1. The subject of a formula
Here is a formula you may know from physics:
V = IR
where V is voltage, I is current and R is resistance.
V is called the subject of the formula.
The subject of a formula always appears in front of the equals sign without any other numbers or operations.
Sometimes it is useful to rearrange a formula so that one of the other variables is the subject of the formula.
Suppose, for example, that we want to make I the subject of the formula V = IR.

2. Changing the subject of the formula
V is the subject of this formula
The formula:
V = IR
can be written as a function diagram: I
× R V
The inverse of this is: I
÷ R V
Ask pupils: what do we do to I to get V and establish that we multiply it by R?
Reveal the first diagram showing the operation × R.
Ask pupils how we can find the inverse of this.
Reveal the second diagram corresponding to V ÷ R = I which gives us the formula I = V/R.
Give a numerical example. For example, ask pupils to give you the value of I when V=12 Volts and R=3 Ohms.
Ask pupils how we could make R the subject of the formula (R = V/I).
So:
I is now the subject of this formula
I = V R

3. Matchstick pattern Look at this pattern made from matchsticks:
Number, n 1 2 3 4
Number of
Matches, m 3 5 7 9
The formula for the number of matches, m, in pattern number n is given by the formula:
Go through each step on the slide and then ask,
If we are given m, in this case m = 47, how can we find n?
What have we done to n?
Establish that we’ve multiplied it by 2 and added 1 and ask,
What is the inverse of this? How do we ‘undo’ times 2 and add 1. Remember, we have to reverse the order of the operations as well as the operations themselves.
Establish that we need to subtract 1 and divide by 2. (47 – 1) ÷ 2 is 23.
We can check that this is correct by verifying that 2 x = 47.
m = 2n + 1
Which pattern number will contain 47 matches?

4. Changing the subject of the formula
m is the subject of this formula
The formula:
m = 2n + 1
can be written as a function diagram: n
× 2
+ 1 m
The inverse of this is: n
÷ 2
– 1 m
We can rearrange this formula using inverse operations.
Writing the formula as n = (m – 1)/2 allows us to find the pattern number given the number of matches.
n is the subject of this formula or
n =
m – 1 2

5. Changing the subject of the formula
To find out which pattern will contain 47 matches, substitute 47 into the rearranged formula.
n =
m – 1 2
n =
47 – 1 2
n = 46 2
Check the solution by substituting 23 into the original formula m = 2n + 1 to get 47.
n = 23
So, the 23rd pattern will contain 47 matches.

6. Changing the subject of the formula
We can also change the subject by performing the same operations on both sides of the equals sign. For example, to make C the subject of
F = 9C 5
subtract 32:
F – 32 = 9C 5
multiply by 5:
5(F – 32) = 9C
This formula converts degrees Celsius to degrees Fahrenheit. This slide demonstrates how to change the subject of the formula by performing the same operations on both sides.
Ask pupils how we could rewrite the formula using functions. We could start with the input C, multiply it by 9, divide it by 5 and add 32. The inverse of this is to start with F, subtract 32, multiply by 5 and divide by 9.
Remind pupils that we are trying to rearrange the formula so that the C appears to the left of the equals sign on its own.
divide by 9:
5(F – 32) 9
= C
5(F – 32) 9
C =

7. Change the subject of the formula 1
Ask a volunteer to come to the board and use the pen tool to change the subject of the given formula.
Ask them to justify each step in their working.

8. Find the equivalent formulae
Ask pupils to decide which of the formulae shown can be rearranged to give the formula shown.

9. Formulae where the subject appears twice
Sometimes the variable that we are making the subject of a formula appears twice. For example,
S = 2lw + 2lh + 2hw
where S is the surface area of a cuboid, l is its length, w is its width and h is its height.
Make w the subject of the formula.
To do this we must collect all terms containing w on the same side of the equals sign.
We can then isolate w by factorizing.

10. Formulae where the subject appears twice
S = 2lw + 2lh + 2hw
Let’s start by swapping the left-hand side and the right-hand side so that the terms with w’s are on the left.
2lw + 2lh + 2hw = S
subtract 2lh from both sides:
2lw + 2hw = S – 2lh
factorize:
w(2l + 2h) = S – 2lh
Notice that we do not factorize 2lw + 2hw completely. We only take out the w to isolate it.
w =
S – 2lh
2l + 2h
divide by 2l + 2h:

11. Formulae involving fractions
When a formula involves fractions we usually remove these by multiplying before changing the subject.
For example, if two resistors with a resistance a and b ohms respectively, are arranged in parallel their total resistance R ohms can be found using the formula, 1 R = a + b aΩ bΩ
Although R is on the left-hand side of this formula it can be rearranged to be in the form R = …
Make R the subject of the formula

12. Formulae involving fractions1 R = a + b
multiply through by Rab: = +
Rab R a b
simplify:
ab = Rb + Ra
factorize:
ab = R(b + a)
divide both sides by a + b:
= R ab
a + b
R = ab
a + b

13. Formulae involving powers and roots
The length c of the hypotenuse of a right-angled triangle is given by
c = √a2 + b2
where a and b are the lengths of the shorter sides.
Make a the subject of the formula
square both sides:
c2 = a2 + b2
subtract b2 from both sides:
c2 – b2 = a2
square root both sides:
√c2 – b2 = a
a = √c2 – b2

14. Formulae involving powers and roots
The time T needed for a pendulum to make a complete swing is
T = 2π l g
where l is the length of the pendulum and g is acceleration due to gravity.
Make l the subject of the formula
When the variable that we wish to make the subject appears under a square root sign, we should isolate it on one side of the equation and then square both sides.

15. Formulae involving powers and roots
divide both sides by 2π: T 2π = l g
square both sides: T2
4π2 = l g
multiply both sides by g:
T2g
4π2
= l
l =
T2g
4π2

16. Change the subject of the formula 2
Ask a volunteer to come to the board and use the pen tool to change the subject of the given formula.
Ask them to justify each step in their working.