EMAC Comparison

ESSENTIAL MATHEMATICS

FOR THE AUSTRALIAN CURRICULUM FOURTH EDITION

Chapter 8 Statistics and probability

8C Column graphs and dot plots LEARNING INTENTIONS • To understand that graphs should include titles and labelled axes, with any numerical values drawn to scale • To know what an outlier is • To be able to construct and interpret a column graph • To be able to construct and interpret a dot plot

Discrete numerical data can be counted and presented as a dot plot, with the number of dots representing the frequency.

Favourite colour red Colour

Categorical data can be counted and presented as a column graph. Each column’s length indicates the frequency of that category. Column graphs can also be useful for labelled continuous data (e.g. height of people).

green yellow blue pink 0

Consider a survey of students who are asked to choose their favourite colour from five possibilities, as well as state how many people live in their household. The colours could be shown as a column graph, and the number of people shown in a dot plot.

5 10 15 20 Number of students As a column graph (horizontal)

Favourite colour 20 Number of students

15 10 5

0

1

2 3 4 Household size

5

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Colour As a column graph (vertical)

Lesson starter: Favourite colours Survey the class to determine the number of people in each student’s family and each student’s favourite colour from the possibilities red, green, yellow, blue and pink.

• •

Each student should draw a column graph and a dot plot to represent the results.

What are some different ways that the results could be presented into a column graph? (There are more than 200 ways.)

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CORE 470

FOR THE AUSTRALIAN CURRICULUM SECOND EDITION

Chapter 8 Statistics and probability

8C 8C Dot plots and column graphs Learning intentions • • • •

To understand that graphs should include titles and labelled axes which should be drawn to scale if they represent numeric data.

To know what an outlier is.

To be able to interpret a column graph and dot plot.

To be able to construct a column graph and dot plot.

Key vocabulary: dot plot, datum, column graph, axis, outlier

Graphs are a good way to illustrate data and highlight special information. For example, if students were surveyed on their favourite colours, the results could be shown as a column graph. The ‘columns’ can be vertical (going up) or horizontal (going across).

Favourite colour

Favourite colour

Key vocabulary identified at the beginning of each lesson.

red

15

green

Colour

Number of students

10

yellow blue

5

pink

nk pi

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5 10 Number of students

15

ye

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ESSENTIAL MATHEMATICS

Colour

The data could also be shown in a dot plot (below).

Refined language supports students with lower literacy levels.

Lesson starter activities may be adapted from the mainstream Essential Mathematics series, with questions modified to better meet the specific needs of students. red

green yellow blue Favourite colour

pink

Lesson starter: Favourite colours Survey the class to determine each student’s favourite colour from the possibilities red, green, yellow, blue and pink.

• Each student should draw a column graph or a dot plot to represent the results.

• Compare your graph with those of other students. Describe any differences you notice.

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Essential Mathematics CORE

ESSENTIAL MATHEMATICS

FOR THE AUSTRALIAN CURRICULUM FOURTH EDITION 8C Column graphs and dot plots

KEY IDEAS ■ A dot plot can be used to display discrete numerical data, where each dot represents one datum.

■ A column graph is a way to show data in different categories, and is useful when more than a

few items of data are present.

■ Column graphs can be drawn vertically or horizontally. Horizontal column graphs are

sometimes also called bar graphs.

■ Graphs should have the following features:

A title explaining what the graph is about

Favourite colour

20 15 10

A label on each axis

Colour

nk pi

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w

5

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Number of students

An even scale for the numerical axis

Category labels for any non-numerical data

■ Any numerical axis must be drawn to scale.

■ An outlier is a value that is noticeably distinct from the main cluster of points.

main cluster an outlier

0 1 2 3 4 5 6 7 8 9 10 ■ Data represented as a dot plot could be described as symmetrical or skewed (or neither).

Negatively skewed Symmetrical

Positively skewed

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

mean = median = mode

mode < median < mean

mean < median < mode

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527

ESSENTIAL MATHEMATICS

CORE

FOR THE AUSTRALIAN CURRICULUM SECOND EDITION

8C Dot plots and column graphs

471

Key ideas A dot plot is a graph that can be used to display data, where each dot represents one datum.

A column graph is a good way to show data in different categories, and is useful when more than a few items of data are present. Column graphs can be drawn vertically (going up) or horizontally (going across).

Graphs should have the following features: A title explaining what the graph is about

An even scale for the numerical axis

Number of students

Favourite colour

15 10 5

Colour

nk pi

ue bl

w

ee

llo ye

gr

A label on each axis

re

d

n

0 Category labels for any non-numerical data

An outlier is a value that is noticeably smaller or larger than the main cluster of points. main cluster

Fewer Key ideas are covered, to focus on helping students meet the minimum requirements of the Achievement Standards.

outlier

0 1 2 3 4 5 6 7 8 9 10

Exercise 8C Understanding

1–3

2, 3

1 Fill in the blanks in the following sentences. is a graph which uses dots to represent data. a A . b A graph showing data in different categories as rectangles is called a c An is a value that is noticeably smaller or larger than the main cluster of points.

Understanding questions are closely linked to the Key ideas section to promote class discussion and to help consolidate the fundamental skills and concepts needed for the topic.

