Reading a journey
Tonight's idea
A distance-time graph shows how far you are from the start at each moment. Time runs along the bottom. Distance from home goes up the side. Here is Maya's bike ride to the library and back.
Read it like a story. In A the line climbs: she is getting further from home. In B it is flat. Time passes but the distance stays at 2 km, so she has stopped. C climbs again, more steeply. D is a long stop at the library, 4 km away. In E the line comes down: the distance from home is shrinking, so she is riding back.
Steeper means faster. Flat means stopped. Downhill means coming home.
Why does steep mean fast? Speed is distance ÷ time. A steep piece gains a lot of distance (a big step up) in little time (a small step across). Compare A and C. Each covers 2 km, but C takes 5 minutes and A takes 10.
Watch one: how fast is part A?
Part A goes from 0 km to 2 km between 0 and 10 minutes.
Speed = distance ÷ time \(= 2 \div 10 = 0.2\) km per minute.
An hour is 60 minutes, so that is \(0.2 \times 60 = 12\) km per hour. Check it another way: 2 km in 10 minutes is 12 km in 60 minutes.
Graphs in grade 6: sheet 15. Next year, steepness gets a name and a formula of its own: grade 8, sheet 2.
Your turn
- Use Maya's graph. (a) How far is the library from her home? (b) For how many minutes is she stopped altogether? (c) At what time does she set off for home?
- Find Maya's speed in km per hour (a) in part C; (b) in part E. (c) Without any calculating, how does the graph show which of A, C and E is fastest?
- Ben's graph joins these points with straight lines: (0 min, 0 km), (20 min, 5 km), (30 min, 5 km), (40 min, 0 km). Sketch it. Then tell the story, with a speed in km per hour for each moving part.
- (a) How far did Maya ride in total, and what was her average speed over the whole hour, stops included? (b) Ben's sister draws a journey graph with a piece that goes straight up. Explain why no real journey can do that.
Stretch: two lines, one meeting
Maya's dad leaves the library at time 0 and jogs home at a steady 7.5 km per hour. Draw his line on her graph. When and where do the two lines cross, and what is Maya doing at that moment?
Answers
- (a) 4 km, the highest the line gets. (b) \(10 + 20 = 30\) minutes (parts B and D). (c) At 45 minutes.
- (a) 2 km in 5 minutes is \(2 \div 5 = 0.4\) km per minute, or 24 km per hour. (b) 4 km in 15 minutes is 16 km in 60 minutes: 16 km per hour. (c) C is the steepest piece, so C is fastest. A is the least steep, so A is slowest.
- Up from (0, 0) to (20, 5), flat to (30, 5), down to (40, 0). Ben travels 5 km in 20 minutes, which is 15 km per hour. He stops for 10 minutes. He comes back 5 km in 10 minutes, which is 30 km per hour, and is home after 40 minutes.
- (a) 4 km there and 4 km back: 8 km in one hour, so 8 km per hour on average. (b) Straight up means the distance changes while no time passes. She would be in two places at the same moment. Speed would be distance ÷ 0, which has no answer.
Stretch. 7.5 km per hour is 1 km every 8 minutes. His line starts at 4 km and falls to 3 km at 8 minutes, 2 km at 16 minutes, and reaches 0 at 32 minutes. At 16 minutes Maya is in part B, stopped 2 km from home. The lines cross there: he meets her while she is resting.