Your child is on track for a grade 6. They understand the content, they practise regularly, and they're working hard. But something's stopping them from breaking into grade 7, 8 or 9 territory.
I'm Aadam, and I've been tutoring GCSE students for over five years at SHLC. Here's what I've learned: the difference between grade 6 and grade 7–9 isn't just about working harder. It's about mastering specific topics that appear repeatedly in the challenging questions at the end of papers.
In this guide, I'm revealing the ten topics that consistently separate grade 7–9 students from everyone else, why they're so important, and exactly how to master each one.
What Makes Grade 7–9 Different?
Before we dive into specific topics, you need to understand what grade 7–9 questions actually test.
They're not harder because they use more advanced content. They're harder because they:
- Combine multiple topics in one question (e.g., trigonometry with Pythagoras with circle theorems)
- Require multi step reasoning where each step builds on the previous one
- Use unfamiliar contexts that test whether you truly understand concepts, not just memorised methods
- Demand precision – one small error cascades through the entire question
To reach grade 9, students need around 90% accuracy across all papers. That means mastering not just individual topics, but how they connect together.
Remember, improving from grade 6 to grade 9 could be worth around £40,000 in lifetime earnings. These topics genuinely matter.
Topic 1: Pythagoras and Trigonometry (Including 3D Applications)
Why it decides grade 7–9: This topic accounts for roughly 15-20% of Higher tier marks. More importantly, it appears in multi step problems throughout the paper. You might use trigonometry within a circle theorem question, or apply Pythagoras twice to find lengths in 3D shapes.
What grade 7–9 students must know:
- Applying Pythagoras and trigonometry in 3D shapes (finding diagonals of cuboids, angles between lines and planes)
- Double applications of the same rule (using trigonometry twice in succession)
- Combining Pythagoras with volume calculations
- Using sine rule, cosine rule and area formula for non right angled triangles
- Solving problems where you must decide which rule applies
- Working with bearings combined with trigonometry
Common grade 7–9 question types: "A pyramid has a square base of side 8cm. The perpendicular height is 12cm. Find the angle the slant edge makes with the base."
This requires: identifying the right triangle, applying Pythagoras to find the slant height, then using trigonometry for the angle. Multiple steps, precise working required.
How to master it: Use Physics and Maths Tutor to access all past paper questions on Pythagoras and trigonometry. Work through 20-30 questions focusing specifically on multi step and 3D applications.
Practice with SHLC past papers, marking carefully to identify exactly where your working breaks down if you get it wrong.
Topic 2: Quadratics (Solving, Graphing and Applying)
Why it decides grade 7–9: Quadratics appear everywhere in grade 7–9 questions, often disguised within other topics. You might need to solve a quadratic equation hidden in a geometry problem, or sketch a graph to find maximum values.
What grade 7–9 students must know:
- Solving quadratics by factorising, completing the square and using the formula
- Knowing when each method is most appropriate
- Sketching quadratic graphs and identifying key features (roots, turning point, y intercept)
- Finding equations of quadratic graphs from key points
- Using quadratics in real world contexts (projectile motion, area problems)
- Solving simultaneous equations where one is linear and one is quadratic
- Understanding the discriminant (b² - 4ac) and what it tells you about roots
Common grade 7–9 question types: "A rectangle has length (x+3) and width (x-1). Its area is 35cm². Find the dimensions."
This requires: setting up the equation (x+3)(x-1)=35, expanding, rearranging to equal zero, factorising or using the formula, then selecting the sensible answer.
How to master it: Quadratics need serious practice volume. Use Maths Genie to work through every quadratics topic systematically. Start with simple factorising, progress to the formula, then tackle graph sketching.
Use my digital revision planner to track which quadratic question types you're confident with and which need more work.
Topic 3: Circle Theorems (All Seven, Applied in Complex Diagrams)
Why it decides grade 7–9: Circle theorems questions are grade 7–9 gatekeepers. They test whether students can spot which theorem applies in complex diagrams with multiple circles, tangents and chords overlapping.
