These constructed examples show how a tutor could use written working and discussion to uncover a difficulty. They illustrate different starting points; the learner’s own work should guide lesson priorities.
Fractions: make the parts the same size
Suppose a learner writes 1/3 + 1/4 = 2/7. Adding both numbers on top and both numbers underneath misses what the fractions represent. Draw two equal-sized bars and split each into twelve equal parts: one third is four twelfths, and one quarter is three twelfths. Now the parts have the same size, so 4/12 + 3/12 = 7/12.
A better explanation is: “I added seven twelfths because I changed both fractions into equal-sized parts.” The tutor could watch the learner mark those parts on paper or on a shared drawing. This fits the equivalent-fraction addition in England’s Year 6 maths programme.
GCSE percentages: check what the percentage is of
A price of £50 rises by 20%, then falls by 20%. It is tempting to say the changes cancel. The increase is £10, taking the price to £60. The decrease is then 20% of £60, or £12, leaving £48. Equivalently, 50 × 1.2 × 0.8 = 48.
Labelling £50 and £60 as the two starting amounts makes the error visible. A better answer explains why the second percentage acts on a different amount. This illustrates multiplicative percentage change in AQA GCSE Mathematics 8300, R9.
A-level calculus: separate height from gradient
For y = x², find the tangent’s gradient at x = 3. Substituting into the original equation gives y = 9: that locates the point (3, 9) on the curve. To find the gradient, differentiate: dy/dx = 2x, then substitute x = 3 to get 6. Here dy/dx denotes the gradient of the curve.
On a sketch, label the point and its tangent separately. The better response is “the tangent gradient is 6”, with the derivative shown in the working. This connects calculation and interpretation within AQA A-level Mathematics 7357, differentiation.
