Chapter 8 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.8 Chapter 8

Graphs from real life

Can a graph tell the story of a walk?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Draw and read a distance time graph of a walk
  • Explain what a flat part of a travel graph means
  • Spot direct proportion as a straight line through (0, 0)

A distance time graph tells the story of a journey. Time goes across, in minutes or hours. Distance from the start goes up, in km. Both axes start at 0.

Amina walks to a spring. She walks 2 km in the first 20 minutes. She rests for 10 minutes. Then she walks 4 km more in 30 minutes. Here is her walk as a graph.

Draw it first

Draw the graph of Amina's walk

  1. A grid with the x axis across and the y axis up. A graph with time (min) across and distance (km) up. Step 1: dots at 0 and 0, 20 and 2, 30 and 2, and 60 and 6. Step 2: straight lines join the dots. Step 3: dashed guides show that after 20 minutes she has gone 2 km.01234560102030405060distance (km)time (min)

    1 Mark a dot for each time and distance: (0, 0), (20, 2), (30, 2) and (60, 6).

  2. A grid with the x axis across and the y axis up. A graph with time (min) across and distance (km) up. Step 1: dots at 0 and 0, 20 and 2, 30 and 2, and 60 and 6. Step 2: straight lines join the dots. Step 3: dashed guides show that after 20 minutes she has gone 2 km.01234560102030405060distance (km)time (min)

    2 Join the dots in order with straight lines.

  3. A grid with the x axis across and the y axis up. A graph with time (min) across and distance (km) up. Step 1: dots at 0 and 0, 20 and 2, 30 and 2, and 60 and 6. Step 2: straight lines join the dots. Step 3: dashed guides show that after 20 minutes she has gone 2 km.01234560102030405060distance (km)time (min)

    3 Read across from 20 minutes. She has gone 2 km.

Worked example

In Amina's graph, how long does she rest?

  1. A rest is a flat part of the graph. The flat part is from 20 minutes to 30 minutes.
  2. Along it the distance stays 2 km. She does not move.
  3. The rest lasts 30 − 20 = 10 minutes.

Answer: 10 minutes

Amina's graph with time (min) across and distance (km) up. The first part is marked walk and goes up to 2 km. The flat part is marked rest. The last part is marked walk and is steeper.01020304050600123456walkrestwalktime (min)distance (km)
The flat part is the rest. The last part is steeper, so she walks faster.

The steepness of a distance time graph is a speed. This is the gradient from chapter 6: the distance gone, divided by the time taken. A steeper line means a faster speed.

Worked example

Kofi cycles at a steady speed. The graph shows that he goes 6 km in 30 minutes. What is his speed in km per hour?

  1. In 30 minutes he goes 6 km. Half an hour is 30 minutes.
  2. In a whole hour he goes twice as far: 6 × 2 = 12 km.
  3. His speed is 12 km per hour.

Answer: 12 km per hour

A graph with time (min) across and distance (km) up. A straight line goes from the origin through the point 30 and 6 to the point 60 and 12. Dashed guides show that after 30 minutes Kofi has gone 6 km.0246810120102030405060distance (km)time (min)
A straight line means a steady speed. In 30 minutes he goes 6 km.

Now a different kind of graph. A tap fills a bucket with 3 litres each minute. Here the bucket starts empty. After 1 minute there are 3 litres, after 2 minutes 6 litres. The points lie on a straight line through (0, 0).

A grid with the x axis across and the y axis up. A graph with min across and litres up. The line y = 3x goes through the origin and the dots (1, 3), (2, 6) and (3, 9).y = 3x123456781234567891011120minlitres
The bucket starts empty, so the line starts at (0, 0). Litres ÷ minutes is 3 every time.

Now start the bucket with 2 litres in it. The line is still straight, but it starts at (0, 2). It misses (0, 0). Litres ÷ minutes is not always the same: 5 ÷ 1 = 5 but 8 ÷ 2 = 4.

Worked example

A straight line goes through (1, 5) and (2, 8). The x number is minutes and the y number is litres. Is the water in direct proportion to the time?

  1. Direct proportion needs a straight line through (0, 0).
  2. For the first point, y ÷ x is 5 ÷ 1 = 5.
  3. For the second point, y ÷ x is 8 ÷ 2 = 4.
  4. The two answers are not the same, so the line cannot go through the origin. If it did, the point with x = 2 would be (2, 10), and 10 ÷ 2 = 5.
  5. The water is not in direct proportion to the time.

