Speed from the graph
Out: 6 km in 1.5 hours
6 ÷ 1.5 = 4 km/h
Back: 6 km in 1 hour: 6 km/h
Distance–time graphs Why does a steeper line on a distance–time graph mean faster?
Foundation and Higher · AQA · 45 minutes
Students work out how fast Ben must go to catch Ana, then read a whole journey from its graph: up, flat and down, with speed as the gradient. Slides to present, a vote that shows who's stuck, practice and homework.
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Ana walks her dog at 2 km/h. Ben sets off after her an hour later at v km/h. Move v: when and where does he catch her? Which speeds never catch her?
They cross at (2, 4)
The idea in one minute
A distance–time graph shows how far someone is from the start at each time. It is not a picture of the route. A line going up means moving away, a flat line means stopped, and a line going down means coming back. The gradient is the speed: distance ÷ time, so a steeper line is faster. Speed in km/h needs the time in hours: 30 minutes is 0.5 hours.
The lesson goes picture first. Students move Ben's speed to see when he catches Ana before anyone explains, then read one journey part by part and work out each speed as the gradient. Only then do they change minutes into hours, find speeds coming back, and compare journeys. The core is for Foundation and Higher; the stretch finds when one person catches another by solving an equation.
By the end, students can
- say what a line going up, a flat line and a line going down mean
- work out a speed as the gradient: distance ÷ time
- change minutes into hours to give a speed in km/h
- read a journey from its graph, and draw one from a story
Key words
- distance–time graph
- speed
- gradient
- km/h
- stationary
- average speed
Before this lesson
Students should read coordinates from a graph and divide with decimals. Gradient as up ÷ across (the straight-line-graphs lesson) helps but is met again here.
Taught in Straight-line graphs.
Next: Velocity–time graphs.
The lesson, step by step
45 minutes in total. The stretch is optional.
5 min 1 · Hook · students think
Catch Ana
Students find how fast Ben must go to catch Ana before anyone explains.
Say“Ana walks her dog at 2 km/h. Ben sets off after her an hour later. How fast must he go to catch her two hours after she left? Write a speed.”
Ask- Where is Ana after 2 hours?
- How long has Ben had to get there?
Watch forAnswering 2 km/h, the same as Ana: he would never catch her. Keep it for the slider.
For the teacher
Give two minutes of silent work. After 2 hours Ana is 4 km away; Ben has had 1 hour, so he needs 4 km/h.
Move v on the slide. A faster Ben has a steeper line and catches her sooner; at v = 2 the lines are parallel and never meet. Don't explain gradient yet: the question is why steeper means faster.
Gather speeds with Ask the class, then move v on the slide: v = 4 crosses Ana's line at 2 hours. Ask what v = 2 and v = 1 show.
Shorter: Move v together; skip the silent working (−2 min).
12 min 2 · Explain · you explain
Read a journey; speed is the gradient
Up, flat and down; each part's speed as distance ÷ time; then minutes into hours.
Say“This graph is not a map. It says how far Jo is from home at each time.”
Ask- Where is Jo after 2 hours? What is she doing then?
- Which part is faster: going or coming back? How can you tell before working it out?
Watch forReading the flat part as walking along a flat road at a steady speed: flat means no distance is covered.
Minutes into hours
3 km in 20 minutes
20 minutes = 20 ÷ 60 = 1/3 hour
3 ÷ 1/3 = 9 km/h
For the teacher
First slide: Jo walks to her gran's, stays an hour, then jogs home. Up: going away from home. Flat, from 1.5 to 2.5 hours: stopped at her gran's, 6 km away. Down: coming back. The graph is distance from home, not the shape of the road.
Second slide: each part's speed is distance ÷ time, its gradient. Out: 6 km in 1.5 hours is 4 km/h. Back: 6 km in 1 hour is 6 km/h, so the line back is steeper. Coming back, the gradient is negative, but the speed is still 6 km/h.
Third slide: y = sx goes s km each hour. Move s: bigger s, steeper line; s = 0 is flat, not moving. Then minutes into hours: divide by 60. 3 km in 20 minutes is 9 km/h, because 20 minutes is a third of an hour.
Vote before moving s (Ask the class): what will the line look like when s is 0?
Shorter: Skip the slide with s to move; keep the journey and minutes (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin, which is almost the same.”
Ask- Is the time in hours yet?
Watch forDividing time by distance: speed is distance ÷ time.
Speed from two points
From (0, 0) to (2, 8): hours and km
8 km in 2 hours
8 ÷ 2 = 4 km/h
Then theirs
From (1, 3) to (3, 9): hours and km
Minutes first
5 km in 30 minutes
30 minutes = 0.5 hours
5 ÷ 0.5 = 10 km/h
Then theirs
2 km in 15 minutes
Coming back
From (3, 6) down to (5, 0)
6 km back in 2 hours
6 ÷ 2 = 3 km/h, back towards home
Then theirs
From (2, 8) down to (4, 0)
For the teacher
Work the first example aloud, writing each line. Then students solve its twin while you watch who finds distance and time from the two points.
Do the same for each pair. The second pair catches students who use 0.15 for 15 minutes; the third, those who think a line going down means slowing down.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
15 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- How far did Kai go in each part, and how long did it take?
Watch forChoosing the highest part of the graph as the fastest.
Show your working on paper. Give speeds in km/h unless the question says otherwise. The stretch is optional.
