gcse maths coursework examples

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Extended Tasks for GCSE Mathematics

This series, which formed a support package for GCSE coursework in mathematics, was developed as part of a joint project by the Shell Centre for Mathematical Education and the Midland Examining Group. The project followed the announcement in January 1984, by Sir Keith Joseph, the then Secretary of State for Education and Science to introduce a new common 16+ examination in England, Wales and Northern Ireland. This new examination, the General Certificate of Secondary Education (GCSE) included coursework which formed part of the school-based assessed element. Teachers were being asked to adopt a different role from that which they had previously, and in most cases successfully, used. There were to be new methods of assessment, carried out in the classroom, and a greater emphasis on practical and investigative work. These materials discuss the changing roles of teachers and students and offer ideas, suggestions and examples of investigations carried out by students along with extensisve teacher's notes and guidance. This resource comprises: *The Teacher's Guide *IMPACT - a departmental development programme *Eight books of investigations classified into four categories *Pure investigations, Statistics, Practical Geometry and Applications. Each of the eight books offer a lead task which is fully supported by detailed teacher's notes, a student's introduction to the problem, a case study, examples of students' work which demonstrate achievement at a variety of levels, together with six alternative tasks of a similar nature. The alternative tasks simply comprise the student's introduction to the problem and some brief teacher's notes. It was intended that these alternative tasks should be used in a similar manner to the lead task and hence only the lead task has been fully supported with more detailed teacher's notes and examples of students' work. The books conclude with comments from an examining board moderator.

• Mathematics
• Open-ended task
• Teacher guidance
• Include Physical Resources

Practical Geometry

These two books from the Shell Centre are part of the Extended Tasks for GCSE Mathematics support material produced for students as they pursued practical geometry tasks within any mathematics scheme. The practical geometry tasks were intended to stimulate students'...

Applications

These two books from the Shell Centre focus on applications. The tasks are intended to stimulate students' interest in, and understanding of, the world in which they live. As they pursue these tasks students will be involved in selecting materials and mathematics to use...

Pure Investigations

These two books from the Shell centre focus on the pure investigations. The pure investigation tasks are, perhaps, rather different from the other two main types of extended task, those of a practical nature and those of an applied nature, in the sense that they allow...

Extended Tasks for GCSE Mathematics: Teacher's Guide and Impact

The teacher’s guide from the Shell Centre which accompanies the series of modules to support school-based assessment is the main guide to the materials. It makes some suggestions as to how the materials might best be used. It was not intended that this guide should be...

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• Open Box Problem

Maths Coursework

For this maths coursework, I will be investigating the volume of different sized open boxes. I will look at the different sizes of the squares to see which gives the biggest volume. I am going to be using both square and rectangular sheets of card for this task.

Here is a diagram to demonstrate what I will be doing and illustrate the layout of the sheet of card. The parts labelled ‘x’  will be the squares that I will cut out which its height and sizes will be identical as the other cut out squares.

      x                                                x

 x                                                           x

     x                                                 x

I will now illustrate a diagram of the box after it has been assembled. The sections labelled x  is the height of the box.

To work out the volume of the box, I will use the following formula:

Length x Width x Height

For the first part of my coursework, I will be looking at square pieces of card. I will be investigating square pieces of card and study the relationships between the different sized square cut outs. I am going to look for the square cut out which gives the highest volume overall out of all the rectangles.

The squares of sizes I will look at are: 12cm x 12cm, 15cm x 15cm and 18cm x 18cm.

The first square I will look at is 12cm by 12cm.

The cut out square which proved to have the biggest volume was 2cm by 2cm.

The second square cut out I will look into is 15cm x 15cm

The cut out square which gave the highest volume is 2.5cm by 2.5cm.

The third cut out square I will observe is 18cm x 18cm.

The cut out square which gave the biggest volume was 3cm by 3cm.

Summary Table

0.16 is 1/6.

The size of the cut out square is approximately 1/6 of the whole square.

From looking at my results table, I predict that the biggest volume for a 14 x 14 square will be completed by cutting a square which is 1/6 of 14.

14cm x 14cm square

1/6 of 14=2.33

I will now check if my prediction is correct by verifying if it is the biggest volume.

This is a preview of the whole essay

Looking at the results in my prediction table, it proves that my prediction was correct; the cut out square with the largest volume was 2.33 x 2.33.    

I will now investigate rectangular pieces of card; I will begin by examining rectangles where the length is twice the size of the width, so the ratio of the width to the length is 1:2.

I have illustrated a diagram below to demonstrate the layout of the rectangle and what I am going to be doing. The parts labelled x will be the cut out squares and will be equal size/height to the other cut out squares.

      x                                                            x

 x                                                                      x

  x                                                                     x

I will now illustrate a diagram to show what the rectangular box will look like when it is assembled. The parts labelled: X=height, W=width and L=length.

                                                                                   X

                                                                        W

                       L

I will look at rectangles of sizes: 14cm x 28cm, 15 cm x 30cm and 16cm x 32cm.

To begin with, I will look at the rectangles with dimensions of 14cm x 28cm.

