Looking at your maths notes from March to June 2025,...
Maths181 views·Updated 23 Aug 2026·12 pagesEssential Foundation Maths Topics
Katie 📝@katiekemp
1 / 10
1
of 10
Basic Algebra and Trigonometry Foundations
Your algebra skills are building nicely with expanding brackets like 4 = 4x+24 and more complex expressions such as x+5$$x+2 = x²+7x+10. These bracket expansions are absolutely crucial for your GCSE, so practising these patterns will save you loads of time in exams.
Trigonometry introduces the famous SOH CAH TOA - your best friend for right-angled triangles! Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, and Tan = Opposite/Adjacent.
When finding missing sides, you'll rearrange these ratios. For example, if you know an angle is 35° and the adjacent side is 10cm, you can find the opposite side using tan35° = opposite/10, giving you approximately 7cm.
Top Tip: Always label your triangle sides as opposite, adjacent, and hypotenuse before choosing which trigonometric ratio to use!
2
of 10
Venn Diagrams and Sector Areas
Venn diagrams help you organise numbers into different categories brilliantly. When dealing with prime numbers and odd numbers from 1-12, you'll place numbers like 2 (prime but not odd) in one section, and numbers like 3, 5, 7, 11 (both prime and odd) in the overlap.
Understanding prime numbers (numbers with only two factors: 1 and themselves) is essential. Remember that 2 is the only even prime number!
Sector area calculations use the formula: (angle/360°) × πr². If you've got a 120° sector with radius 9cm, you'd calculate: × π × 9² = 84.8cm². The key is remembering that you're finding a fraction of the whole circle's area.
Remember: Always check if your angle is given in degrees - most GCSE questions use degrees, not radians!
3
of 10
Compound Shapes and Circle Areas
Breaking down compound shapes into familiar rectangles, triangles, and circles makes complex area problems much simpler. You'll often need to add areas together or subtract one shape from another.
For circle calculations, the area formula πr² is your go-to. When you've got overlapping circles or circles with holes, calculate each area separately then combine them logically.
Your examples show areas like 25π - 9π = 16π cm², which demonstrates how algebra skills connect perfectly with geometry. Don't forget to work with π in your answer unless the question specifically asks for a decimal approximation.
Pro Strategy: Always sketch and label your compound shapes clearly - it prevents silly calculation errors and helps you spot which areas to add or subtract!
4
of 10
Understanding Sequences
Linear sequences like 5, 7, 9, 11 have a constant difference (here it's +2). These are straightforward once you spot the pattern.
Quadratic sequences are trickier but follow a clear rule - the differences between terms aren't constant, but the differences between differences are! In your example 2, 5, 10, 17, 26, 37, the first differences are +3, +5, +7, +9, +11, and the second differences are all +2.
Geometric sequences multiply by the same number each time. The sequence 2, 6, 18, 54, 162 has a common ratio of 3 - you multiply by 3 to get from one term to the next.
Quick Check: If differences are constant, it's linear. If second differences are constant, it's quadratic. If you multiply by the same number, it's geometric!
5
of 10
Arc Lengths and Perimeters
Arc length calculations use a similar approach to sector areas but focus on the curved edge rather than the area. The formula is (angle/360°) × πd, where d is the diameter.
For a 144° arc on a circle with radius 9cm, you'd calculate: × π × 18 = 22.6cm. Remember to double the radius to get the diameter!
When finding the perimeter of sectors, you need the arc length plus the two radii. This gives you the complete distance around the shape, which is often what exam questions are really asking for.
Don't Forget: Arc length questions often need the full perimeter, so add those two radius lengths to your arc calculation!
6
of 10
Finding the nth Term
Recognising quadratic sequence patterns helps you tackle more complex nth term problems. Look at sequences like 12, 13, 16, 21, 28, 37 where first differences are +1, +3, +5, +7, +9 and second differences are constant at +2.
For linear sequences, finding the nth term is more straightforward. The sequence 3, 5, 7, 9, 11 has the formula 2n+1. You can check this works: when n=1, you get 2(1)+1=3, when n=2, you get 2(2)+1=5, and so on.
Using your nth term formula, you can find any term in the sequence without listing them all. For 3n+7, the 100th term would be 3(100)+7 = 307.
Test It: Always substitute n=1 and n=2 back into your formula to check it gives you the first two terms of your sequence!
7
of 10
Rearranging Formulas and Pythagoras
Making variables the subject requires systematic steps. For 2y+w=c, you'd subtract w from both sides to get 2y=c-w, then divide by 2 to get y=/2. Work methodically and do the same operation to both sides.
Pythagoras' theorem helps you find missing sides in right-angled triangles. The hypotenuse (longest side) is always c, and you square the two shorter sides then add them together.
