Part 1 Interpreting graphsmaths.gerrydear.id.au/year09_10/Data/Other... · Part 1 Distance/time graphs 11 Check your response by going to the suggested answers section. Interpreting

Mathematics Stage 5

PAS5.2.5 Graphs of physical phenomena

Part 1 Interpreting graphs

Number: 43664 Title: PAS5.2.5 Graphs of physical phenomena

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Outcomes and indicators from Mathematics Years 7-10 Syllabus © Board of Studies NSW,2002. www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf

Overview, pp. iii-ivPart 1, pp. 3-4Part 2, pp. 3-4

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Part 1 Distance/time graphs 1

Contents – Part 1

Introduction – Part 1..........................................................3

Indicators ...................................................................................3

Preliminary quiz.................................................................5

Reading graphs.................................................................7

Comparing types of graphs .............................................15

Which graph matches?....................................................23

Stories from graphs.........................................................31

Graphs from stories.........................................................39

Suggested answers – Part 1 ...........................................47

Exercises – Part 1 ...........................................................57

2 PAS5.2.5 Graphs of physical phenomena


Introduction – Part 1

Graphs can give specific information, but more importantly they have an

advantage over tables or lists of data. They can give you impressions

about what is happening, show you trends and allow you to sometimes

make predictions.

One important aspect of graphs is that they show you how things are

changing.

In this part you will explore the meaning of graphs, and how the shape

and direction of a graph implies whether something is increasing,

decreasing or staying the same.

Indicators

By the end of Part 1, you will have been given the opportunity to work

towards aspects of knowledge and skills including:

• understanding terms such as gradient, extrapolation, discrete data,

continuous data, dependent variable and independent variable

• determining which variable should be placed on the horizontal axis

• telling a story shown by a graph concentrating on how things are

varying

• sketching informal graphs to model familiar events

• using the relative positions of two points on a graph, rather than a

detailed scale, to interpret information.

By the end of Part 1, you will have been given the opportunity to work

mathematically by:

• describing the meaning of different gradients

• distinguishing between positive and negative gradients from a graph


• matching a graph to a description of a particular event and

explaining the choice

• comparing graphs of the same situation, and deciding which one is

the most appropriate.Source: Adapted from outcomes of the Mathematics Years 7–10 syllabus

<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.


Preliminary quiz

Before you start this part, use this preliminary quiz to revise some skills

you will need.

Activity – Preliminary quiz

Try these.

1 State whether this line has a positive or negative gradient.

y

x

2 Draw a sketch of a decreasing line.

Check your response by going to the suggested answers section.



Reading graphs

Graphs are used to illustrate many different situations. In this section

you will practise reading information from a variety of different types of

graphs.

It is vital to understand what the values on each axis stand for, and what

type of information the whole graph is trying to convey.


Here is a typical graph that might be used in a baby health centre to show

parents what to expect for their healthy baby girl.

23456789

101112131415161718

0 3 6 9 12 15 18 21 2 214 21

2 234 3

Wei

ght

(kilo

gram

s)

Age (months) Age (years)

Girls 0–3 years – weight (kg) by age

97

90

50

103

Perce

ntile

Without the titles for each axis and the title line above the graph, the

information shown would be meaningless. In particular, the scale on the

horizontal axis would be quite strange.

However, once you have read the headings, you can use the graph to

discover specific information. For example, the graph shows that 97% of

girls at 12 months old weigh less than 11.5 kg and that 94% of baby girls

weigh between 2.2 and 4.2 kg when born.

You can also see general trends. For example, as girls grow older, the

difference between the heavier and lighter children increases. Also, since

the curve is steeper at the beginning, this means that babies put on weight

at a much faster rate than 2 and 3–year olds.

Complete the following activity to compare rates at which things change.


Activity – Reading graphs

Try these.

1 Water is pouring into each cylindrical tank at the same steady rate

per minute.

The diagrams show the water levels after one minute.

2 cm

A C

8 cm4 cm

B

a Work out the water levels for the first four minutes in each

cylinder and complete the tables below.

A B C

Time(min)

Level(cm)

Time(min)

Level(cm)

Time(min)

Level(cm)

0 0 0 0 0 0

1 2 1 4 1 8

2 4 2 8 2

3 3 3

4 4 4


b The graph for the water level for cylinder A is shown below.

On the same grid, draw and label graphs for B and C showing

how the water levels rise.

