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
Answer key: GCSE Foundation sample Paper 1 (non-calculator)
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- (b) 4,700 — To round to the nearest 100, look at the digit in the tens column, which decides whether the hundreds column rounds up or stays the same. In 4,685 that digit is 8, and since 8 is 5 or more, the 6 in the hundreds column rounds up to 7, giving 4,700. Simply changing the last two digits to zero without checking the tens digit gives 4,600, which rounds down when it should round up. Rounding to the nearest 10 instead of the nearest 100 gives 4,690. Rounding to the nearest 1,000 instead gives 5,000, one place value too coarse.
- (c) x = 2, y = 3 — Method: no letter cancels straight away, so make one letter the subject of one equation and substitute that expression into the other. Working: 2x + y = 7 gives y = 7 − 2x, so x + 2y = 8 becomes x + 2(7 − 2x) = 8, that is x + 14 − 4x = 8, so −3x = −6 and x = 2; substituting back gives y = 7 − 4 = 3. Answer: x = 2, y = 3, which checks in 2 + 6 = 8. The distractors: x = 3, y = 2 comes from adding the equations to x + y = 5 and then subtracting them the wrong way round, writing x − y = 1 instead of x − y = −1; x = 2, y = 5 comes from substituting x = 2 into 2x + y = 7 as 2 + y = 7, dropping the coefficient; x = −2, y = 11 comes from dividing −3x = −6 as though two minus signs gave a negative answer.
- (d) 2:3 — The highest common factor of 20 and 30 is 10. Divide both parts of the ratio by 10: 20 ÷ 10 = 2 and 30 ÷ 10 = 3, so 20 : 30 = 2 : 3. Dividing by 5 instead of the highest common factor gives 4 : 6, which still shares a common factor of 2, so it is not fully simplified. Dividing only the first part by 10 and leaving the second part unchanged gives 2 : 30, which is not equivalent to the original ratio. Swapping the order of the two parts gives 3 : 2, the ratio the wrong way round.
- (c) 24 cm² — Method: the front elevation of a cuboid is a rectangle formed by the cuboid's length and its height, so its area is length × height. Working: 6 cm × 4 cm = 24 cm². Answer: 24 cm². The distractors: 12 cm² comes from using width × height (3 × 4) instead of length × height, mistaking the side elevation's dimensions for the front's. 18 cm² comes from using length × width (6 × 3), which gives the area of the plan view instead of the front elevation. 20 cm² comes from finding the perimeter of the front face instead of its area: 2 × (6 + 4) = 20.
- (b) 120 — Purple's probability is its angle out of the full circle: 144° ÷ 360° = 2/5. Expected number on purple = 2/5 × 300 = 120. Assuming the three colours are equally likely, ignoring their different angles, gives 300 ÷ 3 = 100. Using orange's angle, 90°, instead of purple's 144° gives a probability of 1/4, so 300 × 1/4 = 75. Treating 144 as a percentage instead of finding its fraction of 360° gives 300 × 0.144 = 43.2, which rounds to 43.
- (b) 1 — The four frequencies are 4, 7, 6 and 3 matches, and the largest of these is 7, which corresponds to 1 goal, so the modal number of goals is 1. Choosing 7 confuses the frequency, how many matches, with the number of goals itself. Choosing 2 uses the second-largest frequency, 6 matches, instead of the largest. Choosing 3 uses the smallest frequency, which belongs to the fewest matches, not the most.
- (a) 0.08 — Method: decimal places are counted from the decimal point, including any zeros straight after it, so rounding to 2 decimal places is decided by the digit in the third decimal place. Working: 0.0759 has 7 in the second decimal place and 5 in the third, and 5 counts as rounding up, so the 7 goes up to 8. Answer: 0.08. The distractors: 0.07 comes from chopping the digits after the second decimal place off instead of rounding them; 0.076 is 0.0759 correct to 2 significant figures rather than 2 decimal places, because the zeros in front of the 7 are not significant figures; 0.1 is 0.0759 rounded to 1 decimal place, a coarser degree of accuracy than the question asks for.
