Sample paper · GCSE Foundation · grades 1–5
GCSE Foundation sample Paper 1 (non-calculator)
The real Paper 1 is 1.5 hour 30 minutes and 80 marks, non-calculator, and all three papers carry equal weight. This sample is 20 original questions in the same content proportions as the Foundation qualification — number 25%, algebra 20%, ratio, proportion and rates of change 25%, geometry and measures 15%, probability 7.5%, statistics 7.5% — with no calculator-only items. AO1 / AO2 / AO3 at this tier: 50% / 25% / 25%.
Non-calculatorGCSE Foundation
GCSE Foundation sample Paper 1 (non-calculator)
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- 1.In a science experiment, the temperature of a liquid is recorded as 18.6 °C, correct to the nearest 0.2 °C. Write down the error interval for the actual temperature, T °C.
- 2.A point has coordinates (x, y), where x + y = 0 and x is not 0. In which two quadrants could this point lie?
- 3.Two mathematically similar cylinders have heights in the ratio 3 : 4. Write the ratio of their volumes in its simplest form.
- 4.A line segment is measured with a ruler and found to be 6.5 cm long. What is this length in millimetres?
- 5.A two-way table records how 80 students travel to school. 45 of the students are boys and the rest are girls. 27 of the boys walk to school. Work out the probability, as a fraction, that a student chosen at random from the boys walks to school.
- 6.A bar chart shows the highest temperature recorded on each of five days. The temperatures were Monday 20 °C, Tuesday 22 °C, Wednesday 18 °C, Thursday 25 °C and Friday 23 °C. Write down the day on which the temperature was highest.
- 7.Two fair six-sided dice are rolled together. One die is red and the other is blue. Work out how many different outcomes are possible.
- 8.The first five square numbers are 1, 4, 9, 16, 25. Write down the next square number in the sequence.
- 9.In a recipe the mass of chocolate to the mass of milk is in the ratio 1:4. Amelia uses 200 g of milk. Work out the mass of chocolate she needs.
- 10.A tent is shaped like a triangular prism: two triangular end faces and three rectangular faces. One of the rectangular faces lies flat on the ground as the groundsheet. How many of the tent's faces are above the ground, not touching the ground?
- 11.Isla travels to school by bus, by bike, on foot or by car, and never by more than one of these on the same day. The probability that she travels by bus is 0.2, the probability that she travels by bike is 0.3 and the probability that she travels on foot is 0.1. Work out the probability that she travels to school by bus, by bike or on foot.
- 12.A pizza is cut into 12 equal slices. Ben eats 5 slices and Mia eats 3 slices. What fraction of the pizza is left, giving your answer in its simplest form?
- 13.Work out the equation of the straight line through the points (−2, 1) and (0, 7).
- 14.The price of a share falls by 10% on Monday and then rises by 10% on Tuesday. Work out the overall percentage change from Monday's starting price.
- 15.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.
- 16.A minibus can carry 16 passengers. A school is taking 179 pupils on a trip. By rounding 179 to the nearest 10, work out an estimate for the number of minibuses needed, given that the school cannot hire part of a minibus.
- 17.Work out the value of n² − n when n = −3
- 18.Two numbers are in the ratio 5 : 8. The larger number is 40. Work out the smaller number.
- 19.Write 0.45 as a fraction in its simplest form.
- 20.A crate exerts a force of 84 N on the floor through a base with an area of 2.4 m². Work out the pressure on the floor, in N/m².
Answer key
- (b) 18.5 ≤ T < 18.7 — Method: the error interval reaches half the rounding unit either side of the recorded value. Working: half of 0.2 is 0.1, so the interval runs from 18.6 − 0.1 to 18.6 + 0.1. Answer: 18.5 ≤ T < 18.7. (18.4 ≤ T < 18.8 comes from using the full rounding unit, 0.2, either side instead of half of it. 18.5 ≤ T ≤ 18.7 comes from including the upper bound with ≤ instead of excluding it with <. 18.6 ≤ T < 18.8 comes from treating the recorded value as the start of the interval and adding the whole rounding unit, 0.2, above it.)
- (c) Second and fourth — If x + y = 0 then y = −x, so x and y always have opposite signs, one positive and one negative. A point with a negative x and a positive y lies in the second quadrant, and a point with a positive x and a negative y lies in the fourth quadrant, so the point lies in the second or the fourth. A candidate who reads x + y = 0 as meaning x and y have the same sign picks First and third, which is where x × y is positive, not where x + y = 0. A candidate who decides that y must be the positive coordinate picks the two quadrants above the x-axis, First and second. A candidate who decides that y must be the negative coordinate picks the two quadrants below the x-axis, Third and fourth.
- (c) 27 : 64 — For similar solids, the ratio of volumes is the ratio of lengths cubed: 3³ : 4³ = 27 : 64. 3 : 4 comes from using the height ratio itself as the volume ratio, without cubing it at all. 9 : 16 comes from squaring each part instead of cubing (3² : 4²) — squaring is the rule for area, not volume. 27 : 4 comes from cubing only the first part of the ratio (3³ = 27), and leaving the second part uncubed.