What grade 7–9 students must know: All seven circle theorems fluently:
- Angle at the centre is twice the angle at the circumference
- Angles in the same segment are equal
- Angle in a semicircle is 90°
- Opposite angles in a cyclic quadrilateral sum to 180°
- Tangent perpendicular to radius at the point of contact
- Two tangents from an external point are equal length
- Alternate segment theorem
But knowing them isn't enough. You must recognise which one applies in cluttered diagrams where multiple theorems interact.
Common grade 7–9 question types: Complex diagrams where you need to apply three or four different theorems in sequence to find a missing angle.
How to master it: Create flashcards for each theorem with clear diagrams. Use Dr Frost Maths to access interactive questions that build from simple to complex applications.
Practice identifying which theorem applies before calculating. This diagnostic skill separates grade 7–9 students from grade 6 students who know the theorems but can't apply them.
Topic 4: Algebraic Fractions and Manipulation
Why it decides grade 7–9: Algebraic manipulation underpins everything in higher maths. Students who struggle with algebraic fractions hit a ceiling at grade 6, whilst those who master it unlock grade 8–9 questions.
What grade 7–9 students must know:
- Simplifying algebraic fractions by factorising and cancelling
- Adding and subtracting algebraic fractions (finding common denominators)
- Multiplying and dividing algebraic fractions
- Solving equations involving algebraic fractions
- Rearranging complex formulas where the subject appears multiple times
- Changing the subject when it's inside a square root or fraction
Common grade 7–9 question types: "Solve: 3/(x+2) + 2/(x-1) = 1"
This requires: finding common denominator (x+2)(x-1), multiplying through, expanding, collecting terms, solving the resulting quadratic.
How to master it: Start with numeric fractions to ensure those skills are automatic. Then systematically work through algebraic fraction operations one at a time.
Use GCSE Maths Questions to find every algebraic fraction question from past papers. Pattern recognition comes from volume practice.
Topic 5: Compound Measures and Proportional Reasoning
Why it decides grade 7–9: Research shows compound measures (speed, density, pressure) account for nearly 20% of marks in the Ratio and Proportion strand. They appear on every exam series and are favourite topics for grade 7–9 questions.
What grade 7–9 students must know:
- Speed = distance ÷ time (and all rearrangements)
- Density = mass ÷ volume (and all rearrangements)
- Pressure = force ÷ area (and all rearrangements)
- Converting between units (km/h to m/s, g/cm³ to kg/m³)
- Combining compound measures with 3D shape volume calculations
- Proportional reasoning in complex contexts
- Direct and inverse proportion including graphs
- Exponential growth and decay (compound percentage problems)
Common grade 7–9 question types: "A cylindrical container has radius 5cm and height 20cm. It's filled with liquid of density 1.2g/cm³. Find the mass of liquid in kg."
This requires: calculating volume (πr²h), converting to mass using density, converting grams to kilograms. Multiple conversions within one question.
How to master it: Create a formula triangle for each compound measure. Practice unit conversions separately until automatic (this is where most students lose marks).
Density frequently appears combined with volume of prisms, cylinders and spheres. Practice these combinations specifically.
Topic 6: Vectors (Geometric Proof and Application)
Why it decides grade 7–9: Vectors are almost exclusively grade 7–9 content. If vectors appear, it's targeting the top grades. Students uncomfortable with vectors immediately lose 8-10 marks.
What grade 7–9 students must know:
- Vector notation (column vectors and lettered vectors)
- Adding and subtracting vectors geometrically and algebraically
- Scalar multiplication of vectors
- Finding magnitude of vectors (using Pythagoras)
- Position vectors and displacement vectors
- Using vectors to prove geometric properties (parallelism, midpoints, ratios)
- Solving vector equations
Common grade 7–9 question types: "OABC is a parallelogram. M is the midpoint of OA. N divides BC in the ratio 2:1. Prove that OMN is a straight line."
This requires: expressing OM and ON in terms of base vectors, showing one is a scalar multiple of the other. Pure algebraic proof.
How to master it: Vectors are conceptually different from other GCSE topics. You need solid foundational understanding before tackling complex proofs.
Use 1st Class Maths vector booklets which build systematically from basics to proof questions.
Get professional feedback on vector proofs through my mock exam marking service – these questions require precise notation that's hard to self mark accurately.