0 1 2 3 4Time (minutes)

Wat

er le

vel (

cm)

5

10

15

20

25

30

35

40

A

c Why does each graph turn out to be a straight line and not a

curve? (Hint: consider how the water is flowing into the

containers.)

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

d The graph for container C has the steepest gradient. Explain

what this means in terms of the problem.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________



Interpreting information from a graph largely depends on understanding

what the axes refer to. Even without scales or specific data, graphs can

show relationships between a variety of things or a trend.


Try these.

2 a Match each person with a point on the graph.

Mass

Hei

ght R

S

T U

P Q

b Vivienne is the same height as Amy and as heavy as Carl. On

the graph, mark a position for Vivienne and label it V.


3 The graph below shows the speed of a car over five minutes.

Time (minutes)

Sp

eed

1 2 3 4 50

When was the car slowing down? Explain your answer.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 Georgie is trying to improve her swimming and so she times herself

every week. This graph shows the times Georgie has taken to swim

100 m over the past six weeks.

Wk1 Wk2 Wk3 Wk4 Wk5 Wk6

Tim

e (s

econ

ds)

a Is she improving? Explain your answer.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


b Can you use this graph to predict her time after one year.

Explain your answer?

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

5 Below is a graph showing the noise level in a stadium at a sporting

event.

Time

Noi

se le

vel

a What happened to the noise level at half time?

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b Each time a goal was scored the crowd went wild. How many

goals were scored?

___________________________________________________

___________________________________________________



To read graphs, you need to understand what each axis represents. The

graph can show trends that are not obvious when you just look at the data

in a table or list.

Demonstrate your ability to read graphs by completing the following

exercise.

Go to the exercises section and complete Exercise 1.1 – Reading graphs.


Comparing types of graphs

In this section you will look at some of the different kinds of graphs you

can use and how you might prefer one type to another in a given

situation.

In the following activity you are asked to compare two graphs to

determine which is most useful.

Activity – Comparing types of graphs

Try these.

1 Here is a table of how a teenager spends their money over one week.

Amusements $10

Sports $5

School needs $3.50

Savings $6.50

You may choose to show this information in a column graph or a

sector graph.

Am

ount

(dol

lars

)

0

Teenager’s weeklybudget plan

amus

emen

ts

10.00

9.00

8.00

7.00

6.00

5.00

4.00

3.00

2.00

1.00

spor

ts

scho

ol n

eed

s

savi

ngs

savings

school needs

sports

amusements


State which graph you would prefer to use to find each of the

following, and give a brief reason for your choice.

a the amount spent on sports each week

___________________________________________________

___________________________________________________

b what takes up the biggest share of the weekly budget

___________________________________________________

___________________________________________________

c the fraction of the weekly income that is saved

___________________________________________________

___________________________________________________


Each type of graph has its strengths and its weaknesses, so deciding upon

a graph design often depends on what it will be used for or the

impression you wish to convey. But sometimes certain types of graphs

should not be used because they are misleading.

Discrete data is data that can only take certain values and not the

number in between, such as the number of children in your family or the

number of cars in a car park. Since the values in between are

meaningless, you should not join the points so a line graph would be

inappropriate for this type of data.

However, this rule is occasionally broken for various reasons. Look at

the following situation.


These two graphs both show the average daily production of oil in the

world from 1982 to 1997 inclusive.

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 9750

52

54

56

58

60

62

64

66

Mill

ion

bar

rels

Year

Average daily world production of oilGraph

A

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 9750

52

54

56

58

60

62

64

66

Mill

ion

bar

rels

Year

Average daily world production of oilGraph

B

Graph A clearly shows how oil production changed from year to year and

the lines are cleaner and easier to draw than a column graph. However,

the only significant points on this line graph those above each year mark.

Points in between have no meaning. For example, you can’t say that the

average daily oil production was ever 55 million barrels/day, Column

Graph B makes this distinction between the discrete values clearer.

So the line graph here is acceptable as it conveys the information and

trends clearly. But you must realise that the points along the line convey

no meaning.


Continuous data is data that can take all values in a range, such as your

age or your height. The graphs below show how continuous data can be

displayed.

Here is a patient’s temperature graph.

36

37

38

39

40

41

Tem

per

atur

e (°

C)

2 am 4 am 6 am 8 am 10 am noon 2 pm 4 pm 6 pm 8 pm 10 am midnightTime

Patient’s temperature graph

This is an example of a continuous function. The patient’s temperature

cannot rise from 38.5°C at 2 am to 49°C at 4 am without going through

all the temperatures between. So it can be useful to join the points to

estimate values in between. It is even better to draw a smooth curve

through the points, as times were chosen at random. The true maximum

temperature was not necessarily exactly at noon.


