Lesson video

Hi, I'm Mrs. Wheelhouse and welcome to today's lesson on checking and securing understanding of changing the subject.

This lesson falls within our unit on algebraic manipulation.

Now algebra can be an incredibly useful thing, so let's get started with our lesson and see how we're going to use it today.

By the end of today's lesson, you'll be able to apply an understanding of inverse operations to a formula in order to make a specific variable the subject.

Now what you see on the screen are some key words we're going to be using in our lesson today.

So we're gonna be talking about the subject of an equation or a formula, and that's a variable that's expressed in terms of other variables.

It should have an exponent of one and a coefficient of one, and you can see an example on the screen now.

Our lesson has two parts, we're gonna begin by rearranging simple formula.

We can use a bar model to show additive relationships.

The formula for finding the perimeter of a rectangle with side lengths L and W can be written as P equals two lots of L plus two lots of W, and we can show this with a bar model.

Now from the bar model, can you write another equation connecting the variables? Or we could say that two lots of L is equal to P, subtract two lots of W, or we could say that two lots of W is equal to P subtract two lots of L.

We can generalise the rule for rearranging additive relationships.

If A plus B is equal to C, then we can say A is equal to C, take away B where A is expressed here in terms of C and B.

We could also say that B is equal to C takeaway A, and here we have B expressed in terms of A and C.

In this form, A is the subject of the equation and in the second form, B is the subject of the equation.

Remember the subject of an equation or formula is a variable that's expressed in terms of other variables.

It should have an exponent of one and a coefficient of one.

For example, solving Y plus 14 equals 21.

We can see from our bar model that we can write this equation in different ways.

So you can say that 14 is equal to 21 takeaway y and y is equal to 21 takeaway 14.

It's in this final form that we have Y being the subject of the equation and in the context of this equation, we can immediately calculate the unknown because it is the subject.

In this case, since y is equal to 21 takeaway 14, y must be seven.

We can also rearrange multiplicative relationships.

So if seven multiplied by eight to 56, what other relationships must be true? Well, it must be true that 56 divided by seven is equal to eight and 56 divided by eight is equal to seven.

If you are unsure how to rearrange an additive or multiplicative relationship, you can always try with numerical values first and then generalise.

We can also identify the subject in multiplicative relationships.

So for example, A multiplied by B is equal to C.

We can therefore write C divided by B equals A and B times A is equal to C and C divided by A is equal to B.

For the first one, C is the subject of the equation.

In the second, A is the subject.

In this form, C is the subject and in the bottom form, B is the subject.

In which of these rearrangements of the equation X plus Y equals Z is X the subject? Pause the video and make your choice now.

There's only one and that's B.

It's the only rearrangement where X is expressed in terms of the other variables.

In which of these rearrangements of the equation, 2X subtract 2Z equals 2Y is Z the subject? Pause and make your choice now.

It's only C.

Z is expressed in terms of the other variables, the subject can be on the right hand side of a formula or equation.

It couldn't be A, because the coefficient of said needs to be one.

And it can't be D, because the coefficient of Z remember has to be one and the coefficient here is negative one.

True or false? In the equation C squared equals B squared plus B squared, the subject is C squared.

Now is that true or false? Pause and make your choice now.

You should have picked false and now you need to justify your answer.

Is it A, C squared does not have a coefficient of one or B, C squared does not have an exponent of one? Pause and make your choice now.

You should of course have picked B.

Remember the subject of an equational formula has to have a coefficient of one and an exponent of one and C squared does not have an exponent of one.

The formula for finding the perimeter of a rectangle with side lengths L and W is P equals 2L plus 2W.

How can we rearrange the formula so that W is the subject instead? Well, we can rearrange equations and formula using inverse operations just we did when we were solving an equation.

So let's consider this.

I'm going to add negative 2L to both sides.

I could of course use a bar model to show how I rearrange this additive relationship if I'm not confident doing this.

That leads to P takeaway 2L is equal to 2W.

Now I'm almost there, but the subject has to have a coefficient of one.

