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 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

3. Graph of monthly mobile phone charges
Explain that graphs can be used to illustrate any function or formula containing two variables.
This graph compares two types of tariff offered by a mobile phone company:
The ‘Pay as you go’ tariff (shown in blue) has no monthly charge and charges a fixed amount per minute. Drag the moving point on the graph to help pupils work out the cost per minute. This is £0.10 per minute.
The ‘Monthly’ tariff (shown in red) has a fixed monthly charge and charges calls at fixed amount per minute. Drag the moving point on the graph to help pupils work out the monthly charge and the cost per minute. This is £7.50 for the monthly charge and £0.05 per minute for calls.
Ask pupils for the significance of the point where the two lines cross. This is the point where the two tariffs cost the same amount for the same amount of time spent on calls.
Ask pupils which tariff is cheaper. Establish that if we spend less than 150 minutes (2½ hours) on calls in a given month, the ‘Pay as you go tariff’ is cheaper. If we spend more than 150 minutes on calls in the month, the monthly tariff is cheaper. (150 minutes a month is about 5 minutes a day.) If we include another month we would have to pay another £7.50 on the ‘Monthly’ tariff.
Discuss whether or not the intermediate points have any practical significance.
Suppose we could zoom in on the graph to read off the charge in pence and the time in seconds. Since calls are rounded to the nearest penny and the time spent on the phone is rounded to the nearest second, intermediate points between pennies and seconds, although they have a practical significance in theory, would have to be rounded up.
Ask pupils to suggest a formula to describe each tariff. For example, for the ‘Pay as you go’ tariff:
Cost in pence = 10 × number of minutes (or C = 10n)
and for the ‘Monthly’ tariff:
Cost in pence = 5 × number of minutes (or C = 5n + 7.5).

4. 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

5. Plotting graphs – using a table of values
When we plot a graph we usually start with a table of values.
The values in the table usually come from a formula or equation or from an observation or experiment.
For example, a car hire company charges £30 to hire a car and then £25 for each day that the car is hired.
This would give us the following table of values:
Number of days, d
Cost in £, c 1 2 3 4 5
Ask pupils if they can give you a formula linking c, the cost, with d, the number of days (c = 25d + 30).
Ask pupils to tell you what kind of graph this will produce (a straight-line graph). Explain that when we are plotting a graph it is very important to know which variable depends on the other. This tells us which variable will go along the horizontal axis and which variable will go along the vertical axis. When we are plotting graphs of functions, for example, the value of y depends on the value of x. 55 80
105
130
155
The cost of the car hire depends on the number of days. The number of days must therefore go in the top row.

6. Plotting graphs – choosing a scale
The next step is to choose a suitable scale for the axes.
Look at the values that we need to plot.
Number of days, d
Cost in £, c 1 2 3 4 5 55 80
105
130
155
The number of days will go along the horizontal axis.
The numbers range from 1 to 5.
A suitable scale would be 2 units for each day.
The cost will go along the vertical axis.
Discuss suitable scale for the range of units.
Explain that when the range is small (as for the days) we usually choose 2, 4, 5 or 10 units (squares) for each whole one.
When the range is large (as for the cost) we usually choose each unit to represent 2, 5, 10, 50, 100, …
If the range starts at a large number, the scale can start at a number other than 0 but this must be shown by a zigzag on the axes.
The cost ranges from 55 to 155.
A suitable scale would be 1 unit for each £10. We could start the scale at £30.

7. Plotting graphs – drawing the axes
We then have to draw the axes using our chosen scale.
We will need at least 10 squares for the horizontal axis and 13 squares for the vertical axis.
When the scale does not start at 0 we must show this with a zigzag at the start of the axis. 30 40 50 60 70 80 90
100
110
120
130
140
150
Number the axes.
Cost (£)
Label the axes, remembering to include units, if necessary. 1 2 3 4 5
Number of days

8. Plotting graphs – plotting the points
Use the table of values to plot the points on the graph.
Number of days, d
Cost in £, c 1 2 3 4 5 55 80
105
130
155
Cost of car hire
150
It is most accurate to use a small cross for each point.
140
130
120
110
If appropriate, join the points together using a ruler.
100
Cost (£) 90 80
Tell pupils that it is most accurate to use a small cross when plotting points on a graph.
Point out that when the points do not lie in a straight line we have to decide whether to use a line of best fit, a smooth curve through the points or to join the points together using straight lines. The one we choose depends on the context from which the graph is generated and whether intermediate points have any significance. 70
Lastly, don’t forget to give the graph a title. 60 50 40 30 1 2 3 4 5
Number of days

