GCSE Maths (Higher / Foundation)

What this covers

• Rearranging and solving equations: linear, simultaneous, quadratics by factorising, formula and completing the square, and changing the subject when the letter appears twice
• Ratio and proportion in context: sharing in a ratio, recipe and best-buy problems, compound interest and reverse percentages, direct and inverse proportion with a constant k
• Geometry and trigonometry: angle chains with reasons, Pythagoras, SOHCAHTOA, exact trig values, and (Higher) sine rule, cosine rule and the ½absinC area formula
• Graphs and their meaning: gradient and y-intercept, parallel and perpendicular lines, recognising quadratic, cubic, reciprocal and exponential shapes, and reading rate from a distance–time or speed–time graph
• Statistics: estimating the mean from grouped data, cumulative frequency and box plots, comparing distributions in words, and (Higher) frequency density in histograms
• Probability: tree diagrams with and without replacement, Venn diagrams and set notation, and the difference between listing outcomes and multiplying probabilities
• Exam technique for multi-step 'problem solving' questions: extracting the maths from wordy contexts, showing enough working for method marks, and checking answers for sensible units

Where learners get stuck

Treating a percentage decrease as reversible by adding the same percentage back

Learners apply the percentage to the wrong starting amount. A £200 item reduced by 20% becomes £160, but adding 20% of £160 returns £192, not £200. Reverse percentage questions need division by 0.8, and this only clicks once the multiplier is written down explicitly.

Expanding (x + 3)² as x² + 9

Index rules for multiplication (x²·x³ = x⁵) get over-generalised to sums. The habit is reinforced because squaring a single term does distribute across a product, so learners assume brackets always work term-by-term. Drawing the grid or writing (x+3)(x+3) in full breaks it.

Using SOHCAHTOA on a non-right-angled triangle

Trigonometry is introduced with right triangles for a whole topic, so the trigger becomes 'triangle with an angle' rather than 'right angle present'. On Higher papers, the sine and cosine rules are needed instead, and the choice depends on which sides and angles are given.

Reading histogram bar heights as frequencies

Bar charts are met years earlier and look similar. In a histogram the vertical axis is frequency density, so frequency is the area of the bar; unequal class widths make the difference visible only if the learner checks the axis label first.

What a session looks like

A typical session runs 25–45 minutes on one topic or one past-paper question type. You talk through your attempt, the tutor asks where each step came from, and errors are traced back to the rule that was misapplied rather than simply corrected. Work is done on paper alongside the conversation, and diagrams, graphs or working can be described aloud. Sessions can be pitched at Foundation content, Higher-only content, or the grade 4–5 overlap where tier choice is still open.

Helpful to know first

• Confident times tables and written methods for multiplication and division without a calculator
• Fractions, decimals and percentages: converting between them and adding or multiplying fractions
• Substituting numbers into a formula and solving a simple one- or two-step equation
• Access to a scientific calculator (Higher work assumes the sin, cos, tan, x² and fraction keys)

Questions

Should my child sit Foundation or Higher tier?

Foundation caps at grade 5; Higher runs 4–9 but a marginal Higher candidate risks a U if they fall below the grade 3 allowance. Sessions can work through Higher-only topics to test whether they are accessible, or consolidate grade 4–5 material so a Foundation entry lands securely at 5 rather than 4.

Can it help with the non-calculator paper?

Yes. Sessions can focus on the skills Paper 1 tests directly: long multiplication and division, fraction arithmetic, standard form, surds and exact trig values, and estimating by rounding to one significant figure.

Does it work through past papers?

You can read out or describe a question you are stuck on and work through it in conversation. The tutor focuses on the method and where marks are awarded; it does not have official mark schemes and is not connected to any exam board.

My child says they 'just don't get algebra'. Where does that get fixed?

Usually at substitution and the meaning of a letter as a varying quantity, not at the topic they are currently failing. Sessions can go back to that point and rebuild through expanding, factorising and solving, which is faster than drilling the surface topic.