This maths revision guide covers essential GCSE topics you'll encounter...
Maths106 views·Updated 16 Aug 2026·9 pagesComprehensive GCSE Math Notes
Charlotte@charlotte26
1 / 9
1
of 9
Inequalities - The Basics
Ever wondered how to show that something is "greater than" or "at least" a certain value? Inequalities are your answer! They're like equations but instead of being exactly equal, they show relationships between values.
The symbols are pretty straightforward: > means greater than, < means less than, and adding an equals sign (≥ or ≤) means "or equal to" as well. When you're graphing these, remember that solid lines represent ≥ or ≤, whilst dotted lines show > or <.
💡 Quick tip: On number lines, use open circles (○) for > or < and closed circles (●) for ≥ or ≤. The length of your line doesn't matter - it's all about showing the direction!
Solving inequalities works just like equations - you can add, subtract, multiply or divide both sides. Just remember to flip the inequality sign if you multiply or divide by a negative number!
2
of 9
Bearings - Finding Your Way
Think of bearings like a compass reading that tells you exactly which direction something is. They're measured clockwise from north and always written as three figures (so 45° becomes 045°).
Here's the key trick: when finding bearings, always draw your north line from the point you're measuring from. If you're finding the bearing of point A from point B, draw north from B, then measure clockwise to A.
💡 Remember: If you need to find the bearing "back" (like B from A when you know A from B), add or subtract 180° to find the opposite direction. It's like turning around completely!
The three-figure rule is crucial for exams - even if your angle is just 45°, you must write it as 045°. Those zeros at the front matter!
3
of 9
Essential Maths Skills - Paper 2 Revision
Error intervals might seem tricky, but they're just about understanding rounding. If 6 is rounded to 1 decimal place, it could have been anything from 5.95 to 6.05 (but we write this as 6.0 ≤ x < 6.1).
Index laws are your best friend for simplifying expressions. Remember that anything to the power of 1 equals itself, and when multiplying powers with the same base, you add the indices. When dividing, you subtract them.
💡 Calculator tip: Always close your brackets properly and write down all the numbers before rounding. This prevents silly mistakes that cost marks!
Expanding brackets follows the same pattern every time - multiply everything in the first bracket by everything in the second. For factorising, look for common factors you can take out, like taking 3 out of 3x + 15 to get 3.
4
of 9
Equations and Quadratic Inequalities
Solving linear equations is all about doing the same thing to both sides. Your goal is to get the letter on its own, so work backwards through the order of operations. If there are letters on both sides, collect them together first.
Word problems become easier when you define your variables clearly. If Chris is x years old and Adam is 8 years older, then Adam is x + 8. Always check your answer makes sense in the original problem!
💡 Parallel lines trick: Lines are parallel when they have the same gradient (the number before x). Rearrange equations into y = mx + c form to spot this quickly.
Quadratic inequalities give you a range of answers rather than just one value. Factorise the quadratic, find where it equals zero, then test which region satisfies your inequality. The answer will be between or outside these values.
5
of 9
Sequences and Algebraic Fractions
Linear sequences have a constant difference between terms. To find the nth term, work out this common difference, then adjust for the first term. For 3, 7, 11, 15... the difference is 4, so the nth term is 4n - 1.
Quadratic sequences are trickier - their differences aren't constant, but the second differences are. Start with n², then adjust to match your sequence. Fibonacci sequences simply add the two previous terms together.
💡 Sequence tip: Always check your nth term formula by substituting n = 1, 2, 3... to see if you get the original sequence back.
Algebraic fractions work just like number fractions. To simplify, factorise the top and bottom, then cancel common factors. When adding or subtracting, find a common denominator first - usually by multiplying the denominators together.
6
of 9
Functions and Real-Life Maths
Inverse functions undo what the original function does. To find f⁻¹, swap x and y in your equation, then rearrange to make y the subject. It's like working backwards through the function's steps.
Composite functions like gf mean "do f first, then g". Substitute the entire f expression into g. Be careful with your algebra when expanding - it's easy to make sign errors!
💡 Function tip: Always substitute carefully and show each step clearly. Examiners love to see your working, especially with composite functions.
Depreciation problems are just repeated percentage decreases. Each year, the car keeps 80% of its value (if it depreciates by 20%). You can multiply by 0.8 repeatedly or use the formula: original value × (multiplier)ⁿ.
