This is a comprehensive GCSE Mathematics Higher Tier exam paper...
Maths264 views·Updated 8 Jul 2026·24 pagesWJEC Maths Unit 1 Practice Paper 2022
Alfie@alfie_revisionguides
1 / 10
1
of 10
Exam Overview
This GCSE Higher Tier mathematics exam lasts 1 hour 45 minutes and is worth 80 marks total. You'll tackle 21 questions without a calculator, so mental maths and written methods are crucial.
The paper covers a brilliant mix of topics that regularly appear in GCSE exams. You'll need basic equipment like a ruler, protractor, and compass for construction work. Remember to use black ink and show all your working clearly.
Top Tip: Each question shows its mark value in brackets – use this to gauge how much detail and time to spend on each part!
The marking scheme ranges from quick 2-mark questions to more complex 6-mark problems. Question 10 specifically assesses your mathematical communication skills, so clear explanations matter.
2
of 10
Essential Formulae
You're given a formula sheet with all the key equations you'll need. The area of a trapezium uses ½h, whilst volume calculations include spheres (⁴⁄₃πr³) and cones (¹⁄₃πr²h).
Trigonometry features heavily with the sine rule, cosine rule, and triangle area formula (½ab sin C). These are absolute lifesavers for triangle problems! The quadratic formula is also provided – memorising the layout helps you substitute values quickly.
Remember: You'll use π = 3.14 throughout this paper, not the calculator value!
The Annual Equivalent Rate formula might seem scary, but it's just compound interest in disguise. Focus on substituting values correctly rather than understanding the complex financial theory.
3
of 10
Probability and Venn Diagrams
Question 1 tests your Venn diagram skills with a real-world scenario about hair colour and glasses. With 200 people total, you need to work systematically through the given information to fill each section correctly.
Start with the intersection (people with black hair AND glasses = 20). Then use the fact that 70 people have black hair total to find those with black hair but no glasses. The key is working step-by-step rather than guessing.
Smart Strategy: Always check your Venn diagram adds up to the total number given!
Part asks for a probability calculation. Once your Venn diagram is complete, simply count everyone wearing glasses and divide by 200. Converting to a fraction in its simplest form usually scores full marks.
4
of 10
Geometric Construction
Question 2 requires accurate geometric construction using only ruler and compass – no protractor allowed! You're constructing triangle ABC where side AC is already drawn, with angles of 60° and 45°.
For the 60° angle, you'll need to construct an equilateral triangle using compass arcs. The 45° angle is trickier – you'll likely need to bisect a 90° angle that you've constructed using perpendicular lines.
Essential Rule: Keep all construction arcs and lines visible – examiners need to see your method!
Remember that construction questions test your geometric skills, not your measuring ability. Accuracy comes from proper compass technique, not trying to estimate angles by eye.
5
of 10
Prime Factorisation and Algebra
Question 3 asks you to express 1575 as a product of prime factors in index form. Start by dividing by small primes (2, 3, 5, 7...) until you're left with 1. Write your answer using powers like 3² × 5³.
The algebra questions test your manipulation skills. For 2p³q × 3p⁴q⁷, multiply the numbers then add the indices for like terms .
Index Law Reminder: When multiplying powers of the same base, add the indices!
The expansion 7a - 2 requires careful bracket expansion followed by collecting like terms. Watch your signs – the minus before the second bracket affects every term inside.
6
of 10
Linear Graphs and Gradients
Question 5 focuses on straight line graphs and their equations. Reading the gradient from a graph means finding how much y increases for every 1 unit increase in x. Count squares carefully or use two clear points.
The y-intercept is where your line crosses the y-axis . Once you have the gradient and y-intercept , you can write the equation as y = mx + c.
Graph Reading Tip: Choose points on grid intersections to avoid reading errors!
This question type appears frequently in GCSE mathematics, so practising graph interpretation is time well spent. Always double-check your equation by substituting a point from the line.
7
of 10
Parallel Lines
Part asks you to prove that y = 3x - 8 and 2y - 6x = 23 are parallel lines. Parallel lines have identical gradients, so rearrange both equations into y = mx + c form.
The first equation already shows gradient = 3. For the second equation, rearrange: 2y = 6x + 23, so y = 3x + 11.5. Both have gradient 3, proving they're parallel.