So I need to divide through by two.

Now W is the subject of this formula.

Sometimes we will need to use multiple operations.

So how could we rearrange this formula so that A is the subject? Well, I could add 2a to both sides or I could start by subtracting B.

Now Aisha prefers the variables to be positive, so she's going to add 2a to both sides.

This leads to B equals 3c plus 2a.

Now I'm going to add negative 3c to both sides.

So I have B subtract 3c equals 2a, I'm now gonna divide through by two.

So B, subtract 3c all divided by two is equal to A.

A is now the subject of the formula.

We can rearrange multiplicative relationships using the same principles.

What is the inverse of multiplying by T? That's right, it's dividing by T.

So if I divide both sides by T, remembering that I must do this to both sides so that I can maintain equality, I get D divided by T is equal to S.

So S is the subject of this formula now.

We start with D expressed in terms of S and T and we've rearranged to have S expressed in terms of D and T.

This is the formula for calculating the average speed when you know the overall distance and the overall time.

How can we rearrange this formula so it shows how to calculate time? Well, we can write this formula as S is equal to D multiplied by one over T and the reciprocal of one over T is just T.

So multiplying both sides by T gives us the result S multiplied by T is equal to D.

D is now the subject of the formula, but I don't want D to be the subject.

I want T to be the subject.

So let's keep going.

I can divide both sides by S, so T is equal to D divided by S, and now T is the subject of the formula.

Aisha has been revising from her notes on trigonometry.

She knows that the sine ratio has the following multiplicative relationship.

So she's going to write this, as sine theta is equal to O divided by H where O represents the opposite side and H is the hypotenuse.

She wants to rearrange this so that O is the subject.

Well Jacob points out that we can use what we know about multiplicative relationships.

So if sine is equal to O divided by H, then by multiplying both sides by H, I can make O the subject.

Let's make X the subject in the following equation.

Or I can subtract 2Y from both sides and then divide through by four and you can simplify if you wish.

I'm gonna make H the subject now.

I multiply both sides by H and then divide through by cos theta.

It's now your turn.

Please make Y the subject in the following equation.

Pause and do this now.

So you should have started by subtracting 6X from both sides and then dividing by three, which you can simplify if you wish to four subtract 2X.

Now please make H the subject for the following formula.

Pause and do this now.

Welcome back.

You should have multiplied by H and then divided through by sine theta.

So H is equal to O divided by sine theta.

Make X the subject in the following.

Well I can group the Y plus Z and treat that as one term.

Subtracting Y plus Z from both sides leads to X is equal to 10 takeaway Y, takeaway Y.

I'm now going to make H the subject in the following formula by grouping the L and the W together.

Remember, I'm multiplying everything, so I'm gonna divide both sides by LW, meaning that V divided by LW is equal to H.

It's now your turn.

Please make Y the subject.

Pause and do this now.

Welcome back.

You should have subtracted 2X and 3Z from both sides.

Which you can do by grouping them as I've shown and then subtracting the entire lot.

We can then write that as Y equals 12 takeaway 2X takeaway 3Z.

Now please make W the subject.

Pause and do this now.

Welcome back.

You should have divided both sides by LH dividing both sides by LH results in V divided by LH equals W.

It's now time for your first task.

For each equation or formula, please change the subject as you are instructive to.

Pause and do this now.

Question two, make O the subject for both questions.

And questions three, make A the subject for both questions.

For question four, which of your formally would be the most useful for finding the adjacent length if you knew the hypotenuse length and an angle size? Pause while you work on these questions now.

Question five, you're given the formula for the volume of a cylinder.

Now Jacob wants to make H the subject.

Suggest why he might want to do this.

In part B, he started to rearrange, what does he need to do next? And then in C, could he have made H the subject in one step? Explain your answer.

Pause and do this now.

Welcome back.

Let's go through the answers.

For question one, you can see how I made Y, U, F and M the subject in their respective questions.

Feel free to pause the video while you check through my working and compare it to yours.

Then for E and F, you can see how I've made R the subject and Y the subject.

