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Essential Foundation Maths Topics

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Katie ๐Ÿ“

15/12/2025

Maths

Foundation Maths

46

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15 Dec 2025

โ€ข

12 pages

Essential Foundation Maths Topics

user profile picture

Katie ๐Ÿ“

@katiekemp

Looking at your maths notes from March to June 2025,... Show more

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Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

Basic Algebra and Trigonometry Foundations

Your algebra skills are building nicely with expanding brackets like 4x+6x+6 = 4x+24 and more complex expressions such as x+5x+5x+2x+2 = xยฒ+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 35ยฐ and the adjacent side is 10cm, you can find the opposite side using tan35ยฐ = 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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/360ยฐangle/360ยฐ ร— ฯ€rยฒ. If you've got a 120ยฐ sector with radius 9cm, you'd calculate: (120/360) ร— ฯ€ ร— 9ยฒ = 84.8cmยฒ. 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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 ฯ€rยฒ 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 25ฯ€ - 9ฯ€ = 16ฯ€ cmยฒ, which demonstrates how algebra skills connect perfectly with geometry. Don't forget to work with ฯ€ 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

Understanding Sequences

Linear sequences like 5, 7, 9, 11 have a constant difference hereitโ€ฒs+2here 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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/360ยฐangle/360ยฐ ร— ฯ€d, where d is the diameter.

For a 144ยฐ arc on a circle with radius 9cm, you'd calculate: (144/360) ร— ฯ€ ร— 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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=cโˆ’wc-w/2. Work methodically and do the same operation to both sides.

Pythagoras' theorem a2+b2=c2aยฒ+bยฒ=cยฒ 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 8ยฒ+6ยฒ=cยฒ, 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 aยฒ+25=169, so aยฒ=144 and a=12cm.

Safety Check: The hypotenuse is always the longest side - if your answer doesn't make sense, double-check your calculation!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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 180ยฐ and angles around a point add to 360ยฐ 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

Angles in Quadrilaterals

Quadrilateral angles always add up to 360ยฐ, 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

Advanced Trigonometry Applications

Using trigonometric ratios to find unknown sides requires careful setup and rearrangement. When the unknown side is in the denominator likecos40ยฐ=25/xlike cos40ยฐ = 25/x, you multiply both sides by x first, then divide by the trigonometric ratio.

The process becomes: cos40ยฐ ร— x = 25, so x = 25/cos40ยฐ = 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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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.

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Best app on earth! no words because itโ€™s too good

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Maths

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15 Dec 2025

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12 pages

Essential Foundation Maths Topics

user profile picture

Katie ๐Ÿ“

@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.

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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Basic Algebra and Trigonometry Foundations

Your algebra skills are building nicely with expanding brackets like 4x+6x+6 = 4x+24 and more complex expressions such as x+5x+5x+2x+2 = xยฒ+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 35ยฐ and the adjacent side is 10cm, you can find the opposite side using tan35ยฐ = 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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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/360ยฐangle/360ยฐ ร— ฯ€rยฒ. If you've got a 120ยฐ sector with radius 9cm, you'd calculate: (120/360) ร— ฯ€ ร— 9ยฒ = 84.8cmยฒ. 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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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 ฯ€rยฒ 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 25ฯ€ - 9ฯ€ = 16ฯ€ cmยฒ, which demonstrates how algebra skills connect perfectly with geometry. Don't forget to work with ฯ€ 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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Understanding Sequences

Linear sequences like 5, 7, 9, 11 have a constant difference hereitโ€ฒs+2here 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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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/360ยฐangle/360ยฐ ร— ฯ€d, where d is the diameter.

For a 144ยฐ arc on a circle with radius 9cm, you'd calculate: (144/360) ร— ฯ€ ร— 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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Join milions of students

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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=cโˆ’wc-w/2. Work methodically and do the same operation to both sides.

Pythagoras' theorem a2+b2=c2aยฒ+bยฒ=cยฒ 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 8ยฒ+6ยฒ=cยฒ, 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 aยฒ+25=169, so aยฒ=144 and a=12cm.

Safety Check: The hypotenuse is always the longest side - if your answer doesn't make sense, double-check your calculation!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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Join milions of students

By signing up you accept Terms of Service and Privacy Policy

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 180ยฐ and angles around a point add to 360ยฐ 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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Access to all documents

Improve your grades

Join milions of students

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Angles in Quadrilaterals

Quadrilateral angles always add up to 360ยฐ, 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!

Wednesday 26th March 2025.
la) 54+343=397
2)4(x+6)=4X+24
b) 119-42=77
C) 18x4= 72
d) 2x5x8=80
3) 5(x-4)=6x-20
4)x(3+x) = 3x + x2
Friday 28 M

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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 likecos40ยฐ=25/xlike cos40ยฐ = 25/x, you multiply both sides by x first, then divide by the trigonometric ratio.

The process becomes: cos40ยฐ ร— x = 25, so x = 25/cos40ยฐ = 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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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

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iOS user

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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.

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Greenlight Bonnie

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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.

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Android user

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Xander S

iOS user

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