Parallel Line Rule: Same gradient = parallel lines, regardless of the y-intercept!
This algebraic proof technique is essential for higher-tier questions. Show each step clearly to demonstrate your understanding of linear equations.
8
of 10
Dimensional Analysis
Question 6 tests your understanding of dimensional analysis – working out whether formulae represent length, area, volume, or none. Each letter represents a length measurement.
For area, you need dimensions that multiply to give length² (like ab or a²). For volume, you need length³ (like abc or a³). When terms are added or subtracted, they must have the same dimensions.
Key Rule: You can only add or subtract quantities with the same dimensions!
Look at each term separately, then check if they're compatible. For example, 5abc - 6bc + b² mixes volume and area terms, so it represents "none of these". This mathematical reasoning skill becomes crucial in advanced mathematics.
9
of 10
Standard Form Calculations
Question 7 tests standard form arithmetic without a calculator. For (3×10⁴) ÷ (6×10⁻³), divide the numbers and subtract the powers .
Your answer becomes 0.5×10⁷, which converts to 5×10⁶ in proper standard form. Remember that the first number must be between 1 and 10.
Standard Form Rule: When dividing powers of 10, subtract the indices!
For addition like (4.78×10⁴) + (1.5×10²), convert to the same power of 10 first. This gives (4.78×10⁴) + (0.015×10⁴) = 4.795×10⁴. Standard form questions are common in GCSE papers, so master these techniques.
10
of 10
Pythagoras and Trigonometry
Question 8 tests your understanding of when to use Pythagoras's theorem versus trigonometry. For finding the hypotenuse x, you need x = √ since you're adding the squares of the two shorter sides.
For the trigonometry part, you have the hypotenuse (25 cm) and need the opposite side y to a 40° angle. Use sin 40° = y/25 because sine relates opposite to hypotenuse.
SOHCAHTOA Reminder: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent
Recognising which method to use is crucial for GCSE success. Pythagoras works with right-angled triangles when you know two sides, whilst trigonometry needs an angle and one side.
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Maths264 views·Updated 8 Jul 2026·24 pagesWJEC Maths Unit 1 Practice Paper 2022
Alfie@alfie_revisionguides
This is a comprehensive GCSE Mathematics Higher Tier exam paper from WJEC, covering essential non-calculator maths skills. You'll find everything from Venn diagrams and prime factorisation to trigonometry and standard form – all the core topics you need to master...
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Exam Overview
This GCSE Higher Tier mathematics exam lasts 1 hour 45 minutes and is worth 80 marks total. You'll tackle 21 questions without a calculator, so mental maths and written methods are crucial.
The paper covers a brilliant mix of topics that regularly appear in GCSE exams. You'll need basic equipment like a ruler, protractor, and compass for construction work. Remember to use black ink and show all your working clearly.
Top Tip: Each question shows its mark value in brackets – use this to gauge how much detail and time to spend on each part!
The marking scheme ranges from quick 2-mark questions to more complex 6-mark problems. Question 10 specifically assesses your mathematical communication skills, so clear explanations matter.
2
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
Essential Formulae
You're given a formula sheet with all the key equations you'll need. The area of a trapezium uses ½h, whilst volume calculations include spheres (⁴⁄₃πr³) and cones (¹⁄₃πr²h).
Trigonometry features heavily with the sine rule, cosine rule, and triangle area formula (½ab sin C). These are absolute lifesavers for triangle problems! The quadratic formula is also provided – memorising the layout helps you substitute values quickly.
Remember: You'll use π = 3.14 throughout this paper, not the calculator value!
The Annual Equivalent Rate formula might seem scary, but it's just compound interest in disguise. Focus on substituting values correctly rather than understanding the complex financial theory.
3
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
Probability and Venn Diagrams
Question 1 tests your Venn diagram skills with a real-world scenario about hair colour and glasses. With 200 people total, you need to work systematically through the given information to fill each section correctly.
Start with the intersection (people with black hair AND glasses = 20). Then use the fact that 70 people have black hair total to find those with black hair but no glasses. The key is working step-by-step rather than guessing.
Smart Strategy: Always check your Venn diagram adds up to the total number given!
Part asks for a probability calculation. Once your Venn diagram is complete, simply count everyone wearing glasses and divide by 200. Converting to a fraction in its simplest form usually scores full marks.