36

37

38

39

40

41

Tem

per

atur

e (°

C)

2 am 4 am 6 am 8 am 10 am noon 2 pm 4 pm 6 pm 8 pm 10 am midnightTime

Patient’s temperature graph

In the following activity, you must consider the impressions gained from

each type of graph. Also consider whether the graphs convey the

information accurately and whether it is valid to use each type of graph

for the data.


Try these.

2 This table shows the number of marks gained by students in a test.

Mark (x )Number ofstudents (f )

3 1

4 4

5 7

6 9

7 8

8 6

9 3

10 2


The information was entered into a computer spreadsheet package

that was used to generate the following graphs.

2

4

6

8

10

3 4 5 6 7 8 9 10x

f

Number of marks in a test

Column graph (histogram)

2

4

6

8

10

3 4 5 6 7 8 9 10x

f


Line graph

Sector graph

103 4

5

67

8

9

Sector graph


3 4 5 6 7 8 9 10x

3D column graph

3 4 5 6 7 8 9 100

2

4

6

8

10

a Which graph or graphs are invalid for this type of data. Give

your reason?

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


b Which graph do you think conveys the most information quickly

and clearly. Give brief reasons.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


Each graph type has its uses. Sector graphs convey proportions quickly

but do not show detailed information. Line graphs easily show trends but

are misleading when used for discrete data. Column graphs also show

trends, convey more details than sector graphs but are time-consuming to

draw. Dot graphs show individual data well, and convey general trends

but are again time-consuming to draw.

It is important to consider what the graph is to be used for so that you can

select the best type of graph for the purpose.

Demonstrate your understanding of these concepts by completing the

following exercise.

Go to the exercises section and complete Exercise 1.2 – Comparing types

of graphs.



Which graph matches?

The steepness of the graph shows how quickly the quantity is changing,

and whether it is increasing or decreasing. This is called the gradient of

the line or curve.

A positive gradient means the quantity is

increasing as you go from left to right.

A negative gradient means the quantity is

decreasing as you go from left to right.

A steep gradient means the change is fast.

A flatter gradient means the change is slower.

From the graph, can you see that:

• A increases quickly at first, then more

slowly.

• B increases slowly, then faster.

A

B

A horizontal part of a line graph means the

quantity stays the same as you go from left to

right (it remains constant).

A vertical part of a line means that the quantity

instantaneously changes from one value to

another without any gradual change.

Most graphs are curves that show a gradual

change from one growth rate to another.


So the slope seen in each part of a graph tells a story about whether the

quantity is increasing or decreasing, and the rate at which this is

happening.

In this section, you will use gradient and the shape of the graph to match

an event to its graph.

Practise describing the rate at which things are changing by completing

the following activity.

Activity – Which graph matches?

Try this.

1 Below is a graph showing the speed of a car over period of eight

minutes. Each part of the line is labelled.

Time in minutes

Sp

eed

1 2 3 4 5 6 7 8

A

B

C

D

E F

0

a What is part A of the graph telling you about the speed of the

car?

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b Over which part or parts of the journey is the car increasing its

speed?

___________________________________________________


c Which part or parts of the graph shows the car slowing down

and is it slowing down quickly or slowly?

___________________________________________________

d At what time does the car come to a stop? _________________

e Describe what the car is doing in section F.

___________________________________________________

___________________________________________________


Continuing with the concept of speed, the following discussion between

students shows how to use the description of how the speed is changing

to find the graph that matches the description.


Try this.

2 A class was asked the following question.

‘Which graph below shows the speed (s) of a car that is travelling at

50 km/h for 10 minutes then at 100 km/h for 10 minutes?’

t

s

Graph Bt

s

Graph Ct

s

Graph A

Three students gave the following responses.


I think it is graph A because the firstpart of the graph goes up and thenthe second part of the graph goes upeven faster.

This matches the car going at 50 km/hfirst then going faster at 100 km/h.

Amanda

I think the second graph, B, is the right one.

The speed stays at 50 km/h for 10 minutes.That means that at 1 minute you put a dot at50, then at 2 minutes you put a dot up at 50,at 3 minutes you put a dot at 50 and so on.

