12 short questions across all six content areas, designed as a lesson starter or a daily warm-up.
⏱️ Ten-minute starters — Foundation
MathsUKwww.geekhero.co.uk
Twelve short questions, two from each of the six content areas: number, algebra, ratio, geometry, probability and statistics. Nothing here needs more than a minute, and nothing needs a calculator. It is built as a lesson starter — print it, hand it out, take ten minutes, mark it together — and it works just as well as a daily warm-up at home in the fortnight before a mock. The value is in the spread rather than the depth: a student who has spent a week on algebra can find out in ten minutes what a week away from statistics has cost. Generate a new sheet whenever you want a fresh twelve on the same pattern.
- 1.The formula for the volume of a cuboid is V = lwh, where l is the length, w is the width and h is the height. Make h the subject of the formula.
- 2.A car travels 90 km using 6 litres of petrol. Work out how far the car can travel, at the same rate, using 9 litres of petrol.
- 3.A radar station is at the origin of a grid, where each unit represents 1 km. A boat is detected at the point (7, 24). Work out the boat's distance from the station, then work out how many hours it will take the boat to reach the station travelling directly towards it at 5 km per hour.
- 4.A sector of a circle has radius 5 cm and angle 72°. Work out the arc length of the sector, in terms of π.
- 5.The point (4, −3) is reflected in the y-axis. Work out the coordinates of the image point.
- 6.A sailmaker cuts two triangular sail panels from a pattern. Panel X has a 10 m side, with angles of 50° and 75° at its two ends. Panel Y also has a 10 m side, with angles of 50° and 75° at its two ends, in the same arrangement. The sailmaker wants to check the panels will match exactly, without cutting a third measurement. Which condition proves the two panels are congruent?
- 7.Four graphs are described below. Each shows one quantity plotted against another. Write down the description of the graph that shows two quantities in direct proportion.
- 8.A sector of a pie chart stands for 25% of the data. Work out the angle at the centre of that sector.
- 9.6 identical printers can print a batch of exam papers in 40 minutes, all working at the same rate. Working at the same rate, work out how many minutes 4 of these printers would take to print the same batch.
- 10.Work out the value of 6² − 4³.
- 11.A school has 1,500 pupils. The head teacher takes a random sample of 150 of them from the school register and asks how long they spend on homework. Rory says the sample is too small for the result to mean anything. Is Rory right? Give a reason for your answer.
- 12.Write down the column vector that describes a translation of 5 units to the left and 3 units down.
Answer key
- (c) h = V / (lw) — Method: undo the multiplication by lw by dividing both sides by lw. Working: V = lwh, so dividing both sides by lw gives h = V / (lw). The value h = V − lw comes from subtracting lw instead of dividing by it. The value h = Vlw comes from multiplying by lw instead of dividing. The value h = lw / V comes from inverting the fraction, dividing lw by V instead of V by lw.
- (a) 135 km — Method: find the distance travelled on one litre, then scale up to 9 litres. Working: 90 ÷ 6 = 15 km per litre, so 15 × 9 = 135 km. Answer: 135 km. 45 km comes from working out the extra distance for the extra 3 litres (15 × 3) but forgetting to add the original 90 km. 60 km comes from using the ratio the wrong way round, 90 × 6 ÷ 9, instead of finding the rate per litre first. 99 km comes from simply adding the number of litres, 9, onto the original distance, 90, instead of scaling the whole journey.
- (c) 5 hours — The distance from the origin to (7, 24) is √(7² + 24²) = √(49 + 576) = √625 = 25 km. Travelling at 5 km per hour, the time taken is 25 ÷ 5 = 5 hours. 25 hours mistakes the distance itself for the time, forgetting to divide by the speed. 125 hours comes from multiplying the distance by the speed, 25 × 5, instead of dividing. 0.2 hours comes from dividing the speed by the distance, 5 ÷ 25, the wrong way round.
- (d) 2π cm — The arc is a fraction of the whole circumference. The fraction is 72 ÷ 360 = 1/5 of the circle, and the full circumference is 2 × π × 5 = 10π cm. So the arc length is 1/5 × 10π = 2π cm. Taking the whole circumference and forgetting the fraction gives 10π cm. Using 72 ÷ 180 instead of 72 ÷ 360 gives 4π cm. Treating the 5 cm as a diameter instead of a radius gives π cm.
- (d) (−4, −3) — Reflecting in the y-axis keeps the y-coordinate the same and changes the sign of the x-coordinate, so (4, −3) maps to (−4, −3). "(4, 3)" changes the sign of the y-coordinate instead, which is what happens when reflecting in the x-axis. "(−4, 3)" changes the sign of both coordinates, which is the result of a rotation of 180° about the origin, not a reflection in the y-axis. "(3, −4)" swaps the two coordinates around instead of reflecting either of them.