Answer: No. The line does not pass through the origin.

A grid with the x axis across and the y axis up. A graph with min across and litres up. The line y = 3x + 2 starts at 2 on the y axis, not at the origin. The dots (1, 5), (2, 8) and (3, 11) are on it.y = 3x + 2123456781234567891011120minlitres
The rule is y = 3x + 2. The bucket starts with 2 litres, so the line starts at (0, 2). It is straight, but it is not direct proportion.

See it three ways

With things

Walk along a path. Every 10 minutes, count how far you are from the start.

As a picture

A grid with the x axis across and the y axis up. A graph with time (min) across and distance (km) up. The dots are at 0 and 0, 20 and 2, 30 and 2, and 60 and 6.01234560102030405060distance (km)time (min)

In symbols

The pairs of time and distance are (0, 0), (20, 2), (30, 2) and (60, 6). The flat part, 2 km at 20 minutes and at 30 minutes, is a rest.

Try it

Graph your own walk

You need: a clock or a watch with seconds, a path or a yard, squared paper and a pencil.

  1. Walk in a straight line at a steady pace. A friend counts your paces and calls the time every 10 seconds.
  2. Stop for 10 seconds, then walk again. Your friend keeps calling the time.
  3. Write down the pairs of time in seconds and distance in paces.
  4. Plot the pairs and join them. Show time across and distance up.

What you will see: Your graph goes up while you walk and goes flat while you stand still. If you walk faster, the line is steeper.

If you cannot try it, this is what happens.
A graph with time (s) across and paces up. The dots are at 0 and 0, 10 and 10, 20 and 20, 30 and 20, and 40 and 35. The part from 20 to 30 seconds is flat, where the walker stood still.010203040010203040pacestime (s)

A wrong idea many people have

A distance time graph is a picture of the path, so a line going up means a hill.

Why it feels right: The graph looks like a map, with a line that climbs and falls.

The test that breaks it: Amina walks on flat ground all the way. Her graph still goes up, because her distance grows as time goes by. It goes flat only while she rests.

What is true: The graph shows distance against time, not the shape of the ground. Up means she is further from the start. Flat means no movement. Steeper means faster.

Key words

distance time graph
a graph of how far something is from the start as time passes. Time goes across and distance goes up.
direct proportion
two amounts where y ÷ x is always the same. When one doubles, so does the other. Their graph is a straight line through (0, 0).

Back to your guess

The answer is B: A flat part of a distance time graph means the person has stopped.

A distance time graph shows how far someone is from the start. A flat part means the distance does not change while time goes on. So the person is not moving. A steeper part means faster.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What does the flat part of Amina's graph mean?

    Amina's graph with time (min) across and distance (km) up. It goes up, then is flat from 20 to 30 minutes, then goes up again.01020304050600123456time (min)distance (km)
    Tap your answer.

    1. Not this one. Walking faster makes the line steeper, not flat. Go to M3.9.8. Try another.

    2. Yes, that is right. In the flat part the distance stays the same while the time goes on. So she is not moving.

    3. Not this one. Walking back would make the line go down. In the flat part the distance does not change. Go to M3.9.8. Try another.

    See the answer

    She is resting. She has stopped.

    In the flat part the distance stays the same while the time goes on. So she is not moving.

    • If you chose A: Walking faster makes the line steeper, not flat. Go to M3.9.8.
    • If you chose C: Walking back would make the line go down. In the flat part the distance does not change. Go to M3.9.8.

  2. How far has Amina gone after 30 minutes?

    Amina's graph with time (min) across and distance (km) up. It goes up to 2 km at 20 minutes, is flat until 30 minutes, then goes up to 6 km at 60 minutes.01234560102030405060distance (km)time (min)
    Tap your answer.

    1. Not this one. That is the distance at 45 minutes. At 30 minutes read up from 30 to the line, then across. Go to M3.9.8. Try another.

    2. Not this one. The 30 is the time, in minutes. The distance is on the axis going up. Go to M3.9.8. Try another.

    3. Yes, that is right. Read up from 30 minutes to the line, then across to the distance axis. The line is at 2 km.

    4. Not this one. That is the distance at 60 minutes, at the end of the walk. Go to M3.9.8. Try another.

    See the answer

    2 km

    Read up from 30 minutes to the line, then across to the distance axis. The line is at 2 km.