So when something stays the same the graphshows a horizontal line.

To get from 50 km/h to 100 km/h the car musthave sped up for a while so that accounts forthe slope in the middle.

The last flat part shows the car stayed at100 km/h for 10 minutes.

Ali

I think graph C is the one because it showsthe car going at the same speed for a whileand then the line goes steeper.

A steeper gradient means the thing is changingfaster.

That fits because the car is now going faster.Colin

Who do you think is correct? _______________________________


When reading a graph, you must consider what is changing. For the car

activity above, you are asked to focus on the speed of the car, and this

only changes for a brief time while the car speeds up. The rest of the

time, the speed is constant (not changing).

However, if you were asked to focus on the distance the car has travelled,

then that is changing all the time. In fact, graph A would be the closest

to describing distance and time.


To complete the following activity, remember to focus on what the graph

is supposed to describe, and how that thing is changing.


Try these.

3 Match each graph with one of the following descriptions.

Time

Tem

per

atur

e

Graph ATime

Tem

per

atur

e

Graph BTime

Tem

per

atur

e

Graph C

a A cake cooling after it is removed from an oven. Graph _____

b Heating a kettle till it boils. Graph _____

c Atmospheric temperature on a summer’s day. Graph _____

4 Water is poured at the same rate into each of these three containers.

A

B

Container I Container 2

A

B

Container 3

B

A

a In which container does the water level rise slowly to A then

more quickly up to B? _________________________________


b The water level is graphed over time for each container. Match

each graph below to its container.

Wat

er le

vel

Time

A

B

This graph belongsto container ______.

Wat

er le

vel

Time

A

B



Wat

er le

vel

Time

A

B

5 Which of these graphs best represents the number of people at a bus

stop during morning peak hour. N represents the number of people

and t represents time. (The lines are drawn to show the trend. As

you know, people represent discrete data and therefore should be

represented by a dot or column graph.)

Graph A Graph B Graph C

N

t

N

t

N

t

_______________________________________________________


By considering the rate at which something changes, you can predict the

shape of its graph.

Specific data can be read from a graph only if the scales give you access

to the numbers. But even without scales, the gradients shown in the

graph give a great deal of information about how something is changing

in relation to something else.


Continue to interpret the information displayed in graphs by completing

the following exercise.

Go to the exercises section and complete Exercise 1.3 – Which graph

matches?



Stories from graphs

Graphs tell stories about how things are changing. The more complicated

the story, the more complicated the graph.

In this section, you will be asked to compose a story that matches the

information in a graph.

The first example deals with a familiar situation: water in a bathtub.

Follow through the steps in this example. Do your own working in the

margin if you wish.

The following graph shows the depth of water in a bath over time.

Describe a scenario (a story) that fits the graph.

2 4 86

8

16

24

32

40

Time in minutes

Dep

th o

f wat

er in

cm

Depth of water in a bathtub

1 3 5 7

4

12

20

28

36

44

0


Solution

Each part of the graph is described below.

2 4 86

8

16

24

32

40

Time in minutes

Dep

th o

f wat

er in

cm

Depth of water in a bathtub

1 3 5 7

4

12

20

28

36

44

0

The water level increasesat a constant rate.

The water level drops at aconstant rate.

The water level rises at aconstant rate.

The water level remainsunchanged.

The water level risesinstantaneously.

The water level remainsconstant.

The story below gives one scenario that fits the information from

the graph.

The plug is put in and the taps are fully turned on for one

minute until the water level rises to 16 cm. The plug comes

out somehow and the water goes out the plughole at a faster

rate than the tap is putting water in. The person does not

notice this for two minutes and the water level drops to

10 cm. The plug is put back in, leaving the tap running at the

same rate as before for another one and a half minutes until

the bath fills to a depth of 34 cm. The taps are turned off and

the person takes one minute to get undressed. At five and a

half minutes after the bath was started, they climb in the tub

and the bath goes up to 40 cm deep. They stay in the bath for

the rest of the time.


Other stories could be created to explain this graph. However, each

aspect of the graph must be discussed, with as much accurate data as

possible included from the graph such as times and depths of water.

Go to the following website to explore this type of graph further.

Access a site related to creating line graphs by visiting the CLI website

<http://www.cli.nsw.edu.au/Kto12>.

Select Mathematics then Stage 5.2 and follow the links to resources for

this unit PAS5 Patterns and algebra then select PAS5.2.5 Graphs of

physical phenomena, Part 1. Select the file called Archimedes bath.