9. Science experiment
A group of pupils are doing an experiment to explore the effect of friction on an object moving down a ramp.
They attach weights of different mass to the object and time how long the object takes to reach the bottom of the ramp.
They put their results in a table and use the table to plot a graph of their results.
Discuss how we can produce a table of values from an experiment.
Before revealing the table ask pupils whether the time or the mass will go on the top row of the table, and hence along the x-axis of the graph.
Establish that the time taken for the object to move down the ramp depends on its mass. It is therefore the mass that goes at the top of the table. The mass is the variable that we are choosing or controlling.
Mass of object moving down ramp (grams)
Time taken for object to move down ramp (seconds)
100 4
150 7
200 12
250 17

10. Science experiment
Mass of object moving down ramp (grams)
Time taken for object to move down ramp (seconds)
100 4
150 7
200 12
250 17
We can join the points using straight lines. 20 16
Do the intermediate points have any practical significance?
Time taken (seconds) 12
Discuss the significance of the intermediate point in this context. Since the variables are both continuous, we can assume that the intermediate points have a practical significance. We can estimate, for example, how long a mass of 175 g would take to slide down the ramp.
Discuss the following:
How the points should be joined. Should each point be joined together using straight lines or should we use a line of best fit?
Would it make sense for the line to meet the horizontal axis?
How are the variables related? (The heavier the object the longer it takes to move down the ramp.)
How long it would take for an object of mass 225 grams to slide down the slope?
How long it would take for an object of mass 300 grams to slide down the slope?
What mass we would use if we wanted the object to side down the slope in 10 seconds?
End the discussion by suggesting that although we may expect a straight line graph in theory, when we do an experiment many factors including human error and rounding can produce points that do not lie on a straight line.
The graph would be more accurate if more points were plotted. Any inaccurate readings would then stand out more easily. 8 4
How could we make the graph more accurate? 50
100
150
200
250
300
Mass of object (grams)

11. 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

12. Plotting a conversion graph
Let’s plot a graph to convert pounds to euros.
A conversion graph for pounds to euros
300
First we need a table of values:
250
200 € 20
100
160
200 30
150
240
300
150 €
100
This gives us the points:
Talk through the example.
Suppose we would like our graph to convert up to £200 to 300 euros. We need some coordinates to plot and so the first thing we should do is draw a table of values.
Let’s find four points.
Explain that 2 points are sufficient to draw a straight line graph but that it is always a good idea to have a few more points to make sure.
We want our graph to convert up £200, so let’s put that at the end of the table. Click to reveal 200 in the table.
£200 equals 300 euros. Click to reveal this too.
Now let’s choose some intermediate amounts that are easy to convert. For example, £20, £100 and £160. We can then put the corresponding number of euros underneath. Click to reveal the values in the table.
This gives us four points that we can plot on our graph. Click to reveal these.
Discuss how to choose appropriate scales for the axes. Looking at our grid we have room to use 20 squares to represent £200 along the horizontal axis. How much will each square represent?
Tell pupils that it is not necessary to number every square. In this example we can number every five squares.
Discuss an appropriate scale for the vertical axis (1 square for every 10 euros) and click to reveal it.
Emphasize to pupils that both axes must increase in equal intervals starting from 0. (If the intervals do not start at 0 then we may use a zigzag in the start of the axis to show this).
Tell pupils that it is best to use a small cross for the points. The points should then be joined together to form a straight line through every point and the origin.
Impress on pupils that they must always give their graphs a title and that they must always label the axes. If they do not do this then their graph will be meaningless.
(20, 30) 50
(100, 150)
(160, 240) 50
100
150
200
(200, 300)

13. Conversion graphs – money
The graph on the board shows a graph for converting Swiss Francs to Pounds Sterling.
Point out that different scales have been used for the horizontal and vertical axes. Ask what one square along the horizontal axis represents and what one square along the vertical axis represents.
Remind pupils that the axes must always be labelled and the graph must always have a suitable title.
Drag the point along the line to demonstrate a variety of Franc to Pound and Pound to Franc conversions.
Ask pupils:
Would it make sense to have negative values?

14. Conversion graphs – temperature
Point out that most conversion graphs are straight lines passing through the origin and ask pupils what this means. Establish that it means that the two quantities are directly proportional and explain what this means.
The conversion graph on the board is an example of a conversion graph in which the units are not directly proportional. This is because 0 degrees Celsius equals 32 degrees Fahrenheit and thus the line passes through the point (0, 32).
Point out that different scales have been used for the horizontal and vertical axes. Ask what one square along the horizontal axis represents and what one square along the vertical axis represents.
Emphasize that the axes must always be labelled and the graph must always have a suitable title.
Drag the point along the line to demonstrate a variety of Fahrenheit to Celsius conversions.
Drag the point on the graph to find approximate answers to the required conversions.

15. 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

16. 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.

17. 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.

18. 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.

19. 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

20. 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?

21. 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.

22. 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)

23. 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.

24. 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.

25. 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

26. 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.

27. 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.