Algebra - Notation, vocabulary and manipulation: understand and use standard mathematical formulae; rearrange formulae to change the subject

The idea

A formula links quantities by a rule. Some formulae are standard and you are expected to recall them and use them as they stand, such as area of a circle = πr² and speed = distance ÷ time. Changing the subject means rewriting a formula so that a different letter stands alone on one side. Work backwards through the operations that were applied to that letter, doing the same thing to both sides each time. To make t the subject of v = u + at, subtract u from both sides to reach v − u = at, then divide both sides by a, giving t = (v − u)/a, which needs a to be non-zero. Undo a square root by squaring both sides, and undo a square by taking the square root, remembering that a positive number has a positive and a negative square root. If the new subject appears twice, gather those terms on one side, factorise it out, then divide by the bracket.

What earns the marks

• Writing down the correct standard formula before any values or any rearranging go into it.
• A correct first step that isolates the term holding the new subject.
• The same operation applied to both sides at each stage, even if the final rearrangement is wrong.
• The new subject standing alone, with everything else on the other side.
• Factorising the new subject out when it appears in more than one term.

Common mistakes and how to put them right

• Moving a term to the other side without changing its sign.

• Dividing only one term on a side, so turning v = u + at into v/a = u + t.

• Taking the square root of each term separately, so writing the square root of a² + b² as a + b.

• Stopping while the new subject is still multiplied by something, leaving an answer such as 2y = x.

Exam tips

• List the operations applied to the new subject in order, then undo them from the outside in.
• Test your rearrangement by putting easy numbers into the original formula and into your version; both should agree.
• When you divide, divide the whole of the other side, using a bracket to hold it together if you need one.