Now it’s time to create your own story in the following activity.

Activity – Stories and graphs

Try these.

1 Write a story to describe what may have happened in the situation

graphed below. Remember to include as much data as possible.

2 4 86

8

16

24

32

40

Time in minutes

Dep

th o

f wat

er in

cm

Kim’s bath

1 3 5 7

4

12

18

28

36

44

9 10 11 120

_______________________________________________________

_______________________________________________________

_______________________________________________________


_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


If the graph does not have a detailed scale, then you can only make

general statements about the situation.


Activity – Stories and graphs

Try these.

2 This graph shows the air pressure in Ali’s front bicycle tyre on a trip.

Write a story to match the graph.

Time

Tyre

pre

ssur

e

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


3 The graph below shows the speed of a car over a short journey.

Write a brief story about the journey.

Time

Sp

eed

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

4 (Harder) The following graph represents the level of water in a

container over time. The water is entering the container at a constant

rate. Beside the graph, draw a picture of what the container might

look like.

Time

Wat

er le

vel



You have used a graph to recreate the situation. In the next section you

will use the story to create some graphs.

Complete the following exercise to demonstrate your ability to recreate a

situation from its graph.

Go to the exercises section and complete Exercise 1.4 – Stories from

graphs.



Graphs from stories

You can use a general description of an event or situation to create the

basic shape of a graph. Look at the following example, paying careful

attention to the setting up of the axes.

Follow through the steps in this example. Do your own working in the

margin if you wish.

Georgie jogs for 10 minutes at a constant pace. She stops and

rests for 5 minutes, and then takes 10 minutes to jog home

again at the same pace. Draw a rough sketch of the graph

showing her speed against time.

Solution

What should go on the axes?

You are asked to focus on speed and time.

The speed Georgie is going depends upon the time you

observe her. This means that speed is the dependent variable

(the thing that varies or changes depending on something else).

In this example, time is the independent variable.

It is a standard rule that the independent variable goes along

the horizontal axis, and the dependent variable goes along the

vertical axis.

Time

Sp

eed


What scale do you use?

You cannot draw an accurate graph with a scale on the vertical

axis because you do not know Georgie’s speed. However, you

can mark off the time to scale.

Time

Sp

eed

10 15 25

Completing the graph

Now you need to draw the line to show her speed.

You are told that:

• it is constant for 10 minutes so you must draw a horizontal

line because the speed stays the same value.

Time

Sp

eed

10 15 25


• she stops so her speed plummets to zero and stays there

for 5 minutes.

Time

Sp

eed

10 15 25

• She then jogs for 10 minutes at the same speed as before.

This will be another horizontal line at the same height as

the first part of the graph.

Time

Sp

eed

10 15 25

You cannot draw any more of the graph because you do not

know what she did next. You cannot even assume she

stopped.

The concept of dependent and independent variables is very important. If

time is one of your variables, then it usually goes on the horizontal axis

because it moves along independent of everything else that is happening.

But occasionally time is the dependent variable because it is the

measurement that depends on something else.

For example, the graph below shows the cooking time for a piece of roast

beef. The length of time you cook beef for depends on the mass of the

beef. So time is the dependent variable in this situation.


1 2 4

60

120

Tim

e to

coo

k in

min

utes

Mass of beef in kg

30

90

150

3

180

210

240

0

In the next activity, you are asked to decide what the two variables are, and

what their relationship is to each other.

Activity – Graphs from stories

Try these.

1 For each situation below, decide what two variables will be graphed

and label each with the word ‘dependent’ or ‘independent’.

a Ivan walks at a constant speed covering 2 km in 20 minutes. He

then turns around and takes 12 minutes to jog home. Draw a

graph showing his distance from home at any time.

The variables are _____________________________________

The independent variable is _____________________________

The dependent variable is ______________________________


b The graph below shows average height to mass relationship for

adult women.

Height

Mas

s




c The cost for each widget bought from a manufacturer reduces

when you buy large numbers. If you buy up to 10 widgets, the

cost is 50 c each, if you order 50 widgets they cost 40c each and

100 widgets cost 30 c each.





Once you have decided what the two variables are, and which one is the

dependent variable, you can sketch a graph to tell the story.

When completing the following activity, remember to concentrate on

how your dependent variable is changing.



Try this.

2 A car travels at 60 km/h then quickly accelerates to 100 km/h to

merge into expressway traffic. The car continues at 100 km/h.