Topic 7: Graphs of Functions (Beyond Linear and Quadratic)
Why it decides grade 7–9: Grade 7–9 students must recognise and sketch various function types: cubic, reciprocal, exponential, trigonometric. Questions test whether you understand graph behaviour, not just can you plot points.
What grade 7–9 students must know:
- Recognising graph types from equations (y=x³, y=1/x, y=2ˣ, y=sin x)
- Sketching graphs with correct shape and key features
- Finding equations from graphs
- Transformations of graphs (translations, reflections, stretches)
- Using graphs to solve equations
- Interpreting real world graphs (distance time, velocity time)
- Finding gradient of curves at points
- Finding areas under curves
Common grade 7–9 question types: "Sketch y=2x³-3 and y=2x+1 on the same axes. Hence find approximate solutions to 2x³-2x-4=0"
This requires: recognising that solving the equation is equivalent to finding intersection points, sketching accurately enough to read off approximate solutions.
How to master it: Create a reference sheet showing the characteristic shape of each function type. Practice sketching from equations without plotting points.
Use graph paper initially, then practice sketching freehand to develop intuition for graph behaviour.
Topic 8: Probability (Tree Diagrams, Venn Diagrams and Conditional Probability)
Why it decides grade 7–9: Probability questions at grade 7–9 combine multiple concepts: tree diagrams with replacement, conditional probability, Venn diagrams with algebraic unknowns. These test logical reasoning under pressure.
What grade 7–9 students must know:
- Tree diagrams with and without replacement
- Conditional probability P(A|B) = P(A∩B)/P(B)
- Venn diagrams including three sets
- Using Venn diagrams with algebraic expressions
- Independent vs dependent events
- Mutually exclusive events
- Frequency trees
- Probability from two way tables
Common grade 7–9 question types: "A bag contains x red balls and 2x blue balls. Two balls are taken without replacement. The probability both are red is 1/6. Find x."
This requires: setting up tree diagram, writing probability expression, forming equation, solving for x, checking answer makes sense.
How to master it: Start simple and build complexity. Master basic tree diagrams before attempting without replacement questions.
Practice verifying probability calculations (probabilities must sum to 1, individual probabilities must be between 0 and 1).
Topic 9: Proof and Reasoning
Why it decides grade 7–9: Proof questions explicitly test whether students understand mathematical reasoning, not just calculation. These appear increasingly in grade 7–9 questions across all topics.
What grade 7–9 students must know:
- Proving algebraic statements ("prove that n²-n is always even")
- Geometric proof using circle theorems, angle properties, congruence
- Proof by exhaustion (testing all cases)
- Proof by contradiction (showing opposite leads to impossibility)
- Counter examples (finding one case that disproves a statement)
- Showing something "must be true" vs "could be true"
Common grade 7–9 question types: "Prove algebraically that the sum of any three consecutive integers is always divisible by 3."
This requires: letting the integers be n, n+1, n+2, showing their sum is 3n+3, factorising to 3(n+1), concluding it's a multiple of 3.
How to master it: Proof questions require clear communication. Your working must show logical steps, not just calculations.
Practice writing proofs in complete sentences. "Therefore..." and "Since..." should appear frequently. This is mathematical writing, not just calculation.
Topic 10: Non Right Angled Triangle Problems
Why it decides grade 7–9: Whilst Pythagoras and basic trigonometry appear throughout Higher tier, sine rule, cosine rule and area formula (½absinC) are almost exclusively grade 7–9 content.
What grade 7–9 students must know:
- Sine rule: a/sinA = b/sinB = c/sinC (and when to use it)
- Cosine rule: a² = b² + c² - 2bccosA (and when to use it)
- Area = ½absinC for any triangle
- Choosing which rule applies in different scenarios
- Finding angles using inverse trig functions
- Applying these in 2D and 3D contexts
- Combining with Pythagoras and basic trig
Common grade 7–9 question types: "A triangle has sides 7cm and 9cm with included angle 65°. Find the area and the length of the third side."
This requires: using area formula with given angle, then using cosine rule for third side. Two separate calculations from one setup.
How to master it: Create decision flowcharts: "Do I have a right angle? → Use Pythagoras/basic trig. No right angle + opposite side and angle given? → Sine rule. No right angle + two sides and included angle? → Cosine rule."