The question that shows who's stuck
- Kai's distance–time graph goes from (0, 0) to (0.5, 4), stays flat to (1.5, 4), then goes to (3.5, 8). Time is in hours, distance in km. When was Kai fastest?
In your Classroom, as they answer (an example) - A8
- B10
- C3
- D4
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- A cyclist rides 36 km in 3 hours at a steady speed. What is her speed in km/h?
- A straight line on a distance–time graph goes from (0, 0) to (4, 10). Time is in hours, distance in km. What is the speed in km/h?
- Mo walks 3 km in 45 minutes. What is his speed in km/h?
- A car travels at 48 km/h for 20 minutes. How many km does it go?
- On a distance–time graph (hours, km from home), a straight line goes down from (2, 6) to (4, 0). What is happening?
- Two friends walk along the same road from the same house. Their lines on one distance–time graph cross at (2, 6). What does the crossing mean?
- Which is faster: 20 km in 30 minutes, or 50 km in 1 hour 15 minutes?
- Sam drives 90 km in 1.5 hours, stops for 30 minutes, then drives 60 km in 1 hour. What is his average speed for the whole journey, in km/h?
- A distance–time graph is a curve that gets steeper and steeper. What is happening?
- Stretch: Ana leaves home walking at 3 km/h. Ben leaves the same place an hour later, along the same path, at 5 km/h. He catches her ___ hours after Ana set off, ___ km from home.
Support · For question 3: change the minutes into hours first, 45 ÷ 60. Then speed = distance ÷ time. For question 5: ask what the distance from home is doing.
For the teacher
Everyone chooses before anyone explains. Then, for each part of Kai's graph, ask: how far, and how long? Ask what the height says (distance from home) and what the steepness says (speed), and how far Kai goes while the line is flat.
Questions 1–2 build fluency; 3–4 change minutes into hours; 5–7 ask for reasons; 8 is exam-style; 9 meets a curve. The stretch is a race like the hook's, solved by an equation.
Students show their working on paper or the board; the quiz keeps their final answers.
If many chose the highest part, cover the y-values and ask which part covers 4 km in less time, before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: draw Ana's and Ben's lines and check the crossing (+10 min).
5 min 5 · Exit · students show
Draw a journey
One sketch from a story and one speed show who reads and draws these graphs.
Say“Sketch the graph of this walk, with units on both axes. Then work out the speed home.”
Watch for45 minutes drawn or used as 0.45 hours.
A sketch on paper, then your speed and one sentence.
- You walk 3 km from home in 1 hour, stop for 30 minutes, then walk home in 45 minutes. Sketch the distance–time graph. What is your speed on the way home? Say how the graph shows which part was fastest.
For the teacher
Walk 3 km from home in 1 hour, stop for 30 minutes, then walk home in 45 minutes. Sketch the distance–time graph.
Then work out the speed home and answer the exit question.
The sketches show who draws the stop flat and the way home going down; the speeds show who changes minutes into hours. Start the next lesson with the most common slip. Set the homework.
Longer: Swap sketches: tell the story of a partner's graph (+5 min).
One to one: let the student walk a journey along a corridor or a table edge while you draw its graph, then swap; ask them to say each part's speed with its units.
Homework: speeds and journeys
Show your working, and give units with every speed. Change minutes into hours first. Draw the graph in question 3 on squared paper, time across and distance up.
- A bus travels 30 km in 45 minutes. What is its speed in km/h?
- How long does it take to walk 15 km at 6 km/h? Give your answer in hours and minutes.
- Lee leaves home at 9 am and cycles 12 km to the park at 8 km/h. He stays for 30 minutes, then cycles home at 12 km/h. Draw the distance–time graph. When does he get home?
- A distance–time graph (hours, km) goes from (0, 0) to (2, 6), is flat to (3, 6), then goes down to (4, 0). Describe the journey, with the speed of each part.
- Ali leaves the house at 12:00, walking at 4 km/h. Bea leaves the same house at 12:30, cycling the same way at 12 km/h. At what time does Bea catch up with Ali, and how far from the house?
- A driver goes 40 km at 80 km/h, then another 40 km at 40 km/h. Show that the average speed for the 80 km is not 60 km/h.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching distance–time graphs: common questions
How do you stop students reading a distance–time graph as a picture of a hill?
Name the axes out loud before anything else: time across, distance from the start up. Then ask what is happening to the distance in each part: growing, staying the same or shrinking. Acting out a journey, or moving a speed and watching the line tilt, makes the point quicker than telling.
Why is the gradient of a distance–time graph the speed?
The gradient is up ÷ across. Up is the distance gone and across is the time it took, so up ÷ across is distance ÷ time: the speed. A steeper line covers more distance in each hour; a flat line covers none.
Why do students get speeds in km/h wrong with minutes?
They write 45 minutes as 0.45 hours. An hour has 60 minutes, so divide by 60: 45 minutes is 0.75 hours. Another way is to scale up to an hour: 3 km in 20 minutes is 9 km in 60 minutes, 9 km/h.
Is this distance–time graphs lesson suitable for Foundation GCSE?
Yes. The core reads journeys, works out speeds as gradients and changes minutes into hours, for students who can read coordinates and divide with decimals. The stretch, solving an equation for when one person catches another, suits Higher.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: A14 (plot and interpret graphs in real contexts, including distance–time), with R11 (speed as a compound unit) and R14 (the gradient of a straight line as a rate of change). The curve in question 9 touches A15 (Higher).
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