7
of 9
Proportionality and Currency
Currency conversion requires careful attention to exchange rates. Always check which currency gives better value by converting both to the same currency. Don't forget that £1 = $1.29 means you multiply pounds by 1.29 to get dollars.
Direct proportion means as one quantity increases, the other increases at the same rate. If y ∝ x, then y = kx where k is your constant. Use given values to find k, then use it to find unknown values.
💡 Proportion tip: Write down your relationship clearly (y = kx or y = k/x) before substituting numbers. This prevents confusion between direct and inverse proportion.
Inverse proportion is the opposite - as one increases, the other decreases. If A ∝ 1/B, then A = k/B. The key is identifying which type of relationship you're dealing with from the context.
8
of 9
Advanced Topics - Pressure and Circle Theorems
Pressure calculations use the formula Pressure = Force ÷ Area. Watch out for unit conversions - you might need to convert cm² to m² by dividing by 10,000 (since 1 m² = 100² cm²).
Circle theorems are essential geometry rules. Angles in the same segment are equal, the angle in a semicircle is 90°, and the angle between a tangent and radius is always 90°.
💡 Circle theorem tip: Always mark equal angles with the same symbols and clearly state which theorem you're using. Examiners want to see you know the rules!
The alternate segment theorem states that the angle between a tangent and chord equals the angle in the alternate segment. These theorems often combine, so look for multiple relationships in complex diagrams.
9
of 9
Statistics - Box Plots and Data Analysis
Box plots show five key values: minimum, lower quartile, median, upper quartile, and maximum. The interquartile range (IQR) is UQ - LQ and shows how spread out the middle 50% of data is.
When comparing box plots, discuss both average (median) and consistency (IQR or range). A higher median means higher average values, whilst a smaller IQR means more consistent data.
💡 Statistics tip: Always relate your statistical comments back to the real-world context of the problem. Don't just say "the median is higher" - explain what this means in practical terms.
Stem and leaf diagrams organise data clearly whilst preserving original values. Always include a key and remember to order the leaves from smallest to largest. They make finding the mode, median, and range much easier.
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Maths106 views·Updated 16 Aug 2026·9 pagesComprehensive GCSE Math Notes
Charlotte@charlotte26
This maths revision guide covers essential GCSE topics you'll encounter in your exams. From inequalities and bearings to functions and statistics, these concepts form the backbone of your mathematical toolkit.
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Inequalities - The Basics
Ever wondered how to show that something is "greater than" or "at least" a certain value? Inequalities are your answer! They're like equations but instead of being exactly equal, they show relationships between values.
The symbols are pretty straightforward: > means greater than, < means less than, and adding an equals sign (≥ or ≤) means "or equal to" as well. When you're graphing these, remember that solid lines represent ≥ or ≤, whilst dotted lines show > or <.
💡 Quick tip: On number lines, use open circles (○) for > or < and closed circles (●) for ≥ or ≤. The length of your line doesn't matter - it's all about showing the direction!
Solving inequalities works just like equations - you can add, subtract, multiply or divide both sides. Just remember to flip the inequality sign if you multiply or divide by a negative number!
2
of 9Sign up to see the content. It's free!
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Bearings - Finding Your Way
Think of bearings like a compass reading that tells you exactly which direction something is. They're measured clockwise from north and always written as three figures (so 45° becomes 045°).
Here's the key trick: when finding bearings, always draw your north line from the point you're measuring from. If you're finding the bearing of point A from point B, draw north from B, then measure clockwise to A.
💡 Remember: If you need to find the bearing "back" (like B from A when you know A from B), add or subtract 180° to find the opposite direction. It's like turning around completely!
The three-figure rule is crucial for exams - even if your angle is just 45°, you must write it as 045°. Those zeros at the front matter!
3
of 9Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
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Essential Maths Skills - Paper 2 Revision
Error intervals might seem tricky, but they're just about understanding rounding. If 6 is rounded to 1 decimal place, it could have been anything from 5.95 to 6.05 (but we write this as 6.0 ≤ x < 6.1).
Index laws are your best friend for simplifying expressions. Remember that anything to the power of 1 equals itself, and when multiplying powers with the same base, you add the indices. When dividing, you subtract them.
💡 Calculator tip: Always close your brackets properly and write down all the numbers before rounding. This prevents silly mistakes that cost marks!
Expanding brackets follows the same pattern every time - multiply everything in the first bracket by everything in the second. For factorising, look for common factors you can take out, like taking 3 out of 3x + 15 to get 3.