The square cut out which gave the biggest volume using the formula L x W x H was 2.95 x 2.95.

Secondly, I will look at the rectangle 15cm x 30cm.

The cut out square which gave the largest volume was 3.15 x 3.15 by using the formula L x W H.

Last of all, I am going to look at 16cm x 32cm.

From using the formula L x W x H to calculate, the square cut out which gave the highest volume was 3.4 x 3.4.

Results Table - Maximum volume- ratio 1:2.

Looking at my results, I can tell that the best square to cut the corners from is almost 105/1000 of the rectangle’s length.

30cm x 60cm

I predict that the cut out square will be 6.3 x 6.3.

As I can see from my results table, the cut out square is 6.3 x 6.3; therefore, my prediction was correct.

The cut out square which gave the highest volume was 2.25 x 2.25.

I have been looking at rectangles in the ratio 1:2, I am now going to be investigating rectangles where the width and the length are in the ratio 1:3 so the length is 3 times the size of the width.

I will look at rectangles of sizes: 10 x 30, 12 x 36, and 14 x 42.

I will begin by looking at rectangle of size 10 x 30.

Secondly, I will look at 12 x 36.

The cut out square which gave the largest volume is 2.7 x 2.7.

Lastly, I will look at the rectangle 14 x 42.

The square cut out which gave the biggest volume is 3.15 x 3.15.

From the results in the table, I can see that the best square to cut the corners from is approximately 75/1000 of the whole of the rectangle’s length.

30cm x 90cm

I will predict that the cut out square will be 6.75 x 6.75.

From my prediction table, I can see that my prediction was 100% correct; the cut out square which gave the highest volume is 6.75 x 6.75.

These were my results for the square and my first two rectangles using the ratios: 1:1 and 1:2, 1:3.

This took me a while to calculate the proportion of the square or rectangle which gave the largest volume. I will now be using differentiation to look for the proportion of rectangle which gives the maximum volume.

My teacher suggested researching this method to extend my investigation further, so I will now look at rectangles where the width and length are in the ratio 1:4 and 1:5.

I will look at a rectangle with the ratio 1:4 where the length is 4 times the size of the width and the rectangle I am going to use is 10cm x 40cm.

Length=40-2x

Width=10-2x

So the volume is (40-2x)(10-2x)x

Y=(4x²-100x+400)x

Y=4x³-100x²+400x

     X     40        -2x

   10     400      -20x

  -2x     -80x      4x²  

Equation of the curve

 = 12x²-200x+400

x=-b±   b² - 4ac

                 2a

    x=-(-200)±   ((-200)²-4(12)(400))

                        2(12)

x=200±   20800

              24

So x will equal:

x=200+  20800               or               200-  20800

           24                                                  24

x=(200+144.222051)                         (200-144.222051)

                24                                               24

=14.34258346                                    =2.324081208

The size of the corner which gives the largest volume is 2.324081208 or 2.3 (to 2 significant figures); it cannot be possible to cut a corner of size 14.3cm from a rectangle of size 10 x 40.

I am now going to look at rectangles with the ratio 1:5 where the length is 5 times the size of the width. The rectangle I am going to use is 20cm x 100cm.

Length=100-2x

Width=20-2x

So the volume is=(100-2x)(20-2x)x

Y=(4x²-240x+2000)x

Y=4x³-240x²+2000x

    X    20          -2x

  100   2000      -200x

  -2x    -40x        4x²

      = 12x²-480x+2000

x=-b± √ b²-4ac

             2a

x= 480± √(480)²-4(12)(2000)  

                      2(12)      

x= 480± √ 134400

             24

x = 480+√134400                        or                   = 480- √134400  

            24                                                                24

x = (480+366.6060556)                                    (480-366.6060556)                                                

                 24                                                             24

       =32.27525232                                            =4.724747683

The size of the corner which gives the biggest volume is 4.724747683 or 4.7 (to 2 significant figures); it is unlikely to cut a corner of size 32.3cm from a rectangle which is 20cm x 100cm.

Differentiation for the ratio 1:10

The rectangle I will look at is 8cm x 80cm.

Length= 80-2x

Width= 8-2x

Therefore the volume is V = (80 – 2x)(8-2x)x

                                     V = (80-2x)(8x-2x²)

                                     V = 640x – 160x² - 16x² - 4x³

                                     V = 4x³ - 176x² + 640x

Equation of the curve:

dv = 12x² - 352x + 640

x = -b± √ b²-4ac

           2a

x = 352±√(-352)²-4(12)(640)

                    2(12)

x = 27.38585602           or          = 1.947477314

Differentiation for 1:20

The rectangle I will look at is 5:100

Length = 100-2x

Width = 5-2x

So the volume is V = (100-2x)(5-2x)x

                         V = (100-2x)(5x-2x²)

                         V = 500x – 200x² - 10x² + 4x³

                         V = 4x³ - 210x² + 500x

dv = 12x³ - 420x + 500

x = -b±√ b²-4ac

x = 420± √-420 ²-4(12)(500)

              2(12)

x = 33.76601775           or                 1.233982253

I am now going to collect my results. I will look at the proportion of both length and width, which gives the maximum volume.