Again, feel free to pause so you can check how I've rearranged and compare it to what you did.

Question two, you had to make O the subject, so you should have reached the following two forms. For question three, you had to make A the subject and you can see what I reached in both cases.

For question four, you had to say which of the formula would be the most useful for finding the adjacent length if you knew the hypotenuse and an angle size.

Well the only formula that involves the adjacent length, the hypotenuse length and the angle size is the one you can see here on the screen.

A is equal to H multiplied by cos theta.

In question five, you had to suggest why Jacob might want to make H the subject of the formula.

If he wants to calculate the height and he knows the volume and the radius, this is a great first step.

Now in part B, he started by dividing both sides by pi, so the next step is to divide by R squared and then he'll have H being the subject.

Now he could have done this in one step, which is what part C asked which grouping pi r-squared together and dividing by this whole expression and that would've got him straight to H is equal to V divided by pi r-squared.

It's now time for the second part of our lesson on further rearranging.

Here's one way to write the formula for the area of a trapezium.

We're going to rearrange this to make H the subject.

Where possible, it is often easiest to start by multiplying so there is no fractional notation in the formula.

So we're going to multiply by two and then we can divide both sides by A plus B.

So we end up with 2a divided by A plus B is equal to H, and H is now the subject.

Of course we could write the formula for the area of a trapezium like this, so instead of dividing everything by two, we're multiplying everything by half.

This means we can group the terms half and A plus B together and rearranging one step by dividing by half of A plus B and Jacob is completely correct.

Now, mathematical convention is to avoid writing fractions as part of the denominator or the numerator.

So we know that dividing by half is the same as multiplying by two.

So actually we can write this as we saw in the very first case.

Now here's one way to write the relationship between temperature in degrees Celsius and temperature in degrees Fahrenheit.

If we wanted to make C the subject, what would be the first step? Well, Aisha think we should start by subtracting 32 and Andeep thinks we should start by multiplying by five.

Who do you agree with? Now you may feel that one of these approaches is a lot easier than the other and if you do think that, then that's the way you should absolutely start.

We are going to have a look at doing it both ways so you can see which one you prefer.

So with Aisha we're gonna subtract 32 from both sides, then multiply by five.

Remember the whole of the formulas we multiplied by five.

So we've used brackets to show this.

We then divide by nine.

We could of course write it as five ninths of F takeaway 32.

Either form is acceptable.

Now Andeep wanted to multiply by five first.

Remember he has to multiply each term by five, which is why the 32 has turned into 160 'cause he had to multiply that by five too in order to maintain the equality.

He then took away 160 from both sides and divided by nine.

Now you might think that what he's got is different to what Aisha got, except if we expand the brackets that Aisha has on the left hand side, If see that you reach 5F, subtract 160 on the numerator.

Therefore the two formula are in fact identical.

Some of the formula we need to rearrange will involve exponents.

So for example, let's make X the subject of Y equals x-squared subtract nine.

We still use inverse operations there's no change here, so we're gonna add nine to both sides.

We then need to do the inverse of squaring.

So remember we need to square root everything on the left hand side and there's our answer.

X is now our subject.

"Oh," says Andeep.

"I wrote my answer as X equals the square of y takeaway nine.

Is this the same thing? Also, I've written a plus minus in front of my square root sign and Andeep hasn't, which one's right? Well, Andeep's rearrangement will not give all the correct relationships for the variables X and Y.

The correct way is to put the plus and the minus in front here.

Now we don't always do this, so why are we choosing to do it here? Well, when we square root a number, there are two possible answers, a positive and a negative value.

For example, if X squared was 16, X could be four or X could be negative four because four times four is 16 and negative four times negative four is 16.

If we look at our original equation and we substitute a value for Y, we can see that X could be five or negative five and therefore we need to make sure we show both possible answers.

We don't have a context to suggest it's only one or the other.

So Andeep points out that if he wants to make X the subject of Y equals 5X squared, it would be X as equal to plus or minus the square root of Y divided by five.

