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Britain & the wider world: 1745 -1901
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0
Charlotte
08/12/2025
Maths
GCSE maths notes
85
•
8 Dec 2025
•
Charlotte
@charlotte26
This maths revision guide covers essential GCSE topics you'll encounter... Show more
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!
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!
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.
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.
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.
Inverse functions undo what the original function does. To find f⁻¹(x), 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(x) mean "do f first, then g". Substitute the entire f(x) expression into g(x). 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.
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 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.
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 .
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.
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.
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.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Quotes from every main character
English LiteratureApp Store
Google Play
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.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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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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!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Inverse functions undo what the original function does. To find f⁻¹(x), 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(x) mean "do f first, then g". Substitute the entire f(x) expression into g(x). 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.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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 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.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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 .
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.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
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.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Quotes from every main character
English LiteratureApp Store
Google Play
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.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Sociology