For example, with sides 8cm and 6cm, you'd calculate 8²+6²=c², which gives 64+36=100, so c=10cm. If you're finding a shorter side, you'd subtract: if c=13cm and one side is 5cm, then a²+25=169, so a²=144 and a=12cm.
Safety Check: The hypotenuse is always the longest side - if your answer doesn't make sense, double-check your calculation!
8
of 10
Calculating Angles
Vertically opposite angles are always equal - when two lines cross, the angles directly opposite each other are identical. This gives you a quick way to find missing angles.
Alternate angles are equal when you have parallel lines cut by a transversal - they're in a 'Z' pattern and this property helps solve many angle problems.
Understanding that angles in a triangle add to 180° and angles around a point add to 360° gives you the foundation for most angle calculations you'll encounter.
Angle Hunter: Look for angle relationships first - often you can find several angles quickly using these rules before doing any complex calculations!
9
of 10
Angles in Quadrilaterals
Quadrilateral angles always add up to 360°, which makes finding missing angles straightforward once you know the others. Whether it's a square, rectangle, rhombus, or irregular quadrilateral, this rule never changes.
Your practice with rearranging equations continues to build here with more complex examples like making y the subject of various formulas. These algebraic manipulation skills connect directly to your trigonometry and other areas.
The mix of angle work and algebra shows how different maths topics support each other - the equation-solving skills you're developing work across all areas of GCSE mathematics.
Connection Point: Notice how the algebra skills from earlier pages help you solve these geometric problems - maths really does fit together!
10
of 10
Advanced Trigonometry Applications
Using trigonometric ratios to find unknown sides requires careful setup and rearrangement. When the unknown side is in the denominator (like cos40° = 25/x), you multiply both sides by x first, then divide by the trigonometric ratio.
The process becomes: cos40° × x = 25, so x = 25/cos40° = 32.6cm. This systematic approach works for all SOH CAH TOA problems where you're finding the hypotenuse or a side that's being divided by.
Labelling triangles properly (opposite, adjacent, hypotenuse relative to the given angle) is absolutely critical. Get this wrong and you'll choose the wrong ratio, leading to incorrect answers despite perfect calculations.
Method Mastery: Always set up your ratio first, then rearrange algebraically - this two-step approach prevents most trigonometry errors!
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Maths181 views·Updated 23 Aug 2026·12 pagesEssential Foundation Maths Topics
Katie 📝@katiekemp
Looking at your maths notes from March to June 2025, we're covering some seriously important topics that'll set you up for GCSE success. From basic algebra and trigonometry to sequences and compound shapes, these concepts build on each other perfectly.
1
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Basic Algebra and Trigonometry Foundations
Your algebra skills are building nicely with expanding brackets like 4 = 4x+24 and more complex expressions such as x+5$$x+2 = x²+7x+10. These bracket expansions are absolutely crucial for your GCSE, so practising these patterns will save you loads of time in exams.
Trigonometry introduces the famous SOH CAH TOA - your best friend for right-angled triangles! Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, and Tan = Opposite/Adjacent.
When finding missing sides, you'll rearrange these ratios. For example, if you know an angle is 35° and the adjacent side is 10cm, you can find the opposite side using tan35° = opposite/10, giving you approximately 7cm.
Top Tip: Always label your triangle sides as opposite, adjacent, and hypotenuse before choosing which trigonometric ratio to use!
2
of 10Sign up to see the content. It's free!
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Venn Diagrams and Sector Areas
Venn diagrams help you organise numbers into different categories brilliantly. When dealing with prime numbers and odd numbers from 1-12, you'll place numbers like 2 (prime but not odd) in one section, and numbers like 3, 5, 7, 11 (both prime and odd) in the overlap.
Understanding prime numbers (numbers with only two factors: 1 and themselves) is essential. Remember that 2 is the only even prime number!
Sector area calculations use the formula: (angle/360°) × πr². If you've got a 120° sector with radius 9cm, you'd calculate: × π × 9² = 84.8cm². The key is remembering that you're finding a fraction of the whole circle's area.
Remember: Always check if your angle is given in degrees - most GCSE questions use degrees, not radians!
3
of 10Sign up to see the content. It's free!
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By signing up you accept Terms of Service and Privacy Policy
Compound Shapes and Circle Areas
Breaking down compound shapes into familiar rectangles, triangles, and circles makes complex area problems much simpler. You'll often need to add areas together or subtract one shape from another.
For circle calculations, the area formula πr² is your go-to. When you've got overlapping circles or circles with holes, calculate each area separately then combine them logically.
Your examples show areas like 25π - 9π = 16π cm², which demonstrates how algebra skills connect perfectly with geometry. Don't forget to work with π in your answer unless the question specifically asks for a decimal approximation.
Pro Strategy: Always sketch and label your compound shapes clearly - it prevents silly calculation errors and helps you spot which areas to add or subtract!