Teacher Reviews

Here's what a teacher thought of this essay.

Cornelia Bruce

*** This is a well structured investigation. It uses high level mathematics to appropriately determine the relationship between length and volume. To improve this the mathematics should be described in more detail and linked to the desired investigation outcomes. Strengths and improvements have been suggested throughout.

Document Details

• Word Count 2763
• Page Count 19
• Subject Maths

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25 GCSE Maths Questions And Answers By Topic And Difficulty

Lindsey ford.

GCSE maths questions have always been a popular part of the online lessons and maths revision resources that Third Space Learning has developed for secondary schools. Here we provide 25 of these to give students and teachers a flavour of the full range of question types you’ll need to be familiar with.

All the exam questions have been inspired by real life questions that have appeared in past papers for the AQA, OCR and Edexcel GCSE. Many of the questions are written in a diagnostic quiz format with several multiple choice options that have been designed to highlight key misconceptions and all questions come with answers.

To accompany the blog there is also a free downloadable worksheet of all 25 GCSE maths questions and answers. Third Space Learning also offers a wide range of GCSE maths worksheets and GCSE maths past papers .

Varied challenge level

1. factorising maths question, 2. surds maths question, 3. algebraic fractions maths question, 4. quadratic graphs maths questions, 5. nth term maths question, 6. algebraic fractions maths question, 7. venn diagram maths question, 8. quadratic equations maths question, 9. area of a circle maths question, 10. rearranging formulae maths question, 11. tree diagrams maths question, 12. circle theorems maths question, 13. direct and inverse proportion maths question, 14. standard form maths question, 15. histograms maths question, 16. straight line graphs maths question, 17. pythagoras’ theorem maths question, 18. parallel lines maths question, 19. lowest common multiple and highest common factor maths questions, 20. venn diagrams maths question, 21. solving equations maths question, 22. trigonometry maths question, 23. order of operations (bidmas) maths question, 24. formulae maths question, 25. density maths question.

GCSE papers need to be accessible to students with a wide range of abilities. This enables the exam boards to produce a distribution of results in order to calculate appropriate GCSE grade boundaries . Therefore, every exam paper will have questions which can present challenges to students.

Elements that can increase the difficulty of the question include: 

• the language in the question, 
• presenting a topic in an unfamiliar context, 
• limited structure and guidance,
• interleaving of skills.

In the following GCSE exam questions, you will find questions for a range of key GCSE maths topics from Grade 9 to foundation. The level of difficulty for each GCSE maths question has been determined by:

• the complexity of the question, 
• common errors students are likely to make on a question, 
• how many students are likely to be able to access the question,
• my own experience preparing students for GCSE maths exams for many years.

Grade 8 and Grade 9 style GCSE maths questions

These GCSE questions require a secure set of mathematical skills and accuracy in applying them. The questions may look simple in their presentation, but they require key skills to be carefully selected and applied. These are skills very often revisited as an introduction to an A Level maths course.

(a) Factorise d^{2} – e^{2}

(b) Hence, or otherwise, simplify fully

Which is the correct answer?

A) (x + 8)(x – 4) B) x^{2} + 12 C) x^{2} + 4 D) 24x^{2} + 48

(a) (d+e)(d-e)

(b) D) 24x^{2} + 48

Answer explanation For b) students may not use the difference of two squares as they did in part a) and proceed to try and expand then simplify the brackets. While this is a correct method, it is a longer process and is therefore subject to potential errors. In order to spot the more efficient method of the difference of two squares, students first have to recognise part b) is in the same form as part a) and that it can be rewritten in the form (x 2 + 8 − x 2 + 4)(x 2 + 8 + x 2 − 4). Next, the algebra needs to be simplified with extra care needed to ensure that the negative and positive signs are correct. Finally, the student needs to decide how to present the fully simplified answer.

For more GCSE maths questions take a look at our free Factorising lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Show that \frac{2\sqrt{14}}{\sqrt{5}}-\frac{\sqrt{7}}{\sqrt{10}} can be written in the form \frac{a\sqrt{b}}{10}

Where a and b are integers.

A) \frac{3\sqrt{7}}{10} B) \frac{3\sqrt{70}}{10} C) \frac{2\sqrt{7}}{10} D) \frac{\sqrt{7}}{\sqrt{10}}

Non-calculator

B) \frac{3\sqrt{70}}{10}

Answer explanation This question requires confidence with subtracting fractions and manipulating surds. There are various methods to completing this question, but the easiest way is to multiply the numerator and denominator of the first fraction by √2 so that both fractions have a common denominator of √10. This can present difficulties as the numerator could be multiplied by 2 rather than √2. Students then need to recognise that once both fractions have common denominators, they can simply subtract the surds on the numerator. Finally, they need to rationalise the denominator by multiplying the numerator and denominator by √10 and simplifying.

For more GCSE maths questions take a look at our free Surds lesson and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

2-\frac{x+2}{x-3}-\frac{2 x-5}{x+3} can be written as a single fraction in the form \frac{ax^{2}+b x+c}{x^{2}-9}

Work out the value of a and the value of b and c .

