1. KS3 Mathematics A6 Real-life graphs
The aim of this unit is to teach pupils to:
Construct functions arising from real-life problems, and plot and interpret their corresponding graphs
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
A6 Real-life graphs

2. A6.4 Distance-time graphs
Contents
A6 Real-life graphs A
A6.1 Reading graphs A
A6.2 Plotting graphs A
A6.3 Conversion graphs A
A6.4 Distance-time graphs A
A6.5 Interpreting graphs

3. Distance-time graphs
In a distance-time graph the horizontal axis shows time and the vertical axis shows distance.
The below distance-time graph shows a journey.
What does the slope of the line tell us?
time
distance
The slope of the line tells us the average speed.
Ask pupils to tell you what is happening at the places where the line is horizontal (the object is not moving).
Ask pupils what it means if the graph slopes downwards (the object is moving in the opposite direction).
Point out that a straight line in a distance-time graph indicates constant speed.
What would it mean if the line was curved?
A curved line would indicate acceleration or change in speed over time.
The steeper the line is, the faster the speed.

4. Label the distance-time graph
Use this activity to discuss the meaning of different parts of the graph.
The correct positions are:
A, accelerating;
B, moving slowly at constant speed;
C, moving rapidly away from the starting point at constant speed;
D, not moving;
E, moving at constant speed towards the starting point;
F, decelerating.

5. Olympic swimmers
This animation shows a distance-time graph being plotted in real time. Start by choosing a different starting speed for each swimmer and pressing go.
Change the speed of the swimmers as the race progresses and note what happens on the graph. To make a swimmer stop altogether change his speed to 0.
You could ask one volunteer to vary the speeds of the swimmers and another to provide a race commentary, stating when one swimmer overtakes another, for example.

6. A6 Real-life graphs Contents A A6.1 Reading graphs A
A6.2 Plotting graphs A
A6.3 Conversion graphs A
A6.4 Distance-time graphs A
A6.5 Interpreting graphs

7. Filling flasks 1
Start by explaining that the we are going to produce a graph of the depth of water in a flask as it fills with water. Note that the water flows out of the tap at a constant rate.
As the first flask fills up the graph of depth against time will be drawn. Ask pupils to tell you how many cm are filled each second for the flask.
Ask pupils to predict the slope of the graph for the new flask compared to the previous flask. Ask pupils to justify why they think the graph will be steeper or less steep than before.
Continue for each flask in turn. Establish the depth in the narrowest flask will increase the fastest and therefore produce the steepest graph. The depth in the widest flask will increase the slowest.
Ask pupils to explain why all the lines pass through the origin.
Ask pupils to explain why the lines are straight.
Ask pupils to explain what would happen if the water from the tap did not flow out at a constant rate. For example, in real life the rate of the water coming out of the tap would speed up as the tap is turned on. How would this affect the shape of the graph?

8. Filling flasks 2
Start by explaining that the we are going to produce a graph of the depth of water in a flask as it fills with water. Note that the water flows out of the tap at a constant rate.
As the first flask fills up the graph of depth against time will be drawn.
Ask pupils to predict the shape of the next graph before it is drawn, justifying their explanations. Establish that the wider the flask is at a given point the loner it will take for the water to increase in depth.
Continue for each flask in turn.
Ask pupils if we can use the graph to work out the capacity of each flask.

9. Interpreting the shapes of graphs
Jessica eats a bar of chocolate. This graph shows how the mass of the chocolate bar changes as it is eaten.
150 50
100 10 20 30 40 60 70 80 90
Eating a bar of chocolate
Mass of chocolate (g)
Time (seconds)
Discuss the graph and ask the following questions:
What do the vertical portions of the graph represent?
What do the horizontal portions of the graph represent?
How many bites did it take to finish the bar? (7)
What was the weight of the biggest bite? (40g)
How long did it take Jessica to eat the first bite? (15 seconds)
What is the original weight of the chocolate bar? (150g)
How long did Jessica take to finish the chocolate bar in seconds/in minutes and seconds? (105 seconds/ 1 minute 45 seconds)

10. Interpreting the shapes of graphs
This graphs shows how the temperature of the water in a pan changes when frozen peas are added.
Time
Temperature of water
Ask a volunteer to locate the point on the graph that shows when the peas are added to the water.
Discuss trends in the graph. Ask pupils to describe what is happening in the portion of the graph that slopes downwards. Ask also what is happening in the portion of the graph that slopes upwards.
Ask pupils to describe what is happening at each turning point in the graph.

11. Which graph is correct?
In an experiment a group of pupils poured water onto a sponge and weighed it at regular intervals.
Each time the sponge soaked up all the water.
Which graph is most likely to show their results?
Mass of sponge (g)
Volume of water (cm3)
Graph A
Graph B
Graph C
Graph D
Ask pupils to justify their choice of graph to fit the situation.