Draw a rough sketch to show the speed of the car over time.


In some situations, there are several graphs that can be drawn to describe

different relationships.

For example, when a car is on a journey, you can measure the distance it

has travelled, the distance it is away from home, the speed it is going, its

acceleration, how much petrol in the tank and even the temperature inside

the car. All of these measurements can then be graphed against time in

separate graphs to create a detailed picture of the journey.

The following activity asks you to create two graphs each describing a

different aspect of the situation.



Try these.

3 John runs a bath for himself. He puts in the plug, turns on the taps

and fills the bath to a reasonable depth then turns off the taps. He

puts his foot in and takes it out quickly because the bath is too hot.

He turns on the cold tap to cool the water. He gets in the bath and

relaxes. As he is relaxing the bath gradually cools down. He gets

out then pulls out the plug.

a Draw a rough sketch a graph showing the depth of water over

time.

b Sketch a graph showing the temperature of the water over time.


4 Students enter an empty classroom. Once all are seated, they sit

quietly watching a video. The video finishes and the class is then

dismissed and they leave the room.

Draw a rough graph to show the noise level in the classroom during

the lesson.


A good story creates pictures in your mind. In this unit you have

discovered how those pictures can be in the form of graphs that show

trends and changes. These graphs, in turn, can allow you to analyse what

is happening and get a detailed or general picture of how things are

changing.

Complete the following exercise.

Go to the exercises section and complete Exercise 1.5 – Graphs from

stories.


Suggested answers – Part 1

Check your responses to the preliminary quiz and activities against these

suggested answers. Your answers should be similar. If your answers are

very different or if you do not understand an answer, contact your teacher.

Activity – Preliminary quiz

1 This line has a positive gradient because it goes up from left to right.

2 Decreasing means the line is going down from left to right. Some

examples are shown below.


1 a Since the water is filling the containers at a constant rate, the

level in A goes up by 2 cm each minute, the level in B goes up

by 4 cm each minute and the level in C goes up by 8 cm.

A B C

Time(min)

Level(cm)

Time(min)

Level(cm)

Time(min)

Level(cm)

0 0 0 0 0 0

1 2 1 4 1 8

2 4 2 8 2 18

3 6 3 12 3 24

4 8 4 16 4 32


b Plot the points from each table, then draw the line to join them.

The completed graph is shown below.

0 1 2 3 4

Time (minutes)

Wat

er le

vel (

cm)

5

10

15

20

25

30

35

40

A

B

C

c You are asked to consider why the data should graph to form a

straight line and not some other shape. Your answer should

have mentioned the fact that the water is pouring in at a

constant rate, so the level of the water will also go up at a

constant rate. This means the slope will not change and

therefore the graph is a straight line.

d Cylinder C is the thinnest, so the same amount of water in C will

fill it to a higher level than containers A and B. This means the

level rises at a faster rate in C than in the other two cylinders. A

faster rate is shown by a steeper line.

2 a Amy (S), Brett (R), Carl (T), Dita (P), Elle (U) and Fred (Q).

In the graph, heavier people will be to the right and tall people

will be higher on the graph.

Amy is the shortest so she must be S.

Brett, Carl and Elle are the same height so they must be R, T

and U. Brett is the lightest so he must be R. Carl is the next

lightest so he must be T and therefore Elle is must be U.


Dita and Fred are the tallest. Dita is the lightest of the two so

she is P and therefore Fred is Q.

b

Mass

Hei

ght R

S

T U

P Q

V

The dot for Viviennegoes here.

3 When a car slows down, its speed drops (decreases). The line has a

negative gradient (is decreasing) from one minute to three minutes.

This is when the car is slowing down.

4 a She got worse for a little while because the curve went up in

weeks 2 and 3 meaning the time went up. Now she is improving

because the curve is going down. This means her times are

dropping so she is swimming faster.

b She is improving, but you cannot predict what her time will be

in one year. If you continue the curve for 52 weeks, it would

drop below the horizontal axis. This means time would go

negative and she would be finished the 100 m before she even

started. Obviously this is silly.

Extending a graph beyond the boundaries of the information

given is called extrapolation. It is always dangerous to

extrapolate.

5 a The noise at half-time dropped quite low then stayed at the same

level for a while. This is shown by the flat part of the graph in

the middle.

b There are three sudden peaks so there were probably three goals.