- (c) ASA — Method: check which condition matches two angles and the side between them, since that is all the sailmaker has measured. Working: the 10 m side lies between the 50° and 75° angles in both panels, so this is two Angles and the included Side, ASA. Options: SAS would need two sides and the angle between them, but only one side has been measured here; SSS would need three sides, but only one is known; RHS needs a right angle and a hypotenuse, and neither panel has a stated right angle. Answer: ASA.
- (c) A straight line through the origin, rising from left to right — Two quantities are in direct proportion when one is a constant multiple of the other, so their graph is a straight line through the origin: when one quantity is 0 the other is 0 as well, and doubling one doubles the other. A straight line crossing the vertical axis at 5 has a fixed amount added on, so when the horizontal quantity is 0 the vertical quantity is 5, not 0 — a straight line on its own is not enough for direct proportion. A curve that falls steeply and then levels off without touching either axis shows inverse proportion: one quantity grows as the other shrinks, and their product stays the same. A horizontal line at a height of 3 shows a quantity that does not change at all as the other one grows, so it is not proportional to it.
- (c) 90° — Method: the sectors of a pie chart share the 360° at the centre of the chart in the same proportion as the data, so a sector's angle is its share of the data multiplied by 360°. Working: a share of 25% is the fraction 25/100, which is one quarter of the whole chart, and one quarter of 360° is 360 ÷ 4 = 90°. Answer: 90°, an angle in degrees rather than a percentage. The distractors: 25° comes from sharing out 100 instead of 360, so the percentage is written straight down as a number of degrees; 14.4° comes from dividing 360 by 25 instead of multiplying 360 by the fraction 25/100, which is the division done the wrong way round; 45° comes from taking a quarter of 180° instead of a quarter of 360°, treating the pie chart as a semicircle.
- (a) 60 minutes — Method: for inverse proportion, printers × time is constant. Working: 6 × 40 = 240 (the constant). With 4 printers: 240 ÷ 4 = 60 minutes. Wrong options: 26.7 minutes comes from treating the relationship as direct proportion, scaling the time down as printers decrease (40 × 4 ÷ 6); 24 minutes comes from multiplying the two printer counts together instead of using the constant; 40 minutes comes from not adjusting the time at all for the change in printers.
- (c) −28 — 6² = 36 and 4³ = 64. Work out 36 − 64 = −28. A candidate who subtracts in the wrong order gets 64 − 36 = 28. A candidate who adds instead of subtracting gets 36 + 64 = 100. A candidate who multiplies the base by the exponent instead of raising the power (6 × 2 − 4 × 3 = 12 − 12) gets 0.
- (a) No, 150 pupils are a tenth of the school, chosen at random — Method: judge a sample on two things, whether every member of the population had the same chance of being chosen, and whether the sample is large enough to carry a pattern. Working: the 150 pupils were drawn from the register of every pupil in the school, so no year group or set is shut out and no pupil chooses to take part; and 150 ÷ 1,500 = 0.1, so one pupil in ten has been asked. A random sample of that share is ample for an estimate of how long the school's pupils spend on homework. Answer: no, because 150 pupils are a tenth of the school and were chosen at random. The distractors: saying a random sample always gives the exact school figure reaches the same verdict for a reason that is false, since a second random sample of 150 would give a slightly different mean; saying 150 pupils cannot be picked at random from 1,500 treats randomness as something only a whole population can have, when drawing names from the register is exactly how a random sample is taken; saying that only asking all 1,500 could show anything rejects sampling altogether, which would leave no way to study any population too large to count.
- (d) $\binom{−5}{−3}$ — Left is a negative horizontal move and down is a negative vertical move, so the vector is $\binom{−5}{−3}$. $\binom{5}{−3}$ comes from treating 'left' as a positive move. $\binom{−5}{3}$ comes from treating 'down' as a positive move. $\binom{−3}{−5}$ comes from swapping the horizontal and vertical components.
What is on this worksheet?
The sheet holds 12 questions drawn from the MathsUK bank — the content areas covered: Number, Algebra, Ratio, proportion and rates of change, Geometry and measures, Probability, Statistics. It is pitched at GCSE Foundation and takes about 10 minutes to work through in full. It works as a class handout, a homework, or a warm-up before a test.
How to use the sheet well
- Print it or open it on screen — both work. Printing is A4; the screen view fits phones and tablets.
- Do all 12 questions before checking — about 10 minutes is the guide, but there is no time pressure.
- Check the answers — press “Show answers” or print the answer page separately.
- Redo the questions you got wrong — twice as effective as doing 12 fresh ones.
- “New questions” — builds a fresh sheet on the same statements, so you can practise again without repeats.
Why this sheet helps
MathsUK worksheets use questions graded by difficulty and a fair spread of correct-answer positions (the answer is not always (a)) — so the student really has to think about each question rather than guess a pattern. Every question is tagged to a DfE content statement and checked before it enters the bank. The answers come with a step-by-step explanation, not just a value — so a wrong answer becomes a lesson.
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