12. Sketching graphs
A group of pupils are conducting an experiment. They fill three beakers with boiling water and record the temperature of the water over time.
Beaker A has no wrapping, Beaker B is wrapped in ice and Beaker C is wrapped in insulation fibre.
Time (minutes)
Temperature (oC)
The temperature graph for beaker A looks as follows:
How would the graphs for beakers B and C compare to this?
Ask pupils to describe what the graph for beaker A shows. Agree that it shows the water cooling rapidly and then cooling more slowly. Ask pupils how the graphs for beakers B and beaker C will differ from the one shown.
Use the next slide to sketch these different graphs.
Beaker A

13. Sketching graphs
Use the pen tool to sketch the graph of beaker A cooling down.
Ask volunteers to sketch the graphs of beaker B and beaker C. Remind pupils that the three beakers all start at the same temperature.

14. Matching graphs to statements
Establish that if something is rising rapidly over time it will have a steeper and steeper gradient. If something is falling rapidly it will have a steeper negative (or downward) gradient. If something is rising or falling steadily then the graph will be straight.
Ask pupils what a horizontal section of graph would represent.
Each of the graphs in this example illustrates trends rather than accurate information.

15. A8.4 Interpreting real-life graphs
Contents
A8 Linear and real-life graphs A
A8.1 Linear graphs A
A8.2 Gradients and intercepts A
A8.3 Parallel and perpendicular lines A
A8.4 Interpreting real-life graphs A
A8.5 Distance-time graphs A
A8.6 Speed-time graphs

16. Real-life graphs
When we use graphs to illustrate real-life situations, instead of plotting y-values against x-values, we plot one physical quantity against another physical quantity.
The resulting graph shows the rate that one quantity changes with another. For example,
This graph shows the exchange rate from British pounds to American dollars.
British pounds
American dollars
It is a straight line graph through the origin and so the equation of the line would be of the form y = mx.
m represents the exchange rate (number of dollars to the pound).
The value of m would be equal to the number of dollars in each pound.
What would the value of m represent?

17. Real-life graphs
This graph show the value of an investment as it gains interest cumulatively over time.
The graph increases by increasing amounts.
time
investment value
Each time interest is added it is calculated on an ever greater amount.
This makes a small difference at first but as time goes on it makes a much greater difference.
This is an example of an exponential increase.

18. Real-life graphs
This graph show the mass of a newborn baby over the first month from birth.
time
mass
The baby’s mass decreases slightly during the first week.
Its mass then increases in decreasing amounts over the rest of the month.

19. Rates of change
For each graph ask pupils whether y is large, small or becomes zero when x is large. Also ask whether y is large, small or becomes zero when x is small.
Establish whether y increases as x increases or whether y decreases as x increases. Then decide whether it is increasing or decreasing by equal, increasing or decreasing amounts.
Ask pupils to describe a real-life situation that could be described by each graph. For example, a graph that shows y increasing by increasing amounts could be showing acceleration if x is time and y is distance travelled.

20. Filling flasks
Start by explaining that the we are going to produce a graph of the depth of water in a flask as it fills with water. Note that the water flows out of the tap at a constant rate.
As the first flask fills up, the graph of depth against time will be drawn.
Ask pupils to predict the shape of the next graph before it is drawn, justifying their explanations. Establish that the wider the flask is at a given point the longer it will take for the water to increase in depth.
Before filling a flask, ask a volunteer to use the pen tool to show the shape of the graph. Play the animation to see how close the pupil’s prediction was.
Continue for each flask in turn.
Ask pupils to explain why all the lines pass through the origin.
Ask pupils to explain what would happen if the water from the tap did not flow out at a constant rate. For example, in real life the rate of the water coming out of the tap would speed up as the tap is turned on. How would this affect the shape of the graph?
Ask pupils if we can use the graph to work out the capacity of each flask.

21. A8.5 Distance-time graphs
Contents
A8 Linear and real-life graphs A
A8.1 Linear graphs A
A8.2 Gradients and intercepts A
A8.3 Parallel and perpendicular lines A
A8.4 Interpreting real-life graphs A
A8.5 Distance-time graphs A
A8.6 Speed-time graphs

22. Formulae relating distance, time and speed
It is important to remember how distance, time and speed are related.
Using a formula triangle can help,
distance = speed × time
DISTANCE
SPEED
TIME
time =
distance
speed
Recall the basic formulae relating distance, time and speed using the formula triangle.
speed =
distance
time

23. Distance-time graphs
In a distance-time graph the horizontal axis shows time and the vertical axis shows distance.
For example, John takes his car to visit a friend. There are three parts to the journey:
John drives at constant speed for 30 minutes until he reaches his friend’s house 20 miles away.
time (mins)
distance (miles) 15 30 45 60 75 90
105
120 5 10 20
He stays at his friend’s house for 45 minutes.
Talk through the various stages of the journey. The distance refers to the distance from the starting point, in this case John’s house.
During the first part of the journey John moves away from the starting point at a constant speed shown by the slope of the graph.
While John stays at his friend’s house his distance from home does not change as time goes on and so the line is horizontal.
In the last part of his journey the graph slopes downwards because as he returns home his distance from the starting point decreases.
Ask pupils to calculate John’s speed for the first and last parts of the journey in miles per hour.
Establish that the speed is given by the gradient of the graph.
He then drives home at a constant speed and arrives home 45 minutes later.