4
of 10Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Geometric Construction
Question 2 requires accurate geometric construction using only ruler and compass – no protractor allowed! You're constructing triangle ABC where side AC is already drawn, with angles of 60° and 45°.
For the 60° angle, you'll need to construct an equilateral triangle using compass arcs. The 45° angle is trickier – you'll likely need to bisect a 90° angle that you've constructed using perpendicular lines.
Essential Rule: Keep all construction arcs and lines visible – examiners need to see your method!
Remember that construction questions test your geometric skills, not your measuring ability. Accuracy comes from proper compass technique, not trying to estimate angles by eye.
5
of 10Sign up to see the content. It's free!
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Prime Factorisation and Algebra
Question 3 asks you to express 1575 as a product of prime factors in index form. Start by dividing by small primes (2, 3, 5, 7...) until you're left with 1. Write your answer using powers like 3² × 5³.
The algebra questions test your manipulation skills. For 2p³q × 3p⁴q⁷, multiply the numbers then add the indices for like terms .
Index Law Reminder: When multiplying powers of the same base, add the indices!
The expansion 7a - 2 requires careful bracket expansion followed by collecting like terms. Watch your signs – the minus before the second bracket affects every term inside.
6
of 10Sign up to see the content. It's free!
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Linear Graphs and Gradients
Question 5 focuses on straight line graphs and their equations. Reading the gradient from a graph means finding how much y increases for every 1 unit increase in x. Count squares carefully or use two clear points.
The y-intercept is where your line crosses the y-axis . Once you have the gradient and y-intercept , you can write the equation as y = mx + c.
Graph Reading Tip: Choose points on grid intersections to avoid reading errors!
This question type appears frequently in GCSE mathematics, so practising graph interpretation is time well spent. Always double-check your equation by substituting a point from the line.
7
of 10Sign up to see the content. It's free!
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Parallel Lines
Part asks you to prove that y = 3x - 8 and 2y - 6x = 23 are parallel lines. Parallel lines have identical gradients, so rearrange both equations into y = mx + c form.
The first equation already shows gradient = 3. For the second equation, rearrange: 2y = 6x + 23, so y = 3x + 11.5. Both have gradient 3, proving they're parallel.
Parallel Line Rule: Same gradient = parallel lines, regardless of the y-intercept!
This algebraic proof technique is essential for higher-tier questions. Show each step clearly to demonstrate your understanding of linear equations.
8
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
Dimensional Analysis
Question 6 tests your understanding of dimensional analysis – working out whether formulae represent length, area, volume, or none. Each letter represents a length measurement.
For area, you need dimensions that multiply to give length² (like ab or a²). For volume, you need length³ (like abc or a³). When terms are added or subtracted, they must have the same dimensions.
Key Rule: You can only add or subtract quantities with the same dimensions!
Look at each term separately, then check if they're compatible. For example, 5abc - 6bc + b² mixes volume and area terms, so it represents "none of these". This mathematical reasoning skill becomes crucial in advanced mathematics.
9
of 10Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Standard Form Calculations
Question 7 tests standard form arithmetic without a calculator. For (3×10⁴) ÷ (6×10⁻³), divide the numbers and subtract the powers .
Your answer becomes 0.5×10⁷, which converts to 5×10⁶ in proper standard form. Remember that the first number must be between 1 and 10.
Standard Form Rule: When dividing powers of 10, subtract the indices!
For addition like (4.78×10⁴) + (1.5×10²), convert to the same power of 10 first. This gives (4.78×10⁴) + (0.015×10⁴) = 4.795×10⁴. Standard form questions are common in GCSE papers, so master these techniques.
10
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
Pythagoras and Trigonometry
Question 8 tests your understanding of when to use Pythagoras's theorem versus trigonometry. For finding the hypotenuse x, you need x = √ since you're adding the squares of the two shorter sides.
For the trigonometry part, you have the hypotenuse (25 cm) and need the opposite side y to a 40° angle. Use sin 40° = y/25 because sine relates opposite to hypotenuse.
SOHCAHTOA Reminder: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent
Recognising which method to use is crucial for GCSE success. Pythagoras works with right-angled triangles when you know two sides, whilst trigonometry needs an angle and one side.
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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