A8 Changing the subject and deriving formulae Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae A8 Changing the subject and deriving formulae This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

A8.1 Changing the subject of a formula Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae A8.1 Changing the subject of a formula

Using inverse operations Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Using inverse operations Andy is 5 years older than his brother, Brian. Their ages are linked by the formula: A = B + 5 where A is Andy’s age in years and B is Brian’s age in years. Using this formula it is easy to find Andy’s age given Brian’s age. Suppose we want to find Brian’s age given Andy’s age. Explain that we are going to look at how we can use inverse operations to rearrange formulae. Go through each step on the slide and then ask pupils questions such as: How old was Brian when Andy was 12? Which formula did you use? How old was Andy, when Brian was 28? Using inverse operations, we can write this formula as: B = A – 5

The subject of a formula Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae The subject of a formula Look at the formula, 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.

Changing the subject of the formula Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Changing the subject of the formula The formula: V is the subject of this formula V = IR can be written as: I × R V The inverse of this is: I ÷ R V We can write the equation V = IR using functions. Ask pupils what do we do to I to get V and establish that we times it by R. Reveal the first diagram showing the operation × R. Compare this to a function diagram. 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 numerical example. For example, ask pupils to give you the value of I when V = 12 and R = 3. Ask pupils how we could make R the subject of the formula (R = V/I). or I is now the subject of this formula I = V R

A8 Changing the subject and deriving formulae Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae 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: 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? Encourage pupils to consider what have we done to n? (We’ve multiplied it by 2 and added 1.) 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. m = 2n + 1 Which pattern number will contain 47 matches?

Changing the subject of the formula Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Changing the subject of the formula The formula: m is the subject of this formula m = 2n + 1 can be written as: 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. or n is the subject of this formula n = m – 1 2

Changing the subject of the formula Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae 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 47 – 1 n = 2 46 2 n = 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.

Changing the subject of the formula Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Changing the subject of the formula To make C the subject of the formula: F = + 32 9C 5 F – 32 = 9C 5 Subtract 32: 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 using inverse operations. Ask pupils how we could write 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 in front of the equals sign without any numbers or operations. 5(F – 32) 9 = C Divide by 9: 5(F – 32) 9 C =

A8 Changing the subject and deriving formulae Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Equivalent formulae The aim of this activity is to correctly identify four equivalent forms of the same function. The correct function appears in four different forms. An example, c = 36 – 9d and d = 36 – c/9 c = 9(4 – d) and d = 4 – c/9 Equivalent formulae can be found by rearranging the given formula. If required ask pupils to find just one of the equivalent formulae. For example, ask pupils to find the formula which gives the other variable as the subject. Pressing the reset button will generate a new set of equivalent formulae.

A8 Changing the subject and deriving formulae Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae A8.2 Deriving formulae

A8 Changing the subject and deriving formulae Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Connecting dots When you click draw lines all red dots will be connected to all blue dots using straight lines. Show two red dots and three blue dots on the board. Ask pupils, how many lines do I need to draw to connect each red dot to each blue dot? Press the “Draw lines” button. L, the number of lines = 6. Demonstrate a few more examples. Ask pupils for a formula to find, L, the number of lines, given R, the number of red dots and B, the number of blue dots. Take some suggestions, asking pupils to justify their formulae in the context of the problem, before pressing the show formula button. Ask pupils for a justification of this formula. Link: A9 Sequences – sequences from practical contexts.

A8 Changing the subject and deriving formulae Boardworks KS3 Maths 2009 A8 Changing the subject and deriving formulae Tiling patterns By changing n and recording the results in the table ask pupils to derive a formula to give the number of tiles given the pattern number. Ask pupils to justify the formula in the context from which it is generated. Each time n increases a square n tiles by n tiles is produced, this is where the n² part of the formula comes from. There are always 4 more tiles around the outside of the pattern, this is where the + 4 part of the formula comes from.