=\frac{2(x-3)(x+3)-(x+2)(x+3)-(2 x-5)(x-3)}{(x+3)(x-3)}

=\frac{2\left(x^2-9\right)-\left(x^2+5 x+6\right)-\left(2 x^2-6 x-5 x+15\right)}{x^2-9}

=\frac{2 x^2-18-x^2-5 x-6-2 x^2+11 x-15}{x^2-9}

=\frac{-x^2+6 x-39}{x^2-9}

a=-1, b=6, c=-39

Answer explanation Students need to be confident with a variety of skills including order of operations, subtracting fractions and integers, expanding brackets and simplifying. There are a variety of mistakes that can be made when expanding brackets and simplifying the fractions. It is important to consider the effect of negative signs when subtracting brackets.

For more GCSE maths questions take a look at our free Algebraic fractions lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Set of 25 Printable GCSE Maths Questions

Try these challenging GCSE maths questions with your students in class. They all include answers on a separate page and are organised by difficulty!

Hard GCSE maths questions

The following GCSE maths questions have been selected to highlight the more unusual way they are presented. This can lead to assumptions being made about the question, which may lead to an incorrect approach.

The diagram shows part of the graph y = x^{2} – 3x + 4 .

(a) By drawing a suitable line, use your graph to find estimates for the solutions of x^{2} – 2x -1 = 0 .

P is a point of the graph y = x^{2} – 3x + 4 where x = 4 .

(b) Calculate an estimate for the gradient of the graph at the point P .

(a) x = 2.4, -0.4

(b) Answers around 4

Answer explanation Students may try to solve the equation by attempting to factorise or use the quadratic formula. However, as the question states, to use the graph, students need to compare the equation of the graph with the equation given in a) and identify what has changed. They need to identify that the x value has decreased by 1 and the constant value has increased by 5. Therefore, they need to accurately plot the line y = − x + 5 on the graph. Finally, they need to remember that the two solutions are found at the two intersection points. Part b) is a more challenging question as they need to find appropriate points either side of the point P to calculate the gradient.

For more GCSE maths questions take a look at our free Quadratic graphs lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

The n th of a sequence is given by an^{2} + bn where a and b are integers.

The 3rd term of the sequence is 3 . The 6th term of the sequence is 42 .

Find the 5th term of the sequence.

A) 29 B) 5 C) 25 D) 18

Answer explanation Students need to construct the two equations by substituting n = 3 and n = 6 in an 2 + bn in order to produce a pair of simultaneous equations which can then be solved. After calculating the values of a and b , students need to remember to finish the question by substituting ‘5’ into the nth term.

For more GCSE maths questions take a look at our free Nth term lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Solve \frac{4x+1}{4}-\frac{2x+1}{3}=\frac{1-x}{6}

A) x = 0.5 B) x = -0.5 C) x = – 5 D) x = 12

Answer explanation Fractions can present difficulties for both higher and foundation students. As this question gives two fractions subtracting to give a third, students may attempt to find a common denominator for all three fractions in the first step. When subtracting the fractions, students must take care to insert brackets and/or multiply every term. E.g. if 4 x + 1 is multiplied by 3, this should be 3(4 x + 1) or 12 x + 3. A common mistake would be to not multiply the term that is outside the bracket by every term inside the bracket.

GCSE higher maths questions

These GCSE maths questions appear on the higher tier paper and draw on a range of skills. They often lack structure and hints, and require the student to confidently work out what is required to work out the solution.

60 people were asked if they spoke French, German or Italian.

Of these people,

• 21 speak French
• 1 speaks French, German and Italian
• 4 speak French and Italian but not German
• 7 speak German and Italian
• 18 do not speak any of the languages
• All 11 people who speak German speak at least one other language.

Two of the 60 people are chosen at random. Work out the probability that they both only speak French.

A) \frac{12}{60} B) \frac{13}{295} C) \frac{1}{25} D) \frac{11}{295}

D) \frac{11}{295}

Answer explanation Students need to recognise that this question requires a 3-way Venn diagram as it could easily be mistaken for a two-way table question. Extracting the correct meaning from the language used in the question can present a challenge in order to place the numbers correctly in the diagram or calculate missing values. To answer the final question, they need to correctly identify the probability that the person speaks only French and multiply these probabilities.

For more GCSE maths questions take a look at our free Venn diagram lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

In the diagram, the square and the trapezium share a common side length of x cm.

The area of the square is equal to the area of the trapezium.

Work out the value of x .

A) 12 cm B) 6 cm C) 64 cm D) 8 cm

Answer explanation Students often find forming equations difficult. After recognising that they need to form a quadratic equation, they need to apply the formula for the area of a trapezium (one which students often struggle to recall). Once they have successfully constructed the correct equation, they then need to rearrange it to make the equation equal to 0 and then factorise. A common mistake that is made when students see terms on each side of the equals sign, is that they attempt to solve as though it is a linear equation. Once the equations have been correctly solved, they need to interpret their solutions by only selecting the positive result. This is because it is impossible to have a negative distance for a side length.

