The subject of a formula Here is a formula you may know from physics: V = IR 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. Photo credit: © fotohunter, Shutterstock.com Sometimes it is useful to rearrange a formula so that one of the other variables is the subject of the formula.

Changing the subject of the formula Suppose, for example, that we want to make I the subject of the formula V = IR. The formula: V is the subject of this formula V = IR can be written as a function diagram: I × R V The inverse of this is: Teacher notes Ask pupils: what do we do to I to get V? (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). V ÷ R I I is now the subject of this formula So: I = V R

Matchstick pattern Look at this pattern made from matchsticks: Pattern 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: Teacher notes 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 23 + 1 = 47. Photo credit: © Eduardo Rivero, Shutterstock.com m = 2n + 1 Which pattern number will contain 47 matches?

Changing the subject of the formula m is the subject of this formula m = 2n + 1 can be written as a function diagram: n + 1 × 2 m The inverse of this is: m ÷ 2 – 1 n Teacher notes 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. or n is now the subject of this formula n = m – 1 2

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 Teacher notes Check the solution by substituting 23 into the original formula m = 2n + 1 to get 47. Photo credit: © Paul O Connell 2010, Shutterstock.com n = 23 So, the 23rd pattern will contain 47 matches.

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, degrees Celsius can be converted to Farenheit by the formula: F = + 32 9C 5 Make a formula to convert Farenheit to Celsius. subtract 32: F – 32 = 9C 5 multiply by 5: 5(F – 32) = 9C Teacher notes 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 =

Change the subject of the formula Teacher notes 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.

Find the equivalent formula

Formulae where the subject appears twice Sometimes the variable that we are making the subject of a formula appears twice. S = 2lw + 2lh + 2hw For example, h S is the surface area of a cuboid, l is its length, w is its width and h is its height. w Make w the subject of the formula. l 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.

Formulae where the subject appears twice S = 2lw + 2lh + 2hw 2lw + 2lh + 2hw = S swap around both sides subtract 2lh from both sides: 2lw + 2hw = S – 2lh factorize: w(2l + 2h) = S – 2lh divide by 2l + 2h: w = S – 2lh 2l + 2h

Formulae involving fractions When a formula involves fractions, we usually remove these by multiplying before changing the subject. For example, if two resistors with resistance a and b ohms respectively, are arranged in parallel their total resistance R ohms can be found using the formula, aΩ bΩ 1 R = a + b Teacher notes 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.

Formulae involving fractions bΩ 1 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

Formulae involving powers and roots The length c of the hypotenuse of a right-angled triangle is given by c a c = √a2 + b2 where a and b are the lengths of the shorter sides. b 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

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 Photo credit: © Vesa Andrei Bogdan, Shutterstock.com When the variable that we wish to make the subject appears under a square root sign, we should isolate the square root on one side of the equation and then square both sides.

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 Photo credit: © Olga Mishyna, Shutterstock.com l = T2g 4π2

Change the subject of the formula Teacher notes 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.

Writing formulae Write a formula to work out, the cost, c, of b boxes of crisps at £3 each the distance left, d, of a 500 km journey after travelling k km the cost per person, c, if a meal costing m pounds is shared between p people the number of seats in a theatre, n, with 25 seats in each row, r the age of a boy Andy, a, if he is 5 years older than his sister Betty, b Teacher notes For each example, substitute some actual values to help pupils determine the correct operations in each formula. 1. c = 3b 2. d = 500 – k 3. c = m/p 4. n = 25r 5. a = b + 5 6. w = (a + b + c)/3 Photo credit: © Ewa Walicka, Shutterstock.com Photo credit: © Josch, Shutterstock.com Photo credit: © Dallas Events Inc, Shutterstock.com Photo credit: © James Steidl, Shutterstock.com Photo credit: © Sean Prior, Shutterstock.com Photo credit: © Jason Stitt, Shutterstock.com the average weight, w, of Alex who weighs a kg, Bob who weighs b kg and Claire who weighs c kg.

Writing formulae A window cleaner charges a £10 call-out fee plus £7 for every window that he cleans. Write a formula to find the total cost C when n windows are cleaned. Using this formula, how much would it cost to clean all 105 windows of Formula Mansion? Teacher notes C = 7n + 10 If n=105, C = 7 × £105 + £10 = £745. If he earns £94 we need to solve: 94 = 7n + 10 and so n = 12 Photo credit: © Jens Stolt, Shutterstock.com At another house the window cleaner made £94. How many windows did he have to clean?

Number grid patterns Teacher notes Get students to write a formula for the sum of the highlighted squares in terms of n, the yellow highlighted square. The squares on the grid can be dragged and dropped to try out other values of n. The squares on the large shape on the right can be clicked on to reveal their general value in terms of n. Click on the notelet to reveal the formula.