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8 Dec 2025
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Mastering Maths Paper 1 Topics: Key Concepts Simplified
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@verity.2009
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Fractions: Converting and Dividing
Ever wondered why converting mixed fractions matters? It's actually the key to solving loads of maths problems! To change a mixed fraction into an improper fraction, multiply the whole number by the denominator, then add the numerator. For example: 1⅛ becomes (1 × 8) + 1 = 9/8.
Dividing fractions is way easier than you think. Just flip the second fraction and multiply instead! So 1 ÷ 4/2 becomes 1/2 × 4/1 = 4/2 = 2.
Quick Tip: Remember that whole numbers like 5 can be written as 5/1 to make fraction division simpler.
The exam loves testing this with real-world problems. Like calculating a sloth's speed when it travels 33/80 of a mile in 2¾ hours. Just use Speed = distance ÷ time, convert the mixed fraction, then flip and multiply!
Grouped Data and Powers
Estimating the mean from grouped data sounds tricky, but it's just a formula you can master. Find the midpoint (Mp) of each group, multiply by frequency (f) to get fx, then use mean = Σfx ÷ Σf. Dead simple once you practise it!
Powers follow brilliant shortcuts that'll save you tons of time. When multiplying powers with the same base, just add the indices: 2³ × 2⁴ = 2⁷. When dividing, subtract the indices: 5⁶ ÷ 5⁴ = 5².
Memory Hook: Think "multiplication = addition" and "division = subtraction" for power rules.
These power rules work with any base - whether it's 2, 5, 7, or even x. Just make sure the bases match before applying the rules. This foundation will be crucial when you tackle more complex algebra later.
Expanding Brackets and Factorising
Expanding brackets is like being a mathematical detective - every term in the first bracket must multiply every term in the second. For , you get x² + 2x + 3x + 6 = x² + 5x + 6.
The key is staying organised with your working. When you see trickier examples like , just methodically multiply each term: 2x² + 4x + x + 2 = 2x² + 5x + 2.
Squared brackets like ² are just shorthand for . Expand them exactly the same way to get 9x² + 6x + 1.
Pro Tip: Factorising is expanding in reverse - find the biggest common factor first.
Factorising means spotting patterns. For 6x - 9, notice that 3 divides into both terms, giving you 3. With algebra like x² + 5x, the common factor is x, so you get x.
Factorising and Cylinder Volume
More factorising examples show how powerful this technique becomes. For 9x + 12x², spot that 3x is the largest common factor, giving 3x. Advanced problems like ² + 3 use as the common factor!
Volume of cylinders uses the trusty prism formula: cross-sectional area × length. Since a cylinder's cross-section is circular, you need πr² for the area, then multiply by the height.
Always remember that radius is half the diameter! For a cylinder with diameter 13cm and length 7.5cm, the volume is π × 6.5² × 7.5 ≈ 995 cm³.
Exam Alert: Questions often ask for answers in litres - just divide cm³ by 1000.
Real exam questions love practical contexts. When a tent packs into a cylindrical bag (diameter 26cm, length 48cm), you calculate π × 13² × 48 ≈ 25,485 cm³ = 25.5 litres.
Sketching Quadratic Graphs
Quadratic graphs always form smooth U-shapes . The equation y = ax² + bx + c creates these distinctive curves that you'll recognise instantly once you've drawn a few.
Creating a table of values is your first step. For y = 2x² - 4x - 2, substitute different x-values into your calculator: when x = 0, you get y = -2; when x = 2, you get y = -2.
Plot these points carefully and join them with a smooth curve - never use straight lines! Your graph should flow naturally through all the points.
Graph Reading Tip: Draw vertical and horizontal lines to find exact coordinate values from your sketch.
Using graphs to solve equations is brilliant once you grasp it. To find where y = 2, draw a horizontal line and read the x-values where it crosses your curve. To solve 2x² - 4x - 2 = 0, find where your graph crosses the x-axis .
Graph Equations and Density Calculations
Sometimes you need to solve different equations using the same graph. If your graph shows y = x² - 4x + 2 but you need to solve x² - 4x - 3 = 1, rearrange to get x² - 4x + 2 = 6, then find where y = 6 on your existing graph.
The mass, density, volume triangle is your best friend for these calculations. Cover the value you want to find - if you cover M (mass), you see D × V, so mass = density × volume.
Density problems often involve finding volume first. For a trapezoidal prism, calculate the cross-sectional area using ½h, then multiply by length. A trapezium with parallel sides 9cm and 11cm, height 6cm, and length 10cm gives volume = 60 × 10 = 600 cm³.
Units Matter: Always check if your final answer needs converting - grams to kilograms means dividing by 1000.
With density 6 g/cm³ and volume 600 cm³, the mass = 6 × 600 = 3600g = 3.6kg. These real-world applications show why these mathematical skills actually matter beyond the classroom.
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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.
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
Mastering Maths Paper 1 Topics: Key Concepts Simplified
V
Verityyyy
@verity.2009
This revision guide covers essential GCSE maths topics that you'll definitely need for your exams. From working with fractions and powers to sketching graphs and calculating volume, these skills build on each other to solve real-world problems.