For more GCSE maths questions take a look at our free Quadratic equations lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

A square with sides of length x mm, is inside a circle. Each vertex of the square is on the circumference of the circle.

The area of the circle is 64 mm ^{2} .

Work out the value of x . Give your answer correct to 3 significant figures.

A) 4.51 mm B) 20.35 mm C) 6.38 mm D) 40.74 mm

Answer explanation The first challenge with this question is the absence of a diagram. Students need to interpret the question correctly and create their own diagram. Students need to recall the formula for area of a circle and rearrange it to calculate the radius, which is also the distance from the centre of the square to one of its vertices . Students are required to know that the diagonal lines within a square are perpendicular and therefore need to apply Pythagoras’ theorem to calculate the side length of the square. Finally, it is important they correctly round their answer to 3 significant figures.

For more GCSE maths questions take a look at our free Area of a circle lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Make b the subject of the formula c=\frac{25(a-3b)}{b}

A) b=\frac{cb-25a}{3} B) b=\frac{25a}{c+75} C) b=\frac{a}{c+75} D) b=\frac{c b-25 a}{-75}

B) b=\frac{25a}{c+75}

Answer explanation A rearranging question where the subject appears twice will be accessible to more able students. However, this question has some unfamiliar elements compared to the usual rearranging questions which require factorising, due to the placing of the brackets and b appearing on its own as the denominator.

For more GCSE maths questions take a look at our free Rearranging formulae lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

David has two spinners, spinner A and spinner B .

Each spinner can only land on blue or green.

The probability that spinner A will land on blue is 0.25.

The probability that spinner B will land on green is 0.8.

The probability tree diagram shows this information.

David spins spinner A and spinner B together.

He does this a number of times.

The number of times both spinners land on blue is 40.

Work out an estimate for the numbers of times both spinners land on green.

A) 800 B) 64 C) 128 D) 480

Answer explanation Tree diagram questions can be straightforward for students to identify what to do, but this question has a twist as students need to understand the calculation within a tree diagram in order to work backwards and calculate how many times the spinner was spun. Relative frequency is disguised in this question rather than a typical part a) complete the tree diagram part b) calculate the probability.

For more GCSE maths questions take a look at our free Tree diagrams lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

W , X , Y and Z are four points on the circumference of a circle.

WCY and XCZ are straight lines.

Prove that triangle CWX and triangle CYZ are similar.

You must give reasons for each stage of your working.

Angle WXC = Angle CYZ because angles in the same segment are equal. Angle XWC = Angle CZY because angles in the same segment are equal. Angle WCX = Angle ZCY because vertically opposite angles are equal.

(We only need two of these angles as we can calculate the third by using the fact that angles in a triangle add to 180 degrees.)

Answer explanation Students need to identify the word ‘similar’ and understand its significance in this question. Students often struggle with ‘prove’ type questions and it is common for them to be unsure how to set out their working. Ensuring students are confident with angle notation and how to recall the circle theorems are essential. Modelling how this style of question should be set out using very logical steps with angle and reasons side by side can be beneficial.

For more GCSE maths questions take a look at our free Circle theorems lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

y is inversely proportional to t^{2} When t = 3 , y = 4 .

t is directly proportional to x^{2} When x = 2 , t = 8 .

Find a formula for y in terms of x .

Give your answer in its simplest form.

A) y=\frac{9}{x^{2}} B) y=\frac{9}{x^{4}} C) y=\frac{18}{x^{2}} D) y=\frac{18}{x^{4}}

B) y=\frac{9}{x^{4}}

Answer explanation Students must be confident with finding the value of the constant for each equation and apply the powers. They need to recognise how to substitute the second equation into the first, taking care to make it ‘ t =’ and not ‘ y =’, then simplify the fraction.

For more GCSE maths questions take a look at our free Direct and inverse proportion lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Use the formula F=\frac{s}{\sqrt{tm}} to find the value of F when

s = 6.2 × 10^{9} t = 4.3 × 10^{8} m = 3.6 × 10^{-3}

Give your answer in standard form, correct to 2 significant figures.

A) 5.0 × 10^{6} B) 4.9 × 10^{2} C) 4.0 × 10^{4} D) 4.9 × 10^{4}

A) 5.0 × 10^{6}

Answer explanation This is a calculator paper question, so if students are confident with efficient use of their scientific calculator and correctly insert brackets where they are needed, they can enter the calculation to work it out. However, this is often subject to errors and if they work through the calculation in stages, they need to consider the order of operations and understand the effect of square rooting on the powers.

For more GCSE maths questions take a look at our free Standard form lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

The histogram shows information about the weights of some letters handled by a delivery company in one week.

“There are more letters weighing between 20 g and 60 g than letters weighing between 60 g and 100 g.”

Is Sam correct?

Show how you decide.

Sam is wrong.

Number of letters between 20 g and 60 g = 20 × 17.5 + 20 × 30 = 950

Number of letters between 60 g and 100 g = 30 × 20 + 10 × 40 = 1000

Answer explanation Students need to understand that the height of the bars is frequency density. Therefore, they need to calculate the area of the bars to calculate the frequencies and then compare the values. Once frequency has been calculated, these sorts of questions can then relate to other frequency-related question types, such as averages from a frequency table and cumulative frequency.

