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

The idea

This skill requires you to understand and use standard mathematical formulae, and to rearrange formulae to change the subject. Understanding a formula means knowing what each variable represents and substituting values correctly. For example, using A = πr²; if r = 5, then A = 25π. Rearranging to change the subject means using inverse operations to isolate a different variable. To make r the subject of A = πr², divide both sides by π to get r² = A/π, then take the square root: r = √(A/π). For formulae with fractions, multiply both sides by the denominator; for formulae with powers, take roots or powers as needed. Always perform the same operation on both sides and check by substitution.

What earns the marks

• Substituting numerical values into standard formulae correctly, including using the correct order of operations and units.
• Rearranging a formula by applying inverse operations to both sides to isolate the required variable.
• Handling formulae involving fractions by multiplying both sides by the denominator before further rearrangement.
• Handling formulae involving powers or roots by taking the appropriate root or power of both sides, e.g. making r the subject of A = πr² gives r = √(A/π).
• Checking the rearranged formula by substituting a known value or by reversing the rearrangement.

Common mistakes and how to put them right

• Applying an operation to only one side of the formula.

  Put it right by performing the same operation to both sides.

• Incorrectly handling negative signs or division when rearranging, such as forgetting to divide all terms.

  Put it right by working step by step and checking each inverse operation.

• Forgetting to take the square root when isolating a squared variable, e.g. leaving r² instead of r.

  Put it right by taking the positive square root where the context requires a positive length.

• Mixing up the subject and another variable, or not fully isolating the required variable.

  Put it right by ensuring the subject appears alone on one side.

Exam tips

• Write down each step of your rearrangement so method marks can be awarded even if the final answer is incorrect.
• Check your rearranged formula by substituting simple numbers back into both the original and rearranged versions to see if they give the same result.
• Pay attention to units and whether the context requires a positive or negative root when taking square roots.