- (a) 65 — There are 10 millimetres in every centimetre, so to convert from cm to mm, multiply by 10: 6.5 × 10 = 65 mm. A candidate who forgets to convert at all writes down the original number, 6.5. A candidate who multiplies by 100 instead of 10, confusing cm-to-mm with m-to-cm, gets 650. A candidate who divides by 10 instead of multiplying gets 0.65. The correct length in millimetres is 65.
- (d) 3/5 — 'Chosen from the boys' restricts the sample space to the 45 boys, of whom 27 walk. So the probability is 27/45 = 3/5. Using all 80 students as the denominator instead of just the boys gives 27/80, which is the probability that a student chosen from everyone is a boy who walks — not what was asked. Using the 18 boys who do NOT walk (45 − 27) as the numerator instead of the 27 who do gives 18/45 = 2/5. Dividing the 27 walking boys by the number of girls (80 − 45 = 35) instead of by the number of boys gives 27/35.
- (c) Thursday — Method: on a bar chart the tallest bar belongs to the greatest value, so compare the five temperatures and then read off the day that the greatest one belongs to. Working: the temperatures are 20 °C, 22 °C, 18 °C, 25 °C and 23 °C; in order of size these are 18, 20, 22, 23 and 25, so the greatest temperature is 25 °C, and the day recorded with 25 °C is Thursday. Answer: Thursday — the answer is a day, not a temperature. The distractors: Friday comes from stopping at the final temperature listed instead of comparing all five; Wednesday comes from picking out the shortest bar, 18 °C, and so answering for the lowest temperature rather than the highest; Monday comes from writing down the first day in the chart without comparing any of the temperatures at all.
- (c) 36 — Method: set the outcomes out in a grid with one die along the top and the other down the side, so that every cell of the grid is one outcome, and count the cells. Working: the red die can land in 6 ways, so the grid has 6 columns, and the blue die can also land in 6 ways, so the grid has 6 rows; the number of cells is 6 × 6 = 36. Answer: 36. The distractors: 12 comes from adding 6 and 6 instead of multiplying them; 6 comes from counting the outcomes of a single die and forgetting that the second die also has to land; 21 comes from treating the two dice as indistinguishable, so that a red 2 with a blue 3 and a red 3 with a blue 2 are counted as one outcome.
- (a) 36 — Square numbers are n² for n = 1, 2, 3, ...; the fifth term, 25, is 5². The sixth square number is 6² = 36. A candidate who mislabels 25 as the sixth square number would compute 7² = 49 instead. A candidate who repeats an earlier difference between terms (5, from 4 to 9) rather than the correct next difference (11, since the differences are the odd numbers 3, 5, 7, 9, 11) would reach 30. A candidate who instead adds 10 would reach 35.
- (d) 50 g — Method: the milk is 4 parts of the ratio, so use the milk to find the value of one part, then read off the chocolate, which is 1 part. Working: one part = 200 ÷ 4 = 50, and the chocolate is one part. Answer: 50 g. The distractors: 40 g comes from treating the 200 g as the total mass of the mixture and splitting it into 1 + 4 = 5 parts; 250 g is the total mass of the finished mixture, the 200 g of milk plus the chocolate, rather than the chocolate on its own; 800 g comes from multiplying 200 by 4 instead of dividing, which scales the milk up rather than down to the chocolate.
- (b) 4 — A triangular prism has 5 faces in total: 2 triangular ends and 3 rectangular faces. One rectangular face is the groundsheet, lying on the ground, so the other 5 − 1 = 4 faces are above the ground. A candidate who forgets one of the triangular ends when counting the remaining faces answers 3. A candidate who forgets to subtract the groundsheet at all answers 5. A candidate who only counts the two sloped rectangular faces, forgetting the two triangular ends, answers 2. The number of faces above the ground is 4.
- (c) 0.6 — Method: no two of these ways of travelling can happen on the same day, so they are mutually exclusive and the probability that one of them happens is the sum of their probabilities; they need not add to 1, because travelling by car is a fourth way. Working: 0.2 + 0.3 = 0.5, and 0.5 + 0.1 = 0.6, the decimal points lined up at each step. Answer: 0.6, which also shows that the probability of being taken by car is 0.4, since all four ways together must make 1. The distractors: 0.4 comes from carrying on past the question and giving the probability of the remaining way, by car; 0.5 comes from adding only the bus and the bike and leaving the smallest of the three probabilities out of the total; 0.006 comes from multiplying the three probabilities together instead of adding them.
- (a) 1/3 — Method: find the total fraction eaten, then subtract it from the whole pizza. Working: together they eat 5/12 + 3/12 = 8/12, so the fraction left is 12/12 − 8/12 = 4/12 = 1/3. Answer: 1/3. 2/3 comes from giving the fraction eaten instead of the fraction left. 7/12 comes from only subtracting Ben's slices and forgetting Mia's. 1/2 comes from comparing the 4 slices left with the 8 slices eaten, 4/8, a part-to-part comparison instead of comparing with the whole pizza of 12 slices.