For more GCSE maths questions take a look at our free Histograms lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

A straight line graph goes through points

( a , b ) and ( c , d ), where

a + 3 = c b + 6 = d

Find the gradient of the line.

A) 2 B) 3 C) 6 D) ½

Answer explanation Students need to know how to calculate the gradient between two coordinate points and then correctly identify the x and y coordinates. This question can easily become overly complicated, but with some simple rearranging to find d – b = 6 and c – a = 3, it is possible to calculate the gradient in a few steps. With four unknowns, students could easily become confused and think that the gradient is given as a formula. However, it may also be that students can guess the answer to this question. Other question types could involve calculating the midpoint of two coordinates, or working out the equation of the straight line that passes through them, or is parallel or perpendicular to them.

For more GCSE maths questions take a look at our free Straight line graphs lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Grade 4 and Grade 5 style GCSE maths questions

The following GCSE maths questions are all questions that appear on both the higher and foundation paper, so are targeted at Grade 4 and Grade 5 students. Questions that solely appear on the foundation paper have not been selected as they tend to have simpler underlying structures or provide structure.

Read more: Question Level Analysis Of Edexcel Maths Past Papers (Foundation)

This rectangular frame is made from 5 straight pieces of wood.

The weight of the wood is 2.5 kg per metre. Work out the total weight of the wood in the frame.

A) 5 kg B) 17.5 kg C) 19 kg D) 47.5 kg

Answer explanation The numbers and calculations required for this question are straightforward. However, as it is a non-calculator paper, it presents some extra challenges. Students must identify that use of Pythagoras’ theorem is required, which can be unexpected on a non-calculator paper, despite the use of Pythagorean triples here. This question uses multiple skills including Pythagoras’ theorem, perimeter and multiplying decimals.

For more GCSE maths questions take a look at our free Pythagoras theorem lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

The equation of the line L_{1} is y = 5x – 1 . The equation of the line L_{2} is 5y – 25x + 4 = 0 .

Show that these two lines are parallel.

Rearrange L_{2} to make y the subject.

L_{1} and L _{2} have the same gradient so they are parallel.

Answer explanation Students may make a mistake in thinking that the x coefficient always indicates the gradient and when given two equations, if these coefficients are equal, then the lines are parallel. In this question, the fact that the first equation has ‘5 x ’ term and the second has ‘5 y ’ term could be mistaken as an indication that the two lines have the same gradient. The second equation requires rearranging and dividing the whole equation by 5, which presents a fraction for the y -intercept value.

For more GCSE maths questions take a look at our free Parallel lines lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

(a) Find the lowest common multiple (LCM) of 30 and 54.

A) 270 B) 1620 C) 6 D) 540

(b) Write down the highest common factor (HCF) of A and B .

A) 60 B) 54000 C) 900 D) 120

Answer explanation Students will often try to list multiples of each number until they find a common multiple. This can be a laborious process and is therefore subject to arithmetic errors. The most efficient method is to write each number as a product of its prime factors.

For more GCSE maths questions take a look at our free Lowest common multiple and Highest common factor lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

This diagram represents students in a year group who study Geography (G) or History (H).

How many students study both Geography and History?

A) Not enough information to tell B) All of them C) None of them D) Half of them

C) None of them

Answer explanation This style of question can be unfamiliar to students as Venn diagrams are almost always presented as overlapping circles. Students may not recognise this as a Venn diagram, and therefore may be unlikely to identify that the sets are independent of each other and share no data.

For more GCSE maths questions take a look at our free Venn diagrams lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Work out the value of b .

A) 1 B) 5 C) 2 D) 25

Answer explanation Students need to be confident using square numbers, cube numbers and how to correctly apply square and cube roots. The question can appear to be challenging upon first inspection, but once the powers and roots have been simplified, it is an easy question to solve.

For more GCSE maths questions take a look at our free Solving equations lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

XYZ is a right-angled triangle.

XZ = 15 cm. Angle Z = 90° . Size of angle ZYX : size of angle YXZ = 3:2 .

Work out the length of XY . Give your answer correct to 3 significant figures.

A) 18.5 cm B) 12.1 cm C) 14.5 cm D) 25.5 cm

Answer explanation With the absence of angles YXZ and XYZ on the diagram, students need to firstly identify this as a trigonometry question and calculate the missing angles using the ratio. This then allows them to decide which trigonometric function to use. Finally, students need to correctly round their answer to 3 significant figures. Higher graded questions on trigonometry can include using the Sine Rule, the Cosine Rule and calculating the area of any triangle using the formula ½ abSinC .

For more GCSE maths questions take a look at our free Trigonometry lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

Sam is using these numbers to make a new number.

He can only use brackets, +, −, ×, ÷ once. He cannot use any number more than once. He cannot use powers. He cannot put numbers together e.g. he cannot use ‘147’.

What is the biggest number he can make? Show how he can make this number.

