is 0.
75.
Substitute G is 0.
75.
We end up with a square root of H is 0.
75 divided by 0.
5, H is equal to 1.
5 squared.
H is 2.
25.
Fill that in our table and finally we can find G when H is 64.
Substitute H is 64 into the equation of proportionality.
We end up with G is four and now you can have a go at this question.
G is directly proportional to the cube root of H.
Find the missing value in this table.
Pause a video and have a go at this question.
Super work.
Let's check, G is proportional to cube root of H.
G is equal to K cube root of H substituting in G is 21, H is 343.
We end up with a value of K of three, so the equation linking them is G equals three, cube root of H.
When G equals 0.
3, we can solve that equation we end up with H is 0.
01 or one over a thousand.
If you need to, you can pause the video and have a look through my steps of working.
Now we can do task B or I should say you can do task B.
Pause the video, give this question a go and then come back when you're ready.
And question number two, again, pause the video, come back when you're done.
And question number three, pause the video, come back when you've filled in the two missing values in the table.
Well done on those.
Let's check our answers.
Question number one, the answer is to A is E equals 12 cube root of F, B is 96, C is 216.
Pause the video and look through the workings if you need to.
Question two, A is Y equals 0.
6 root X.
B is 13.
8 and C is 1.
96.
G was proportional to the square root of H, not the cube root of H that gave the equation once you'd substituted in your values, G equals four root H, so the missing values in the table when H was 64 over nine, G was 32 over three and when G was 200, H was 2,500.
Let's summarise our learning from today's lesson.
Direct proportion equations always take the form Y equals KX to the power of N.
Remember that N may be a fraction, if we know the values of X and Y, we're able to find the value of K, which is the constant of proportionality, and then we can use this equation to find other variables and there's an example there of one that we went through during this lesson.
Thank you for joining me today.
Hope you've really found today's lesson useful.
Take care of yourself and I look forward to seeing you again really soon.
Goodbye.