However, it is possible that any or all of these peaks may have

been caused by attempted tries that were not allowed, or some

other exciting event like a fight. You must be careful when

making statements from data.



1 a The column graph because it is the only one that shows the

actual amounts.

b Both graphs shows this easily so either graph can be used.

c The sector graph shows this best because it is the only one that

shows the total amount of money, represented by the full circle.

Savings represents about a quarter of the total.

2 a The data is discrete because you cannot have parts of marks or

parts of students. So you should not use a line graph.

b There are many answer to this. You may have selected any

graph other than the line graph provided you backed up you

selection with a valid reason. Some suggestions are shown

below.

Column graph because it shows the most details at a quick

glance.

Sector graph because is shows fractions of the whole so you can

compare the results quickly.

The dot diagram because it is simple to create and shows the

pattern of the marks clearly.

3D column graph because it is clear, attractive and shows the

pattern clearly.


1 a Part A shows that the speed does not change. So the car goes at

a constant speed for 1 minute.

b Increasing means a positive gradient. Parts B and F.

c Part D shos the car slowing down (speed decreasing) and

because it is steep then the car is slowing down quickly.

d At four and a half minutes (speed = 0)

e The car is speeding up slowly.


2 Ali is correct.

3 a Graph C (temperature is dropping gradually)

b Graph A (temperature is rising then plateaus at boiling point)

c Graph B (day warms up then cools down)

4 a Container 3 because it takes longer to fill the wide section.

b

Wat

er le

vel

Time

A

B


Wat

er le

vel

Time

A

B



Wat

er le

vel

Time

A

B

3 1 2

5 Graph B because is shows the number of people gradually rising

then a sudden drop to zero when the bus takes them away. The next

group of people start arriving and the patterns repeats.

Activity – Stories from graphs

1 You can embellish your story with explanations such as phone calls,

hair washing, etc but the bare facts are listed below. Check that you

have included similar information in your story.

• At time zero the plug is in and the bath fills for 30 seconds to a

depth of 2 cm.

• The bath drains for one minute (perhaps a leak or to clean out

dirt from the bath before bathing).

• At one and a half minutes, the plug goes back in, the bath begins

to fill slowly (perhaps getting the temperature correct) up to a

depth of 2 cm.

• At two and a half minutes the bath starts filling more quickly

(Kim turns the taps on harder) and continues to fill at the same

rate for two and a half minutes to a depth of 28 cm.


• At five minutes the bath level increased instantaneously by 8 cm

up to a level of 36 cm (probably because Kim got in the bath).

• The bath level stays the same for three minutes (the taps are off

and Kim is washing).

• The water level increases to 40 cm in 30 seconds (perhaps Kim

needed to heat the water up).

• The level stays at 40 cm for a further minute (more washing or

just relaxing).

• The water level drops instantaneously by 8 cm (same as increase

at five minutes so perhaps Kim steps out).

• The water level drops steadily for two and a half minutes till the

bath is empty (Kim pulls the plug out).

2 Your reasons behind the facts may be different to the story shown

here but your facts should be the same.

Ali rode for a while then got a puncture in her tyre which went flat

very quickly. She took some time to tape up the puncture. She

pumped up the tyre and rode on but the tyre gradually started to go

down again. She pumped it up again and closed the valve more

tightly this time. The tyre stayed at full pressure after that.

3 Your reasons behind the facts may be different to the story shown

here but your facts should be the same.

The car sped up quickly then drove at a constant speed till it

reached traffic lights. It stopped quickly and waited for a few

minutes. The lights turned green and the car accelerated quickly

back to normal speed. It then accelerated quickly to merge with

expressway traffic then kept going on the expressway at a constant

speed.


4 The water level rises as three different rates shown by the three

different gradients. That means there are three sizes of cylinders in

the container.

Time

Wat

er le

vel


1 a The variables are distance and time.

The independent variable is time.

The dependent variable is distance from home.

(The distance Ivan is away from home depends on the time he

has been walking.)

b The independent variable is height (the horizontal axis).

The dependent variable is mass (the vertical axis).

c The cost depends on the number you buy. So the independent

variable is number of widgets you order and the dependent

variable is the cost of each widget.

2

Time

Sp

eed


3 Your sketches should be similar to the one below. You did not need

to include the comments.

a

Dep

th o

f bat

h

Time

Filling the bath

Puts foot in and out

Puts in cold water

John gets in the bath John in the bath relaxing

John gets out of bath

Water drainsout of bath

b The graph below is acceptable for this course. It uses a straight

line to show the water cooling down. A straight line means the

rate of cooling is steady.