A) 144 B) 132 C) 336 D) 51

Answer explanation There are multiple aspects to this question that need to be interpreted. and it requires a good understanding of the order of operations and place value. This must then be communicated clearly to obtain full marks. It could be assumed that multiplying all the numbers will find the largest result but the presence of the number ‘1’ is what alters this and requires the application of the order of operations.

For more GCSE maths questions take a look at our free Order of operations (BIDMAS) lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

A puma is running with a velocity of 2 m/s. It then accelerates at 3 m/s ^{2} for 5 seconds.

Use the formula

to work out the final velocity of the puma.

A) 13 m/s B) 17 m/s C) 11 m/s D) 10 m/s

Answer explanation The first challenge is that students have to identify the values of u , a and t from the information, and this requires an understanding of the formula and velocity. Compound measures such as speed and density are often covered at foundation GCSE, but rarely expressed as acceleration and velocity.

For more GCSE maths questions take a look at our free Formulae lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

See also: GCSE maths formulas

The density of orange juice is 1.02 grams per cm ^{3} . The density of fruit syrup is 1.5 grams per cm ^{3} . The density of lemonade is 0.95 grams per cm ^{3} .

30 cm ^{3} of orange juice is mixed with 20 cm ^{3} of fruit syrup and 150 cm ^{3} of lemonade to make a drink.

Work out the density of the drink. Give your answer correct to 2 decimal places.

A) 3.47 g/cm ^{3} B) 203.1 g/cm ^{3} C) 1.02 g/cm ^{3} D) 0.98 g/cm ^{3}

C) 1.02 g/cm ^{3}

Answer explanation At a glance, this looks simple as students may think that they just need to total the densities of each part. However, they need to work out the mass of each part in order to calculate the total mass and total volume of the drink before calculating the total density. It is important to set out the calculations clearly. Organising the information as a two-way table is an effective way of simplifying the problem. Students need to correctly round their solution to 2 decimal places.

For more GCSE maths questions take a look at our free Density lessons and for more personalised support schools can plug gaps with our targeted one-to-one interventions .

The GCSE maths topics are: • Number including fractions, decimals and percentages  • Algebra including equations, nth term etc. • Geometry and measure including trigonometry • Ratio and proportion including exchange rates • Probability including frequency • Statistics including pie chart

No, GCSE maths is not difficult if you are sitting the right exam and you have prepared. There are two GCSE exams – the foundation paper and the higher paper. The higher paper is more difficult. GCSE maths papers need to be accessible to students with a wide range of abilities. Therefore, every exam paper will have questions that can present difficulties to students. The main elements that can increase the difficulty of a question include:  • The language of the question • Presenting a topic in an unfamiliar context • Limited structure and guidance • Interleaving of skills To help overcome initial difficulties, it is useful to be aware of these different elements when you first approach a GCSE question. Try to overlook your first concerns and identify the question they are asking you.

To answer a GCSE maths question the first thing you need to do is read the question properly. It is so important to take your time and really make sure you understand the intention behind the question. It is easy to get caught up in the language of the question and lose sight of what it is actually asking! So, take your time, and make sure you understand what the question is asking before putting the maths into practice.

A great way to practice GCSE maths is by using a range of different resources, such as practice questions, summary worksheets and quick quizzes. You want to find resources that are both engaging and effective, getting you to familiarise yourself with the material and layout of a GCSE Maths question. Third Space Learning has a growing resource library of GCSE Maths lessons and GCSE maths revision support created by maths experts to help prepare for maths GCSE. Take a look, see what works for you and get practising!

To summarise, students need to be secure with some essential skills in order to access the sorts of GCSE maths questions that have been included above. Focus on these to start with:

• How We Developed Our GCSE Maths Revision Programme
• GCSE Intervention Strategies
• How To Revise For GCSE
• Exam Techniques
• Analysis of GCSE Maths Paper 1 Topics With Recommended Revision List For GCSE Maths Paper 2 and Paper 3 (2023)

Do you have students who need extra support in maths? Every week Third Space Learning’s maths specialist tutors support thousands of students across hundreds of schools with weekly online 1-to-1 lessons and maths interventions designed to address learning gaps and boost progress. Since 2013 we’ve helped over 162,000 primary and secondary students become more confident, able mathematicians. Learn more or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

Secondary school tuition designed to plug gaps and prepare KS3 and KS4 students for the challenges ahead.

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FREE Essential GCSE Worksheet Pack

Help your students prepare for their maths GCSE with this free pack of our most frequently downloaded worksheets.

Contains skills-based, applied and exam style questions that are suitable for foundation and higher GCSE maths revision for AQA, OCR and Edexcel exam boards.

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Which GCSEs Have Coursework?

In GCSE by Think Student Editor September 23, 2022 Leave a Comment

If you’re currently trying to decide which GCSEs you should take, it’s important to know whether the option you’re considering will involve coursework. Coursework is a useful way of showing your ability outside of taking written exams. Coursework can allow you to: take more responsibility for what you study, study a topic in more depth, and have more control over the pace at which you study.

To understand which subjects involve coursework and learn the percentage of coursework and exams in these subjects, keep reading this article.

Table of Contents

Do GCSEs still have coursework?