Practice choosing the correct rule before calculating. This diagnostic skill is crucial.
How to Actually Master These Topics
Knowing which topics matter is useless without a system to master them. Here's the approach I use with grade 7–9 students at SHLC:
Step 1: Diagnostic Assessment (Week 1) Complete a full Higher tier paper. Identify which of these ten topics caused the most lost marks. Rank them from weakest to strongest.
Step 2: Systematic Practice (Weeks 2-16) Target one topic per week in order of weakness. For each topic:
- Watch video explanation on Maths Genie
- Complete 15-20 questions increasing in difficulty
- Attempt 5 grade 7–9 specific questions from past papers
- Review worked solutions for every mistake
Use my digital revision planner to track progress from red (struggling) to amber (improving) to green (confident).
Step 3: Integration Practice (Weeks 17-24) Now these topics must connect. Complete mixed topic questions where multiple grade 7–9 topics appear in one question.
Use 1st Class Maths extension materials specifically designed for grade 8–9 students.
Step 4: Timed Paper Practice (Weeks 25-32) Complete at least two full Higher tier papers per week. Your targets:
- Paper 1 (non calculator): 60/80 marks minimum for grade 7, 70/80 for grade 8-9
- Papers 2 & 3 (calculator): 65/80 marks minimum for grade 7, 72/80 for grade 8-9
Track scores over time. You should see steady upward movement.
Step 5: Professional Feedback Self marking misses subtle errors. Use my mock exam marking service every 4-6 weeks to get expert analysis of where marks are still being lost and why.
Common Mistakes Students Make
Mistake 1: Focusing on Easier Topics Grade 6 students often repeatedly practice topics they already know (fractions, percentages, basic algebra) because it feels comfortable. But grade 7–9 requires mastering uncomfortable topics.
Mistake 2: Learning Topics in Isolation Grade 7–9 questions combine topics. Mastering quadratics separately from graphing, separately from simultaneous equations, leaves you unprepared for questions that blend all three.
Mistake 3: Not Practising Problem Solving Grade 7–9 isn't about memorising methods, it's about choosing which method applies in unfamiliar contexts. This requires extensive problem solving practice.
Mistake 4: Insufficient Past Paper Practice Research shows students need at least 15-20 full papers before achieving grade 7–9 consistency. Use SHLC past papers to build this volume.
Mistake 5: Starting Too Late Building grade 7–9 skills takes 6-9 months of systematic practice. Starting in March for May exams leaves insufficient time.
For Parents: Supporting Grade 7–9 Ambitions
Set Realistic Expectations Moving from grade 6 to grade 9 is a significant jump. Grade 7 is often more realistic initially, with grade 8–9 following after.
Celebrate Process, Not Just Results "You completed all ten circle theorem questions today" deserves celebration, even if the paper score hasn't jumped yet. The scores follow the practice.
Consider Professional Support Grade 7–9 is where tutoring makes the biggest difference. These topics require expert explanation that many parents can't provide.
At SHLC, I specialise in supporting students targeting grade 7–9. I've helped dozens of students break through that grade 6 ceiling to achieve the top grades that unlock university options and future careers.
Monitor Burnout Pursuing grade 7–9 requires intensive practice. Ensure your child isn't sacrificing sleep, exercise or social time. Sustainable progress beats sprints that end in burnout.
The Bottom Line
These ten topics consistently separate grade 7–9 students from everyone else. Master them, and grade 7–9 becomes achievable. Neglect them, and you'll stay stuck at grade 6 regardless of effort.
The students I tutor who reach grade 7–9 have three things in common:
- They start early (at least 6 months before exams)
- They practice systematically, tracking weak areas precisely
- They seek expert feedback to catch subtle errors
You now know which topics matter most. The question is: will you put in the systematic practice needed to master them?
Remember: each grade improvement is worth around £23,000 in lifetime earnings. Moving from grade 6 to grade 9 could be worth nearly £70,000. These topics are genuinely life changing.
Start today. Your future self will thank you.
Want expert support mastering these grade 7–9 topics? Get in touch with SHLC to discuss how targeted tutoring can help you achieve the top grades.