Time

Tem

per

atur

e

Temperature is constantwhile bath fills

Temperature drops quicklywhen cold water is added

Temperature drops slowlyas John relaxes and thebath drains


A more accurate graph uses curves to show the two occasions

when the water is cooling down.

Time

Tem

per

atur

e

Temperature is constantwhile bath fills

Temperature drops quicklywhen cold water is added

Temperature drops slowlyas John relaxes and thebath drains

4 Your graph should show similar features as the one below.

Time

Noi

se le

vel

Students enter the room,move chairs, chat etc.

Class quietens preparing for video.

The noise level fromthe video varies slightly.

Students stand andprepare to leave.

Room quietens quicklyas students leave.



Exercises – Part 1

Exercises 1.1 to 1.5 Name ___________________________

Teacher ___________________________

Exercise 1.1 – Reading graphs

1 Sian and Tran made model aeroplanes. Sian’s is both lighter and

faster than Tran’s.

S shows Sian’s plane’s position on a graph. Plot a point T that could

be Tran’s plane.

Max

imum

sp

eed

Mass

S

2 Below is a graph showing the speed of a car during a five minute

journey.

Time in minutes

Sp

eed

1 2 3 4 50


When was the car standing still? Explain your answer.

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

3 The graph shows the number of hours of sleep for a group of

children of different ages on a certain day.

4 8 16

8

16

Time in minutes

Dep

th o

f wat

er in

cm

4

12

20

120

a Describe the trend shown in this graph and what it implies.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


b One person was sick and spent more time than usual asleep.

Circle the dot that represents this person and explain your

choice.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

c Can you use this graph to predict how much sleep a 40 year old

will need? Explain.

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________


Exercise 1.2 – Comparing types of graphs

Sydney Water has worked out figures for how much water an average

household (three people + garden) uses each week.

Below are three different ways to show this information.

Average household water use

bathbasin

kitchen

laundry

shower toilet

outdoors

sector graph

0 5 10 15 20 25 30

Percentage household water use

bath

basin

kitchen

laundry

shower

toilet

outdoors

3D bar graph


0

5

10

15

20

25

30

Percentage household water use

bat

h

bas

in

kitc

hen

laun

dry

show

er

toile

t

outd

oors

column graph

Which graph do you prefer and why?

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

Which graph is least useful and why?

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________


Exercise 1.3 – Which graph matches?

1 Choose the graph that best represents the following:

a the speed (s) of a car travelling at a constant speed as it goes up

a hill.

Graph A Graph B Graph C

s

t

s

t

s

t

___________________________________________________

b Water is poured into both containers at the same constant rate

into both.

Container BContainer A

Which graph below correctly shows the water level in each

container at a given time?

Graph 1

Time

Leve

l of w

ater A

B

Graph 2

Time

Leve

l of w

ater B

A

Graph 3

Time

Leve

l of w

ater A

B

___________________________________________________


2 The tank shown has a tap at its base. The tap is turned on and the

water drains out at a constant rate.

Which graph below best describes the volume of water in the tank at

any given time?

Graph 1Time

Volu

me

of w

ater

Graph 2Time

Volu

me

of w

ater

Graph 3Time

Volu

me

of w

ater

_______________________________________________________


Exercise 1.4 – Stories from graphs

1 The graph below shows the petrol in the tank of a car during a

journey. Write a brief story describing the trip.

Distance travelled (in km)

Am

ount

of p

etro

l in

tank

(in

litre

s)

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


2 The graph shows the pulse rate of an athlete, Wendy, as she prepares

for and participates in an 800 m race. Write a story from the graph.

Time

Pul

se r

ate

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________


Exercise 1.5 – Graphs from stories

1 Mary is ill. Her temperature rises steadily for two hours. She takes

some medication and 15 minutes later her temperature falls quickly

back to normal. However, three hours later her temperature rises

rapidly again. Draw a rough sketch to show Mary’s temperature

over time.

2 Water is poured into this container at a constant rate.

a Draw a rough sketch showing the water level over time.


b (Harder) Draw a rough sketch showing the volume of water in

the container over time.

Documents

Part 1 Interpreting graphsmaths.gerrydear.id.au/year09_10/Data/Other... · Part 1 Distance/time graphs 11 Check your response by going to the suggested answers section. Interpreting