4
of 10Sign 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
Understanding Sequences
Linear sequences like 5, 7, 9, 11 have a constant difference (here it's +2). These are straightforward once you spot the pattern.
Quadratic sequences are trickier but follow a clear rule - the differences between terms aren't constant, but the differences between differences are! In your example 2, 5, 10, 17, 26, 37, the first differences are +3, +5, +7, +9, +11, and the second differences are all +2.
Geometric sequences multiply by the same number each time. The sequence 2, 6, 18, 54, 162 has a common ratio of 3 - you multiply by 3 to get from one term to the next.
Quick Check: If differences are constant, it's linear. If second differences are constant, it's quadratic. If you multiply by the same number, it's geometric!
5
of 10Sign up to see the content. It's free!
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Arc Lengths and Perimeters
Arc length calculations use a similar approach to sector areas but focus on the curved edge rather than the area. The formula is (angle/360°) × πd, where d is the diameter.
For a 144° arc on a circle with radius 9cm, you'd calculate: × π × 18 = 22.6cm. Remember to double the radius to get the diameter!
When finding the perimeter of sectors, you need the arc length plus the two radii. This gives you the complete distance around the shape, which is often what exam questions are really asking for.
Don't Forget: Arc length questions often need the full perimeter, so add those two radius lengths to your arc calculation!
6
of 10Sign up to see the content. It's free!
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Finding the nth Term
Recognising quadratic sequence patterns helps you tackle more complex nth term problems. Look at sequences like 12, 13, 16, 21, 28, 37 where first differences are +1, +3, +5, +7, +9 and second differences are constant at +2.
For linear sequences, finding the nth term is more straightforward. The sequence 3, 5, 7, 9, 11 has the formula 2n+1. You can check this works: when n=1, you get 2(1)+1=3, when n=2, you get 2(2)+1=5, and so on.
Using your nth term formula, you can find any term in the sequence without listing them all. For 3n+7, the 100th term would be 3(100)+7 = 307.
Test It: Always substitute n=1 and n=2 back into your formula to check it gives you the first two terms of your sequence!
7
of 10Sign up to see the content. It's free!
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Rearranging Formulas and Pythagoras
Making variables the subject requires systematic steps. For 2y+w=c, you'd subtract w from both sides to get 2y=c-w, then divide by 2 to get y=/2. Work methodically and do the same operation to both sides.
Pythagoras' theorem helps you find missing sides in right-angled triangles. The hypotenuse (longest side) is always c, and you square the two shorter sides then add them together.
For example, with sides 8cm and 6cm, you'd calculate 8²+6²=c², which gives 64+36=100, so c=10cm. If you're finding a shorter side, you'd subtract: if c=13cm and one side is 5cm, then a²+25=169, so a²=144 and a=12cm.
Safety Check: The hypotenuse is always the longest side - if your answer doesn't make sense, double-check your calculation!
8
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- Join milions of students
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Calculating Angles
Vertically opposite angles are always equal - when two lines cross, the angles directly opposite each other are identical. This gives you a quick way to find missing angles.
Alternate angles are equal when you have parallel lines cut by a transversal - they're in a 'Z' pattern and this property helps solve many angle problems.
Understanding that angles in a triangle add to 180° and angles around a point add to 360° gives you the foundation for most angle calculations you'll encounter.
Angle Hunter: Look for angle relationships first - often you can find several angles quickly using these rules before doing any complex calculations!
9
of 10Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
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Angles in Quadrilaterals
Quadrilateral angles always add up to 360°, which makes finding missing angles straightforward once you know the others. Whether it's a square, rectangle, rhombus, or irregular quadrilateral, this rule never changes.
Your practice with rearranging equations continues to build here with more complex examples like making y the subject of various formulas. These algebraic manipulation skills connect directly to your trigonometry and other areas.
The mix of angle work and algebra shows how different maths topics support each other - the equation-solving skills you're developing work across all areas of GCSE mathematics.
Connection Point: Notice how the algebra skills from earlier pages help you solve these geometric problems - maths really does fit together!
10
of 10Sign up to see the content. It's free!
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Advanced Trigonometry Applications
Using trigonometric ratios to find unknown sides requires careful setup and rearrangement. When the unknown side is in the denominator (like cos40° = 25/x), you multiply both sides by x first, then divide by the trigonometric ratio.
The process becomes: cos40° × x = 25, so x = 25/cos40° = 32.6cm. This systematic approach works for all SOH CAH TOA problems where you're finding the hypotenuse or a side that's being divided by.
Labelling triangles properly (opposite, adjacent, hypotenuse relative to the given angle) is absolutely critical. Get this wrong and you'll choose the wrong ratio, leading to incorrect answers despite perfect calculations.
Method Mastery: Always set up your ratio first, then rearrange algebraically - this two-step approach prevents most trigonometry errors!
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.
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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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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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.
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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.
AnnaiOS user