- (b) £14 — Method: read the fixed charge (the cost at 0 miles) and the rate (the cost per extra mile) from the graph, then use them to work out the cost for a distance beyond the part that is plotted. Working: the graph shows a fixed charge of £2 at 0 miles, and the cost rises by £2 for every extra mile, so for 6 miles the cost is £2 + (£2 × 6) = £2 + £12 = £14. Answer: £14. Distractor refutation: £12 comes from multiplying the rate by the distance and leaving out the £2 fixed charge. £8 comes from misreading the rate as £1 per mile instead of £2 per mile. £24 comes from adding the fixed charge to the rate first and then multiplying the total by the distance, instead of multiplying the rate by the distance and then adding the fixed charge.
- (a) 15 km/h — Speed = distance ÷ time, so 45 ÷ 3 = 15 km/h. Working out 45 × 3 = 135 multiplies distance and time together instead of dividing. Working out 45 + 3 = 48 and 45 − 3 = 42 both combine the two values by addition or subtraction, which does not give a speed at all. The cyclist's average speed is 15 km/h.
- (c) (2, 4) and (6, 4) — Method: two points lie on the same horizontal line when they stand at the same height above the x-axis, and a point's height is given by its y-coordinate, the second number in the bracket. Working: compare the second numbers in each pair. For (3, 5) and (3, 8) they are 5 and 8, which differ. For (1, 2) and (2, 1) they are 2 and 1, which differ. For (0, 0) and (1, 1) they are 0 and 1, which differ. For (2, 4) and (6, 4) they are 4 and 4, which agree, so only that pair lies on a horizontal line. Answer: (2, 4) and (6, 4). The distractors: (3, 5) and (3, 8) comes from swapping horizontal for vertical, since equal x-coordinates place two points one above the other; (1, 2) and (2, 1) comes from looking for the same pair of numbers rather than for the same y-coordinate; (0, 0) and (1, 1) comes from reading 'horizontal' as simply 'in a straight line', which any two points are.
- (a) 1/29 — The probability that Priya's ticket wins is 6/30. Since her ticket is not returned, there are now only 5 winning tickets left out of 29 tickets in total, so the probability that Tom's ticket also wins is 5/29. Multiplying these, 6/30 × 5/29 = 30/870 = 1/29. A candidate who answers 1/25 has treated Priya's ticket as returned, using 6/30 twice. A candidate who answers 1/30 has correctly reduced the winning tickets to 5 for Tom but forgotten to reduce the total number of tickets, using 5/30 instead of 5/29. A candidate who answers 11/59 has added the numerators and added the denominators, (6+5)/(30+29), instead of multiplying.
- (c) 27 — Each of the 3 digits can be chosen independently for each of the 3 positions, so multiply: 3 × 3 × 3 = 27. 9 comes from multiplying only two of the three positions, 3 × 3, and forgetting the third. 6 comes from working out 3 × 2 × 1 = 6, which counts codes where digits do not repeat, but the question allows repeated digits. 3 comes from considering only one digit position.
- (c) 6 — Method: substitute the value, work out the top of the fraction first, then the division, and add the 3 last. Working: the top gives 10 − 4 = 6, dividing by 2 gives 6 ÷ 2 = 3, and adding 3 gives 3 + 3 = 6. Answer: 6. The distractors: 11 comes from dividing only the 4 by 2 instead of the whole of the top, giving 10 − 2 + 3; 4.5 comes from dividing the + 3 by 2 as well, giving (10 − 4 + 3) ÷ 2; 0 comes from subtracting the wrong way round on the top, giving (4 − 10) ÷ 2 = −3 and then −3 + 3.
- (b) 6 litres — Method: find the volume of the cuboid in cm³, then change cm³ into litres using 1 litre = 1000 cm³. Working: 30 × 20 × 10 = 6000 cm³, and 6000 ÷ 1000 = 6. Answer: 6 litres. The distractors: 60 litres comes from using 1 litre = 100 cm³; 600 litres comes from using 1 litre = 10 cm³; 0.6 litres comes from using 1 litre = 10 000 cm³.
