Algebra - Notation, vocabulary and manipulation: substitute numerical values into formulae and expressions, including scientific formulae

The idea

Substitution means putting a given number in place of each letter, then working the expression out. Replace every copy of a letter, and keep the order of operations: brackets and indices first, then multiplication and division, then addition and subtraction. In v = u + at with u = 3, a = 4 and t = 5, the multiplication comes first, so at is 20 and v = 23. Write a negative value inside brackets before you use it. If x = −3, then 2x² means 2 × (−3)², which is 18, because the index acts on the x alone. Scientific formulae behave the same way: density = mass ÷ volume, so a mass of 240 g in a volume of 30 cm³ gives 8 g/cm³. Check which letter stands for which quantity before you start.

What earns the marks

• Substituting the given values into every letter correctly, even if the arithmetic that follows is wrong.
• Evaluating in the correct order, with indices and brackets dealt with before multiplication.
• The correct final value, carrying the units the scientific formula produces.
• A value correctly carried forward into a later part after an earlier slip.

Common mistakes and how to put them right

• Reading 3x with x = 4 as thirty-four rather than as 12.

• Squaring the coefficient as well as the letter, so evaluating 4x² at x = 3 as 12² rather than as 4 × 9.

• Losing the minus sign, so treating (−2)³ as 8 when it is −8.

• Mixing units, such as substituting a length in centimetres into a formula that expects metres.

Exam tips

• Copy the formula out, then copy it again with the numbers in place, before you touch the calculator.
• Put brackets round every negative number you type in, so the calculator applies the index to the sign as well.
• If a question gives the rule in words, write it as a formula with letters first, then substitute.