Fractions: Converting and Dividing
Ever wondered why converting mixed fractions matters? It's actually the key to solving loads of maths problems! To change a mixed fraction into an improper fraction, multiply the whole number by the denominator, then add the numerator. For example: 1⅛ becomes (1 × 8) + 1 = 9/8.
Dividing fractions is way easier than you think. Just flip the second fraction and multiply instead! So 1 ÷ 4/2 becomes 1/2 × 4/1 = 4/2 = 2.
Quick Tip: Remember that whole numbers like 5 can be written as 5/1 to make fraction division simpler.
The exam loves testing this with real-world problems. Like calculating a sloth's speed when it travels 33/80 of a mile in 2¾ hours. Just use Speed = distance ÷ time, convert the mixed fraction, then flip and multiply!
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Grouped Data and Powers
Estimating the mean from grouped data sounds tricky, but it's just a formula you can master. Find the midpoint (Mp) of each group, multiply by frequency (f) to get fx, then use mean = Σfx ÷ Σf. Dead simple once you practise it!
Powers follow brilliant shortcuts that'll save you tons of time. When multiplying powers with the same base, just add the indices: 2³ × 2⁴ = 2⁷. When dividing, subtract the indices: 5⁶ ÷ 5⁴ = 5².
Memory Hook: Think "multiplication = addition" and "division = subtraction" for power rules.
These power rules work with any base - whether it's 2, 5, 7, or even x. Just make sure the bases match before applying the rules. This foundation will be crucial when you tackle more complex algebra later.
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Expanding Brackets and Factorising
Expanding brackets is like being a mathematical detective - every term in the first bracket must multiply every term in the second. For , you get x² + 2x + 3x + 6 = x² + 5x + 6.
The key is staying organised with your working. When you see trickier examples like , just methodically multiply each term: 2x² + 4x + x + 2 = 2x² + 5x + 2.
Squared brackets like ² are just shorthand for . Expand them exactly the same way to get 9x² + 6x + 1.
Pro Tip: Factorising is expanding in reverse - find the biggest common factor first.
Factorising means spotting patterns. For 6x - 9, notice that 3 divides into both terms, giving you 3. With algebra like x² + 5x, the common factor is x, so you get x.
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Factorising and Cylinder Volume
More factorising examples show how powerful this technique becomes. For 9x + 12x², spot that 3x is the largest common factor, giving 3x. Advanced problems like ² + 3 use as the common factor!
Volume of cylinders uses the trusty prism formula: cross-sectional area × length. Since a cylinder's cross-section is circular, you need πr² for the area, then multiply by the height.
Always remember that radius is half the diameter! For a cylinder with diameter 13cm and length 7.5cm, the volume is π × 6.5² × 7.5 ≈ 995 cm³.
Exam Alert: Questions often ask for answers in litres - just divide cm³ by 1000.
Real exam questions love practical contexts. When a tent packs into a cylindrical bag (diameter 26cm, length 48cm), you calculate π × 13² × 48 ≈ 25,485 cm³ = 25.5 litres.
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Sketching Quadratic Graphs
Quadratic graphs always form smooth U-shapes . The equation y = ax² + bx + c creates these distinctive curves that you'll recognise instantly once you've drawn a few.
Creating a table of values is your first step. For y = 2x² - 4x - 2, substitute different x-values into your calculator: when x = 0, you get y = -2; when x = 2, you get y = -2.
Plot these points carefully and join them with a smooth curve - never use straight lines! Your graph should flow naturally through all the points.
Graph Reading Tip: Draw vertical and horizontal lines to find exact coordinate values from your sketch.
Using graphs to solve equations is brilliant once you grasp it. To find where y = 2, draw a horizontal line and read the x-values where it crosses your curve. To solve 2x² - 4x - 2 = 0, find where your graph crosses the x-axis .
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Graph Equations and Density Calculations
Sometimes you need to solve different equations using the same graph. If your graph shows y = x² - 4x + 2 but you need to solve x² - 4x - 3 = 1, rearrange to get x² - 4x + 2 = 6, then find where y = 6 on your existing graph.
The mass, density, volume triangle is your best friend for these calculations. Cover the value you want to find - if you cover M (mass), you see D × V, so mass = density × volume.
Density problems often involve finding volume first. For a trapezoidal prism, calculate the cross-sectional area using ½h, then multiply by length. A trapezium with parallel sides 9cm and 11cm, height 6cm, and length 10cm gives volume = 60 × 10 = 600 cm³.
Units Matter: Always check if your final answer needs converting - grams to kilograms means dividing by 1000.
With density 6 g/cm³ and volume 600 cm³, the mass = 6 × 600 = 3600g = 3.6kg. These real-world applications show why these mathematical skills actually matter beyond the classroom.
We thought you’d never ask...
What is the Knowunity AI companion?
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Students love us — and so will you.
4.9/5
App Store
4.8/5
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