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Try this example for an older primary learner or someone revisiting fractions. The question is 1/3 + 1/4. A learner writes 2/7, adding the top numbers and then the bottom numbers. The difficulty is that thirds and quarters are different-sized parts of a whole.

Imagine two equal-length bars, each divided into twelve equal pieces. One third covers four pieces; one quarter covers three. This gives 1/3 = 4/12 and 1/4 = 3/12, so 4/12 + 3/12 = 7/12. We add the number of pieces and keep their size the same.

A useful check is size: adding a positive quarter to a third must give more than a third. The proposed answer 2/7 is smaller than 1/3, so it cannot be right.

For a follow-up, try 1/2 + 1/3. Using sixths gives 3/6 + 2/6 = 5/6. A stronger response explains why the pieces must be equal-sized and can show that on a diagram. England’s Year 6 curriculum explicitly connects adding fractions with different denominators to equivalent fractions.

Consider this secondary-school or everyday maths question: a price falls from £80 to £68. What is the percentage decrease?

The decrease is £80 − £68 = £12. To express that change as a percentage, compare it with the original £80: 12 ÷ 80 × 100 = 15%. A clear answer is: “The price has decreased by 15% of its original value.” Dividing by £68 instead compares the change with the reduced price, which answers a different question.

Check the result by calculating 80 × 0.85 = 68: after a 15% reduction, 85% of the original price remains. For a more demanding follow-up, reverse the problem: if £68 is the price after a 15% reduction, the original is 68 ÷ 0.85 = £80.

The useful teaching point is identifying what represents 100% before selecting a calculation. A learner who can explain that choice has something to use when the numbers or wording change.

For a learner studying differentiation, consider this example: find the tangent to y = x² at x = 3. A tangent’s gradient describes the slope of the curve at that particular point.

The point is (3, 9) because 3² = 9. Differentiating y = x² gives dy/dx = 2x; dy/dx means the gradient function. At x = 3, the gradient is 6. A line with gradient 6 through (3, 9) has equation y − 9 = 6(x − 3), which simplifies to y = 6x − 9.

One possible mistake is to take the points (3, 9) and (4, 16) and calculate (16 − 9) ÷ (4 − 3) = 7. That is the gradient of the straight line joining the two points, called a chord. The tangent question asks for the gradient at x = 3.

Try the same question at x = 2. The point becomes (2, 4) and the gradient is 4, so the tangent is y − 4 = 4(x − 2), or y = 4x − 4. Explaining the difference between a chord and a tangent shows how the calculation connects to the graph. This is one pure-maths example; take statistics or mechanics difficulties to the tutor as separate priorities.

Latimer offers online one-to-one tuition. Microsoft Teams is the default lesson platform, and you can agree another platform with the tutor. For maths, agree a way for both people to see the working clearly: this might be a shared whiteboard, a tablet or paper shown to a camera. Check that fractions, indices and diagrams remain readable.

Tutors set their own rates, shown on their profiles, and Latimer uses pay-as-you-go billing. Confirm the lesson length and price before booking, then agree a frequency that fits your budget, priorities and time for practice. There is no single lesson count to promise for learning maths.

A focused goal makes the cost easier to assess: for example, being able to identify the starting value in percentage questions without prompting. Discuss what preparation and follow-up work the tutor proposes, as well as the live lesson itself.

For remote lessons, compare online Maths tutors and discuss how you will share working during the lesson.