After new education plans were introduced in 2015, most GCSEs no longer include any coursework that count towards students’ final grades. Before this, there would be coursework tasks even in subjects such as maths and English.

In some subjects coursework was done through long written tasks, whereas in maths this was done through a handling data project and an applying mathematics task. In English Language, 40% of the end grade used to be from coursework. This was through assessment of speaking, listening and written assignments.

Despite the recent changes to the GCSE system, all creative and practical subjects do still have some level of coursework. This is because in certain subjects, like Art for example, coursework is necessary for students to demonstrate their talent at particular skills. The subjects that have coursework are Food Preparation & Nutrition, Drama, Art, Music, DT (Design Technology), and PE (Physical education).

What percentage of creative or practical GCSEs is coursework?

No GCSE is currently 100% coursework. There will always be some weighting placed on final exams. All of these final exams are written, apart from Art which is instead a creative project done under time pressure.

Also, it’s important to note that for the same subject, different exam boards may require different amounts of coursework. Make sure to find out which exam board your school uses for the particular subject you’re considering. If your school offers IGCSEs, have a read of this Think Student article to understand the difference between them and normal GCSEs.

Have a look at the table below which has information outlining what percentage of the GCSEs are coursework and exams. This data is from AQA’s website .

In each of these subjects, the type of task to be completed for coursework is completely different. Most exam boards refer to coursework as a non-exam assessment (NEA).  

What does GCSE coursework involve?

In the Food Preparation and Nutrition GCSE, the non-exam assessment mainly consists of a cooking practical. Students will have to prepare, cook and present a final menu of three dishes. The students will then have to write a report about their work and include photographic evidence. To find out more about the Food Preparation and Nutrition course, visit the AQA page .

For the coursework in Drama, there are two different components. One involves performing a group devised mini play and keeping a log of the creation process. The other involves performing two extracts from a play. To find out more about GCSE Drama, visit the AQA page .

In GCSE Art, the coursework component consists of selecting and presenting a portfolio representative of their course of study. The portfolio must include one main project as well as a selection of other work from activities such as experiments, skills-based workshops, or responses to gallery visits. To find out more about GCSE Art, visit the AQA page .

In GCSE Music, students must do both an ensemble performance and a solo performance using the instrument of their choice (which can be voice). They must also create two different music compositions. To find out more about GCSE Music, visit the AQA page .

For coursework in GCSE DT, students must design and produce a product. This will involve investigating design possibilities, planning, creating their idea, and evaluating the end result. At school, students will have to use special equipment such as machines and saws. To find out more about GCSE DT, visit the AQA page .

For coursework in GCSE PE, students will be assessed through their performance in three different sports or physical activities of their choice. One has to be a team activity, one an individual activity, and the third either a team or individual activity. Students will also be assessed on their analysis and evaluation of their improvements in performance.  To find out more about GCSE PE, visit the AQA page .

Does GCSE Science have coursework?

GCSE Science doesn’t involve any graded coursework. However, there is a list of required practicals that students are supposed to complete. These science practicals will involve following instructions set out by the teacher to investigate materials or scientific principles.  Students will often have to write up the method and conclusion. It’s important that students try their best to understand these practicals as there will be questions about them that are worth several marks in the exams.

Does GCSE English have coursework?

GCSE English technically doesn’t have any coursework that has a weighting on the final grade. However, in English Language there’s a compulsory spoken language assessment that isn’t done at the same time as normal GCSE exams. It’s reported as a separate grade (either Pass, Merit, Distinction or Not Classified) and doesn’t contribute to the result of the GCSE English Language qualification. To learn more about the spoken language assessment, have a look at this AQA page .

For English Literature, despite there not being any coursework tasks, there are of course novels and poems that students need to become familiar with in order to pass the GCSE. This will have to be done throughout Year 10 and Year 11. Students might be set the homework of reading a couple of chapters for example.

What are some tips for completing GCSE coursework?

If you’re deciding to do one or multiple GCSE subjects that involve coursework, it’s crucial that you can be organised enough to complete them to the best possible standard. As seen from the table shown previously, coursework makes up a significant proportion of the final grade. To have the best chance at getting a high grade, you’ll need to put an adequate amount of time into the task and not treat it as trivial.

In GCSE Art in particular, there is a lot of work that will need to be completed throughout the two-year course. A lot of homework will end up being graded for coursework as they will go in your portfolio. GCSE Art is often said to be one of the most stressful GCSEs because of this constant pressure of getting work done on time out of school.

If you love art and want to continue studying it, it’s important to manage your time well and not post-pone completing tasks until the last minute. As soon as you start getting behind with work that needs completing, that’s when you’ll really start to struggle and make things harder for yourself. If you want to learn how to get a good grade in GCSE Art in general, check out this Think Student article .

Make sure that you always note down what you’ve got to get done and by what date. This could be in a physical planner, on an app on your phone, or on a digital calendar. Perhaps set a specific time each weekend to complete any remaining work that you didn’t manage to get done during the week. If you remain on schedule, you’ll significantly reduce any stress plus you’ll have a higher chance of producing your best quality work.