These constructed examples show the kind of reasoning to look for in teaching. They illustrate content from England’s primary and KS3 maths curriculum, AQA GCSE Maths and AQA A-level Maths. Use the learner’s own course when agreeing exam preparation. These are illustrations, not accounts of a Latimer lesson.
Primary fractions: keep the pieces the same size
For 1/3 + 1/4, a learner might add the top and bottom numbers to get 2/7. The problem is that thirds and quarters are different-sized pieces. Draw two equal-length bars, divide each into twelve equal parts, then show that one third covers four parts and one quarter covers three. Now 4/12 + 3/12 = 7/12: seven pieces, each one twelfth of the whole. A useful follow-up is 1/2 + 1/3. The learner can draw sixths and explain why 3/6 + 2/6 = 5/6.
KS3 algebra: unpack a bracket without losing a term
In 3(x + 4) = 21, the bracket means three lots of the whole quantity x + 4. Dividing both sides by three gives x + 4 = 7, then subtracting four gives x = 3. Substituting it back checks the answer: 3 × (3 + 4) = 21. If expanding first, both terms must be multiplied: 3x + 12, not 3x + 4. The useful question is why those two routes give the same answer.
GCSE percentages: identify what counts as 100%
A price is £92 after a 15% increase. Subtracting 15% of £92 gives £78.20, but that takes the percentage of the new price. The original price is 100%; the new price is 115%. If p is the original price, 1.15p = 92, so p = £80. Check forwards: 15% of £80 is £12, giving £92. For a follow-up, suppose a price is £72 after a 10% discount. Here £72 represents 90% of the original price, so 0.9p = 72 gives p = £80. The learner should be able to explain why the multiplier changed.
A-level calculus: connect a derivative to a tangent
For the curve y = x² + 3x, the derivative dy/dx = 2x + 3 gives the gradient at each value of x. At x = 2, the gradient is 7 and the point on the curve is (2, 10). A tangent line needs both pieces of information: y − 10 = 7(x − 2), giving y = 7x − 4. Stopping at “7” finds the gradient but does not answer a question asking for the tangent’s equation. Asking the learner to explain the roles of the point and gradient reveals whether the method makes sense.