4
of 9Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Equations and Quadratic Inequalities
Solving linear equations is all about doing the same thing to both sides. Your goal is to get the letter on its own, so work backwards through the order of operations. If there are letters on both sides, collect them together first.
Word problems become easier when you define your variables clearly. If Chris is x years old and Adam is 8 years older, then Adam is x + 8. Always check your answer makes sense in the original problem!
💡 Parallel lines trick: Lines are parallel when they have the same gradient (the number before x). Rearrange equations into y = mx + c form to spot this quickly.
Quadratic inequalities give you a range of answers rather than just one value. Factorise the quadratic, find where it equals zero, then test which region satisfies your inequality. The answer will be between or outside these values.
5
of 9Sign up to see the content. It's free!
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- Improve your grades
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Sequences and Algebraic Fractions
Linear sequences have a constant difference between terms. To find the nth term, work out this common difference, then adjust for the first term. For 3, 7, 11, 15... the difference is 4, so the nth term is 4n - 1.
Quadratic sequences are trickier - their differences aren't constant, but the second differences are. Start with n², then adjust to match your sequence. Fibonacci sequences simply add the two previous terms together.
💡 Sequence tip: Always check your nth term formula by substituting n = 1, 2, 3... to see if you get the original sequence back.
Algebraic fractions work just like number fractions. To simplify, factorise the top and bottom, then cancel common factors. When adding or subtracting, find a common denominator first - usually by multiplying the denominators together.
6
of 9Sign up to see the content. It's free!
- Access to all documents
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Functions and Real-Life Maths
Inverse functions undo what the original function does. To find f⁻¹, swap x and y in your equation, then rearrange to make y the subject. It's like working backwards through the function's steps.
Composite functions like gf mean "do f first, then g". Substitute the entire f expression into g. Be careful with your algebra when expanding - it's easy to make sign errors!
💡 Function tip: Always substitute carefully and show each step clearly. Examiners love to see your working, especially with composite functions.
Depreciation problems are just repeated percentage decreases. Each year, the car keeps 80% of its value (if it depreciates by 20%). You can multiply by 0.8 repeatedly or use the formula: original value × (multiplier)ⁿ.
7
of 9Sign up to see the content. It's free!
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- Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Proportionality and Currency
Currency conversion requires careful attention to exchange rates. Always check which currency gives better value by converting both to the same currency. Don't forget that £1 = $1.29 means you multiply pounds by 1.29 to get dollars.
Direct proportion means as one quantity increases, the other increases at the same rate. If y ∝ x, then y = kx where k is your constant. Use given values to find k, then use it to find unknown values.
💡 Proportion tip: Write down your relationship clearly (y = kx or y = k/x) before substituting numbers. This prevents confusion between direct and inverse proportion.
Inverse proportion is the opposite - as one increases, the other decreases. If A ∝ 1/B, then A = k/B. The key is identifying which type of relationship you're dealing with from the context.
8
of 9Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Topics - Pressure and Circle Theorems
Pressure calculations use the formula Pressure = Force ÷ Area. Watch out for unit conversions - you might need to convert cm² to m² by dividing by 10,000 (since 1 m² = 100² cm²).
Circle theorems are essential geometry rules. Angles in the same segment are equal, the angle in a semicircle is 90°, and the angle between a tangent and radius is always 90°.
💡 Circle theorem tip: Always mark equal angles with the same symbols and clearly state which theorem you're using. Examiners want to see you know the rules!
The alternate segment theorem states that the angle between a tangent and chord equals the angle in the alternate segment. These theorems often combine, so look for multiple relationships in complex diagrams.
9
of 9Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Statistics - Box Plots and Data Analysis
Box plots show five key values: minimum, lower quartile, median, upper quartile, and maximum. The interquartile range (IQR) is UQ - LQ and shows how spread out the middle 50% of data is.
When comparing box plots, discuss both average (median) and consistency (IQR or range). A higher median means higher average values, whilst a smaller IQR means more consistent data.
💡 Statistics tip: Always relate your statistical comments back to the real-world context of the problem. Don't just say "the median is higher" - explain what this means in practical terms.
Stem and leaf diagrams organise data clearly whilst preserving original values. Always include a key and remember to order the leaves from smallest to largest. They make finding the mode, median, and range much easier.
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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Students love us — and so will you.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
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