And we can now use inverse operations to check if Andeep is correct here.

And Andeep is correct.

Jacob points out he's been revising the relationship between sides and a right angled triangle.

He knows the formula, A square plus B square equal C squared where C is the hypotenuse.

Now he wants C to be the subject, so he needs the positive or negative square root.

Is Jacob right? Because Laura points out, well, hang on, I just want the positive square root.

So have I written something wrong? Well actually she hasn't.

Laura is in fact correct.

Jacob does not need to have the negative square root here.

Can you explain why? Remember, A and B represent the side lengths of a triangle.

So they have to be positive.

The sum of their squares will also be positive.

The square root of this sum could be positive or negative, but because we are dealing with a triangle, we know the side lengths must be positive.

So therefore the length of hypotheses is found by squaring the sum of the two smaller side squared.

So in general, the length hypotenuse can be found by calculating the positive square root of A square plus B squared.

Now because the square root sign implies the positive root, we don't need to write the plus sign.

So it can be written like this.

Quick check now.

An equation for a parabola is Y equals X squared plus eight.

Which of these is the correct rearrangement to make X the subject? Pause the video and make your choice now.

Welcome back.

You should have picked B, and that's because X could be a positive or negative value for this situation.

The formula for calculating the repayment on a loan after two years is shown.

Andeep has rearranged so that I is the subject.

Can you explain what he's done at each step? And then for part B, can you suggest why he doesn't need the plus minus symbol in front of the square root? So Andeep hasn't shown it.

Why does he not need to? Pause and work on this now.

Welcome back.

So to explain what Andeep's done at each step.

The first thing he did was divide everything by C.

He then found the square root of both sides and then subtracted one.

Now, why did he not need to show the plus minus symbol? So remember, I is an interest rate written as a decimal, interest rates are positive percentages.

Their decimal equivalents are therefore positive values.

So I plus one must be the positive square root.

For our final task.

Question one, I'd like you to rearrange each formula to make A the subject.

Pause and do this now.

Question two, rearrange each formula to make X the subject.

Pause and do this now.

In question three, the formula for the volume of a cylinder is given.

Jacob uses an online equation rearrangement tool to get the following.

Now, Laura got a different answer.

Is the answer from the internet equivalent to Laura's? And then for part B, what is incorrect about the internet answer and why do you think this might have happened? Pause and do this now.

It's time to go through our answers.

For 1a, I've rearranged the A is equal to B over four and then subtract five.

In 1b, I've rearranged that A is equal to B subtract five all divided by four.

And then in 1c, I've rearranged that A is equal to 2C divided by B.

For 1d, I've rearranged that A is equal to 4B subtract five.

In 1e, A is equal to 4B subtract 20 and in 1f, A is equal to five lots of B subtract C all over four.

For question two, you can see how I've rearranged to make X the subject.

You'll notice that in 2c, I did not need the plus minus symbol here because I'm cube rooting and therefore there is only one value it could be.

Feel free to pause the video and check your rearrangements against mine.

Question two parts D, E, and F.

Again, you can see how I've rearranged.

So do feel free to pause the video and check your working against mine.

And finally, question three.

So is the answer from the internet equivalent to Laura's? Well to square root a fraction, you can square root both the numerator and denominator separately.

So that's fine, but the internet version allows for the positive and negative square roots, whereas Laura's is just positive, so they're not equivalent.

Now the internet answer was incorrect and that's because the radius cannot be negative.

Now the internet has applied a process without being able to account for the context of the question.

It's now time to sum up what we've learned today.

The subjects of an equation or formula is a variable that is expressed in terms of other variables.

It should have an exponent of one and a coefficient of one.

Additive and multiplicative relationships can be rearranged to change the subject.

Some formula take multiple steps to rearrange and we can use inverse operations in the same way as we solve an equation.

When we square root a value there are two answers, a positive and a negative one, and the context of the formula or equation dictates whether both values are valid.

Well done, you've a great job today.

I look forward to seeing you for more lessons in our algebraic manipulation units.