- (a) Isosceles trapezium: base angles are equal — WX is parallel to ZY and the two non-parallel sides WZ and XY are equal in length, so WXYZ is an isosceles trapezium. In an isosceles trapezium the two angles at each of the parallel sides are equal, so angle W = angle X (and angle Z = angle Y). A student who answers with the parallelogram property has quoted a fact that is true of a parallelogram, but WZ and XY are given as non-parallel so WXYZ is not a parallelogram — and in a parallelogram "opposite angles" pairs W with Y, not W with X. A student who answers with the kite property has again taken a true fact about the wrong shape: a kite's equal sides are two pairs of adjacent sides, not the two non-parallel sides of a trapezium. A student who answers with the rhombus property has used something no rhombus has — a rhombus has two pairs of parallel sides and only two pairs of equal angles; all four angles are equal only in a square.
- (b) 9/4 — Method: write the whole part as a fraction with the same denominator, then add the fraction part to it. Working: there are 4 quarters in 1 whole, so 2 wholes are 2 × 4 = 8 quarters; adding the 1 quarter that is already there gives 8 + 1 = 9 quarters over a denominator of 4. Answer: 9/4. The distractors: 3/4 comes from adding the whole number to the numerator, as 2 + 1, instead of multiplying it by the denominator first; 7/4 comes from multiplying correctly but then subtracting the numerator, as 2 × 4 − 1; 5/4 comes from multiplying the numerator by the denominator instead of the whole number, as 1 × 4 + 1.
- (d) 4 — Method: collect the x terms on one side and the number terms on the other. Working: subtract 3x from both sides: 4x − 5 = 11. Add 5 to both sides: 4x = 16. Divide by 4: x = 4. Answer: 4. 0.6 comes from adding the x terms instead of subtracting when collecting them, 7x + 3x = 10x, and also subtracting the constants the wrong way round, 11 − 5 = 6, giving 10x = 6. 1.5 comes from correctly collecting the x terms as 4x but subtracting the constants the wrong way round, 11 − 5 instead of 11 + 5. −4 comes from moving the x terms to the wrong side, giving 3x − 7x instead of 7x − 3x, along with a matching sign error on the constants.
- (c) −0.2, the car uses 0.2 litres of fuel for each mile — Method: the gradient is the change in the vertical value divided by the change in the horizontal value, which on this graph is a number of litres for each mile, and a negative gradient means the vertical quantity is going down. Working: from (0, 45) to (150, 15) the fuel changes by 15 − 45 = −30 litres while the distance changes by 150 − 0 = 150 miles, so the gradient is −30 ÷ 150 = −0.2, which says the tank loses 0.2 litres for every mile driven. Answer: −0.2, the car uses 0.2 litres of fuel for each mile. The distractors: '0.2, the car gains 0.2 litres of fuel for each mile' comes from subtracting the fuel values the other way round, 45 − 15 = 30, which drops the minus sign and reverses what the graph says; '−5, the car uses 5 litres of fuel for each mile' comes from dividing the change in distance by the change in fuel, 150 ÷ (−30), turning the gradient upside down; '−30, the car uses 30 litres of fuel for each mile' is the change in fuel on its own, never divided by the 150 miles travelled.
- (c) £0.45 — Method: find the total cost, then subtract from £20. Working: 4 × £3.20 = £12.80. £12.80 + £6.75 = £19.55. Change = £20.00 − £19.55 = £0.45. Answer: £0.45. (£7.20 comes from forgetting to include the compost and subtracting only the plants' cost from £20. £1.45 comes from dropping the carry when adding the pence: 80p + 75p = £1.55, but only the 55p is written down, giving £18.55 instead of £19.55. £10.05 comes from buying only one plant instead of four, using £3.20 + £6.75.)
- (a) 4 : 15 — Convert to the same unit first: 1.5 kg = 1500 g, since 1 kg = 1000 g. This gives the ratio 400 : 1500. Divide both parts by their highest common factor, 100, to get 4 : 15. Giving 40 : 150 divides by 10 only, which is not the highest common factor, so it is not fully simplified. Giving 15 : 4 swaps the order. Giving 4 : 1.5 has not converted 1.5 kg into grams, so the two parts are not measured in the same unit.
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.