A lorry's distance-time graph, with time in hours and distance in km, is made of five straight sections: (0, 0) to (1, 45), horizontal to (1.5, 45), then to (2, 75), horizontal to (2.75, 75), then to (4, 100).
For how many minutes in total was the lorry stationary?
Real-Life Graphs - Edexcel GCSE Higher Maths
Typically 4-8 marks per paper, often as one longer multi-part question · Spec 1MA1
Real-life graphs are the part of Edexcel's 1MA1 algebra content where the axes carry units instead of just and . Spec point A14 covers plotting and interpreting graphs in real contexts, including the kinematic ones: distance against time, velocity against time. A15 is Higher only, and it goes further, asking you to calculate or estimate the gradient of a graph and the area underneath it, then say what those numbers mean for the situation described. Neither point involves calculus. Two sentences carry most of the topic: the gradient tells you the rate at which the vertical quantity is changing, and the area under the graph multiplies the two axis units together.
Edexcel sets this across all three papers, and the shape of the question is fairly settled. A four or five mark version gives a speed-time curve as a set of readings, asks for an estimate of the distance using strips of equal width, and then asks whether that estimate is too big or too small with a reason. Elsewhere you might meet a distance-time graph split into three or four straight sections, a conversion graph between litres and gallons, or a graph of cost against usage where the intercept is a standing charge and the gradient is the price per unit. The area of a trapezium is printed on the Exam Aid formulae sheet issued with 1MA1 papers, so on the strips question the formula is handed to you; the reasoning mark is not. On Paper 1 the readings are kept to whole or half units so the arithmetic stays doable without a calculator.
Marks go missing in a small number of predictable places. Answering with an acceleration when the question asked for a distance means the gradient was calculated instead of the area, and that scores nothing however tidy the working. On the over or under estimate part, a reason that talks about the number of strips gets no credit; the mark is for saying where the straight tops of the trapezia sit relative to the curve. And every answer needs its unit, because a bare 310 could be metres, seconds or metres per second. Practise reading a graph description straight into a sketch of your own, marking the coordinates of each corner before you calculate anything.
Worked example
A car accelerates from rest. The speed of the car, in metres per second, is recorded at time seconds after it starts to move.
At the speed is 0, at it is 12, at it is 18, at it is 21 and at it is 22.
The speed-time graph through these points is a smooth curve that rises steeply at first and then flattens off.
(a) Using 4 strips of equal width, work out an estimate for the distance the car travels in the first 20 seconds.
(b) Is your estimate an underestimate or an overestimate of the actual distance? Give a reason.
[4 marks]- Each strip is 5 seconds wide. First strip, from to : area .M1: strips of the right width and one trapezium area correctly formed
- The other three strips: , then , then .M1: all four areas attempted, each using its own pair of readings
- Total area , so the estimated distance is 310 m.A1: 310 with the unit; m/s multiplied by s gives metres, which is why the area is a distance
- It is an underestimate: the curve is flattening off, so the straight top of every trapezium lies below the curve and each strip leaves out a thin sliver of area.C1: the reason must place the chords relative to the curve, not mention how many strips were used
Practice questions
These are original questions in Edexcel 1MA1 Higher style. There are no pictures here, so every graph is described by its corner coordinates or by a list of readings, which is worth sketching on paper before you calculate; that is closer to what you do in the exam anyway. The set runs from single-section distance-time graphs through velocity-time areas to strip estimates and the over or under estimate reasoning that carries the last mark.
No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.
Question 1 Warm-up [2 marks] Question 2 Exam pace [2 marks] The last section of a train's velocity-time graph is a straight line from (30, 24) to (40, 0), where time is in seconds and velocity is in m/s. Work out the deceleration of the train, in metres per second squared.Question 3 Exam pace [3 marks] A runner's speed-time graph, with time in seconds and speed in m/s, is a straight line from (0, 0) to (4, 8), then horizontal from (4, 8) to (10, 8), then a straight line from (10, 8) to (16, 0). Work out the total distance run, in metres.Question 4 Exam pace [1 mark] The runner's speed readings are 0 m/s at 0 s, 5 m/s at 2 s and 8 m/s at 4 s, and the speed-time graph through them is a curve that is flattening off. An estimate of the distance is found using 2 trapezium strips. Which statement about that estimate is correct?Question 5 Exam pace [3 marks] Water flows into a tank. The flow rate , in litres per second, is recorded at time seconds: 0 at , 3 at , 4 at and 4.5 at . Using 3 strips of equal width, work out an estimate for the volume of water that enters the tank in the first 30 seconds, in litres.Question 6 Stretch [4 marks] A velocity-time graph, with time in seconds and velocity in m/s, is a straight line from (0, 0) to , then horizontal from to , then a straight line from to (35, 0). The total distance travelled is 550 m. Work out the value of .
Common mistakes examiners see
Reading a horizontal section on a velocity-time graph as the object having stopped, so a middle section that should contribute a rectangle of 20 m/s for 20 seconds contributes nothing.
Check what the vertical axis measures before you interpret anything. Horizontal on a velocity-time graph means the velocity is not changing, so the object carries on at a steady speed and the area under that section is a rectangle. Only on a distance-time graph does horizontal mean stationary.
Working out a gradient when the question asked for a distance. On a speed-time graph that turns a 310 m answer into 1.1 m/s per second.
Decide first which of the two operations the question needs, then write the units next to your answer as a check. Divide the axis units for a gradient, multiply them for an area: m/s divided by s is an acceleration, m/s multiplied by s is a distance.
Justifying an underestimate with 'because I only used 4 strips' or 'because it is only an estimate'.
Say where the straight tops sit. A curve that flattens off has its chords below it, so the estimate is too small; a curve that gets steeper has its chords above it, so the estimate is too big. That single comparison is the whole reasoning mark.
Treating the strips as rectangles by using only the left-hand reading of each one, which on a rising curve loses roughly half a strip of area every time.
Use both readings for every strip: area , where and are the heights at the two ends and is the width. The trapezium formula is on the Edexcel formulae sheet, so there is no reason to approximate it.
Giving a negative speed for the return leg of a distance-time graph, answering -30 km/h because the line slopes downwards.
On a graph of distance from home, a negative gradient means travelling back towards the start. The speed is the size of the gradient, so write 30 km/h. Keep the minus sign only when the question asks for a rate of change, such as a tank losing 20 litres per minute.
Frequently asked questions
- Are real-life graphs on the Foundation paper too?
- Partly. Spec point A14, which covers interpreting graphs in real contexts including distance-time and speed-time problems, appears on both tiers. A15, estimating gradients and areas under graphs and interpreting the results, is Higher only, so the strips estimate and the tangent gradient questions belong to Higher tier alone.
- What does the area under a velocity-time graph tell you?
- The distance travelled. The quickest way to remember it is to multiply the axis units: metres per second times seconds gives metres. The same trick works on any rate graph, so the area under a graph of litres per second against seconds is a volume in litres.
- How do I know if my estimate is an overestimate or an underestimate?
- Look at where the straight tops of the trapezia lie compared with the curve. If the curve is flattening off, the chords sit below it and your total is an underestimate. If the curve is getting steeper, the chords sit above it and your total is an overestimate. Saying that more strips would be more accurate is true but earns nothing.
- Do I get the trapezium formula in the exam?
- Yes. Area of a trapezium is printed on the Exam Aid formulae sheet issued with 1MA1 papers for the 2025 to 2027 assessment windows, alongside the quadratic formula, the sine and cosine rules and the compound interest formula. What is not given is the meaning of a gradient or an area in context, so those you have to know.