24. Finding speed from distance-time graphs
How do we calculate speed?
Speed is calculated by dividing distance by time.
In a distance-time graph this is given by the gradient of the graph.
gradient =
change in distance
change in time
time
distance
change in distance
= speed
The steeper the line, the faster the object is moving.
change in time
A zero gradient means that the object is not moving.

25. Interpreting distance-time graphs
Move the points to change the graph and ask pupils to interpret it. For example, ask pupils at what speed the object is traveling for different stage in the journey.
Ask pupils to make up a story for a given graph.

26. Distance-time graphs
When a distance-time graph is linear, the objects involved are moving at a constant speed.
Most real-life objects do not always move at constant speed, however. It is more likely that they will speed up and slow down during the journey.
Increase in speed over time is called acceleration.
acceleration =
change in speed
time
It is measured in metres per second per second or m/s2.
When speed decreases over time is often is called deceleration.

27. Distance-time graphs
Distance-time graphs that show acceleration or deceleration are curved. For example,
This distance-time graph shows an object accelerating from rest before continuing at a constant speed.
time
distance
This distance-time graph shows an object decelerating from constant speed before coming to rest.
time
distance
Point out that the first graph gets steeper as time goes on. If the gradient is increasing then the object is speeding up.
The second graph gets shallower as time goes on. If the gradient is decreasing then the object is slowing down.

28. Contents A8 Linear and real-life graphs A A8.1 Linear graphs A
A8.2 Gradients and intercepts A
A8.3 Parallel and perpendicular lines A
A8.4 Interpreting real-life graphs A
A8.5 Distance-time graphs A
A8.6 Speed-time graphs

29. Speed-time graphs
Travel graphs can also be used to show change in speed over time.
For example, this graph shows a car accelerating steadily from rest to a speed of 20 m/s.
It then continues at a constant speed for 15 seconds.
time (s)
speed (m/s) 5 10 15 20 25 30 35 40
The brakes are then applied and it decelerates steadily to a stop.
Talk through the various stages in the graph.
Stress that when the graph slopes downwards a decrease in speed is indicated. Unlike in a distance-time graph it is still moving in the same direction.
The car is moving in the same direction throughout.

30. Finding acceleration from speed-time graphs
Acceleration is calculated by dividing speed by time.
In a speed-time graph this is given by the gradient of the graph.
gradient =
change in speed
change in time
time
speed
change in speed
= acceleration
The steeper the line, the greater the acceleration.
Remind pupils that acceleration is measured in metres per second per second or m/s2.
change in time
A zero gradient means that the object is moving at a constant speed.
A negative gradient means that the object is decelerating.

31. Finding distance from speed-time graphs
The following speed-time graph shows a car driving at a constant speed of 30 m/s for 2 minutes.
time (s)
speed (m/s) 15 30 45 60 75 90
105
120 5 10 20
What is the area under the graph?
The area under the graph is rectangular and so we can find its area by multiplying its length by its height.
Remind pupils that the distance travelled can be found by multiplying the average speed by the time taken.
Use this to establish that the area under the graph corresponds to the distance travelled in metres.
Area under graph = 20 × 120 =
240
What does this amount correspond to?

32. The area under a speed-time graph
This area under a speed-time graph corresponds to the distance travelled.
time (s)
speed (m/s) 5 10 15 20 25 30 35 40
For example, to find the distance travelled for the journey shown in this graph we find the area under it.
The shape under the graph is a trapezium so, 15
Area = ½( ) × 20
Point out that we could also find the area under the graph by dividing it into a rectangle with two triangles either side. Demonstrate, if necessary, that this leads to the same answer.
The units in the answer depend on the units used in the graph. Here we have m/s multiplied by s to give the answer in metres.
= ½ × 55 × 20
= 550
So, distance travelled
= 550 m

33. Interpreting speed-time graphs
Move the points to change the graph and ask pupils to interpret it. For example, ask pupils to calculate the acceleration and distance travelled over various stages in the journey.
Ask pupils to find the total distance travelled by finding the area under the graph.
Ask pupils to make up a story for a given graph.