- (c) y = 3x + 7 — Method: the gradient is the change in y divided by the change in x with both taken in the same order, and a point whose x-coordinate is 0 gives the constant straight away because it lies on the y-axis. Working: m = (7 − 1) ÷ (0 − (−2)) = 6 ÷ 2 = 3; the point (0, 7) lies on the y-axis, so c = 7 and the line is y = 3x + 7. Answer: y = 3x + 7. The distractors: y = −3x + 7 comes from taking the y-difference as 1 − 7 while taking the x-difference as 0 − (−2), so the two subtractions run in opposite orders; y = 3x + 1 comes from using the y-coordinate of (−2, 1) as the constant instead of the point that actually lies on the y-axis; y = (1/3)x + 7 comes from writing the gradient upside down as the change in x over the change in y, 2 ÷ 6.
- (b) −1% — Method: write each change as a multiplier and multiply them. A 10% fall is × 0.9 and a 10% rise is × 1.1. Working: 0.9 × 1.1 = 0.99, so the final price is 99% of the original, which is 1% less. Answer: an overall change of −1%. The distractors: 0% comes from assuming a 10% fall and a 10% rise cancel — they do not, because the rise is 10% of a smaller amount; +1% has the size right but the sign wrong, from reading the multiplier 0.99 as 1% above 1 instead of 1% below it; −2% comes from finding the 1% fall and then counting it once for each of the two changes.
- (b) Minor segment — Method: compare the sizes of the two regions cut off by the chord, and recall the term used for the smaller one. Working: the chord creates two segments; the smaller region is called the minor segment and the larger one the major segment. A student who answers major segment has picked the larger region by mistake instead of the smaller one. A student who answers minor arc has named the curved boundary rather than the two-dimensional region it encloses. A student who answers semicircle has wrongly assumed the chord must pass through the centre. Answer: minor segment.
- (b) 12 — Method: round the number of pupils to the nearest 10, divide by the number of passengers each minibus can carry, then round up because a part-full minibus still needs a whole vehicle. Working: 179 rounds to 180 (nearest 10); 180 ÷ 16 = 11.25; 11 minibuses only carry 176 passengers, so a 12th minibus is needed for the rest. Answer: 12. 11 comes from rounding 11.25 to the nearest whole number in the usual way, without checking that the leftover pupils still need transporting. 10 comes from rounding 179 down to 170 instead of to the nearest 10, 180. 180 comes from stopping after rounding the number of pupils, without dividing by the number of passengers each minibus carries at all.
- (d) 12 — Method: substitute the value into both terms, remembering that subtracting a negative number has the same effect as adding the matching positive number. Working: n² = (−3) × (−3) = 9, and subtracting n means subtracting −3, which adds 3, so the calculation is 9 + 3 = 12. Answer: 12. The distractors: 6 comes from subtracting 3 rather than subtracting −3, giving 9 − 3; −6 comes from squaring −3 as −9 while still adding the 3, giving −9 + 3; −3 comes from reading n² as 2n, giving 2 × (−3) = −6 and then −6 + 3.
- (c) 25 — Method: divide the larger number by its ratio part to find the value of one part, then multiply by the smaller number's ratio part. Working: 40 ÷ 8 = 5 (value of one part). Smaller number = 5 × 5 = 25. Wrong options: 64 comes from dividing by the smaller ratio part instead of the larger (40 ÷ 5 × 8); 45 comes from adding the value of one part onto 40 instead of scaling down (40 + 5); 35 comes from subtracting the value of one part from 40 (40 − 5) instead of multiplying it by the smaller ratio part.
- (a) 9/20 — Method: write the decimal over 100 using its two decimal places, then simplify. Working: 0.45 = 45/100 = 9/20 (dividing both numerator and denominator by 5). Answer: 9/20. 9/100 comes from dividing only the numerator by 5 and leaving the denominator as 100. 9/2 comes from writing the decimal over 10 instead of 100, as if there were only one decimal place, then simplifying 45/10. 9/200 comes from writing the decimal over 1000 instead of 100, as if there were three decimal places, then simplifying 45/1000.
- (b) 35 N/m² — Pressure = force ÷ area. 84 ÷ 2.4 = 35 N/m². 201.6 N/m² comes from multiplying the force by the area instead of dividing (84 × 2.4). 81.6 N/m² comes from subtracting the area from the force (84 − 2.4) instead of dividing. 0.03 N/m² comes from dividing the area by the force instead of the force by the area (2.4 ÷ 84).
How the 20 questions are shared out
- Number — 5 questions (25% of the qualification)
- Algebra — 4 questions (20% of the qualification)
- Ratio, proportion and rates of change — 5 questions (25% of the qualification)
- Geometry and measures — 3 questions (15% of the qualification)
- Probability — 2 questions (7.5% of the qualification)
- Statistics — 1 question (7.5% of the qualification)
Where an area has fewer printable questions than its share, the shortfall is filled from the other areas. These are original questions, not past papers.