    • If you chose A: That is the distance at 45 minutes. At 30 minutes read up from 30 to the line, then across. Go to M3.9.8.
    • If you chose B: The 30 is the time, in minutes. The distance is on the axis going up. Go to M3.9.8.
    • If you chose D: That is the distance at 60 minutes, at the end of the walk. Go to M3.9.8.

  3. Which part of Amina's walk is the steepest?

    Tap your answer.

    1. Yes, that is right. Speed is the distance divided by the time. At first she goes 2 km in 20 minutes, which is 6 km per hour. Later she goes 4 km in 30 minutes, which is 8 km per hour. So the last part is steepest.

    2. Not this one. In the first 20 minutes she goes 2 km. In the last 30 minutes she goes 4 km, which is faster. Go to M3.9.8. Try another.

    3. Not this one. The rest is flat. It is the least steep part of the graph. Go to M3.9.8. Try another.

    See the answer

    The last 30 minutes

    Speed is the distance divided by the time. At first she goes 2 km in 20 minutes, which is 6 km per hour. Later she goes 4 km in 30 minutes, which is 8 km per hour. So the last part is steepest.

    • If you chose B: In the first 20 minutes she goes 2 km. In the last 30 minutes she goes 4 km, which is faster. Go to M3.9.8.
    • If you chose C: The rest is flat. It is the least steep part of the graph. Go to M3.9.8.

  4. A straight line goes from the origin to (60, 12), where x is minutes and y is km. What is the speed in km per hour?

    Tap your answer.

    1. Not this one. 12 ÷ 60 = 0.2 is the km in one minute. Multiply by 60 to get the km in one hour. Go to M3.9.8. Try another.

    2. Not this one. You divided 60 by 12. The speed is the distance divided by the time. Go to M3.4.6. Try another.

    3. Not this one. You added 60 and 12. The speed is distance ÷ time, using 60 minutes as one hour. Go to M3.4.6. Try another.

    4. Yes, that is right. 60 minutes is 1 hour. In 1 hour the line shows 12 km. So the speed is 12 km per hour.

    See the answer

    12 km per hour

    60 minutes is 1 hour. In 1 hour the line shows 12 km. So the speed is 12 km per hour.

    • If you chose A: 12 ÷ 60 = 0.2 is the km in one minute. Multiply by 60 to get the km in one hour. Go to M3.9.8.
    • If you chose B: You divided 60 by 12. The speed is the distance divided by the time. Go to M3.4.6.
    • If you chose C: You added 60 and 12. The speed is distance ÷ time, using 60 minutes as one hour. Go to M3.4.6.

  5. Which of these lines shows direct proportion?

    A grid with the x axis across and the y axis up. Two parallel straight lines. The first, line A, goes through the origin. The second, line B, starts at 3 on the y axis, above the origin.line Aline B123451234567890xy
    Tap your answer.

    1. Yes, that is right. Line A is straight and goes through (0, 0). Line B is straight too, but it misses the origin.

    2. Not this one. Line B is straight, but it starts at 3 on the y axis. For direct proportion the line must go through (0, 0). Go to M3.9.8. Try another.

    3. Not this one. A straight line is not enough. It must also go through the origin. Go to M3.4.5. Try another.

    See the answer

    Line A

    Line A is straight and goes through (0, 0). Line B is straight too, but it misses the origin.

    • If you chose B: Line B is straight, but it starts at 3 on the y axis. For direct proportion the line must go through (0, 0). Go to M3.9.8.
    • If you chose C: A straight line is not enough. It must also go through the origin. Go to M3.4.5.

  6. A cyclist goes 15 km in 3 hours at a steady speed. What is the speed in km per hour?

    See the answer

    5

    Speed is the distance divided by the time. 15 ÷ 3 = 5 km per hour.

  7. On a distance time graph the flat part runs from 15 minutes to 40 minutes. How many minutes is the rest?

    See the answer

    25

    The rest lasts from 15 to 40 minutes. 40 − 15 = 25 minutes.

  8. From an earlier chapter

    What is the gradient of the line y = 7x − 3?

    See the answer

    7

    In y = mx + c the number in front of x is the gradient. Here it is 7.

Keep it

Copy this into your exercise book.

On a distance time graph, up means further away, flat means stopped, and steeper means faster. A straight line through (0, 0) is direct proportion. A line that misses (0, 0) is not.

Draw: Draw time in minutes across and distance in km up. Plot (0, 0), (20, 2), (30, 2) and (60, 6). Join them and write "rest" on the flat part.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (761 KB). It opens on any phone.