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Essential Mathematics CORE

471

ESSENTIAL MATHEMATICS

FOR THE AUSTRALIAN CURRICULUM FOURTH EDITION

530

Chapter 8 Statistics and probability

SOLUTION

EXPLANATION First decide which scale goes on the vertical axis.

Height (cm)

Height chart 160 140 120 100 80 60 40 20 0

Maximum height = 150 cm, so axis goes from 0 cm to 160 cm (to allow a bit above the highest value).

Remember to include all the features required, including axes labels and a graph title.

Tim Phil Jess Don Nyree Name

Now you try Draw a column graph to represent the following people’s arm spans.

Name

Matt

Pat

Carly

Kim

Tristan

Arm span (cm)

180

190

150

160

170

Exercise 8C FLUENCY Example 5

Example 5

1

1–6

The dot plot on the right represents the results of a survey that asked some people how many times they had flown overseas. a Use the graph to state how many people have flown overseas once. b How many people participated in the survey? c What is the range of values? d What is the median number of times flown overseas? e What is the outlier? f What is the mode?

2 In a Year 4 class, the results of a spelling quiz are presented as a dot plot. a What is the most common score in the class? b How many students participated in the quiz? c What is the range of scores achieved? d What is the median score? e Identify the outlier.

2–7

2, 3, 5–7

Overseas flights

0 1 2 3 4 5 6 7 Number of times o/s Spelling quiz results

0 1 2 3 4 5 6 7 8 9 10 Score out of 10

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CORE 472

8C

FOR THE AUSTRALIAN CURRICULUM SECOND EDITION

Chapter 8 Statistics and probability

2 This column graph shows the height of four boys.

Answer true or false to each of the following statements. a Mick is 80 cm tall. b Vince is taller than Tranh. c Peter is the shortest of the four boys. d Tranh is 100 cm tall. e Mick is the tallest of the four boys.

Height (cm)

72

ESSENTIAL MATHEMATICS

110 100 90 80 70 60 50 40 30 20 10 0

Height graph

Mick Tranh Vince Peter Name

Two separate working programs (rather than three) are subtly embedded in every exercise.

3 Which dot plot describes the data 1, 1, 1, 2, 2, 3, 5, 5?

A B

0 1 2 3 4 5

0 1 2 3 4 5

Fluency

4–8

4, 6–9

Example 5 Interpreting a dot plot This dot plot represents the results of a survey that asked some children how many pets they have at home. a Use the graph to state how many children have 2 pets. b How many children participated in the survey? c What is the range of values? d What is the median number of pets? e What is the outlier? f What is the mode?

Pets at home survey

0 1 2 3 4 5 6 7 8 Number of pets

Solution

Explanation

a 4 children

There are 4 dots in the ‘2 pets’ category, so 4 children have 2 pets.

b 15 children

The total number of dots is 15.

c 8−0=8

Range = highest − lowest In this case, highest = 8, lowest = 0.

Worked examples are carefully selected to suit the needs of this cohort and often contain fewer new skills in each example. They appear in the exercises directly above the related questions.

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Essential Mathematics CORE

ESSENTIAL MATHEMATICS

FOR THE AUSTRALIAN CURRICULUM FOURTH EDITION 8C Column graphs and dot plots

SOLUTION

EXPLANATION

a 4 children

There are 4 dots in the ‘2 pets’ category, so 4 children have 2 pets.

b 22 children

The total number of dots is 22.

c 8−0=8

Range = highest − lowest In this case, highest = 8, lowest = 0.

d 1 pet

As there are 22 children, the median is the average of the 11th and 12th value. In this case, the 11th and 12th values are both 1.

e the child with 8 pets

The main cluster of children has between 0 and 3 pets, but the person with 8 pets is significantly outside this cluster.

f

The most common number of pets is 1.

1 pet

Now you try The dot plot shows the number of visits a sample of pet owners have made to the vet in the past year. a How many people were surveyed? b How many people made more than three visits to the vet in the past year? c What is the range? d What is the median? e What is the outlier? f What is the mode?

Example 6

Visiting the vet

0 1 2 3 4 5 6 7 Number of visits

Constructing a column graph

Draw a column graph to represent the following people’s heights.

Name

Tim

Phil

Jess

Don

Nyree

Height (cm)

150

120

140

100

130

Continued on next page

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529

ESSENTIAL MATHEMATICS

CORE

FOR THE AUSTRALIAN CURRICULUM SECOND EDITION

8C Dot plots and column graphs

d 1 pet

473

Write the values in order: 0, 0, 0, 1, 1, 1, 1, 1 , 2, 2, 2, 2, 3, 3, 8 Middle value = median = 1

e 8 pets

The main cluster of values is from 0 pets to 3 pets. The dot showing 8 pets is significantly outside this cluster.

f 1 pet

The most common number of pets is 1 pet.

Now you try

This dot plot represents the number of fish inside a number of fish tanks in a shop. a How many tanks have 3 fish? b How many tanks in the shop were surveyed? c What is the range of values? d What is the median number of fish? e What is the outlier? f What is the mode.

Number of fish inside fish tanks

0 1 2 3 4 5 6 7 8 9 Number of fish

4 The favourite after-school activity of a number of Year 7 students is recorded in this dot plot. a How many students have chosen television as their favourite activity?

Favourite after-school activities of Year 7 students

b How many students have chosen surfing the internet as their favourite activity?

video games sport television internet

c What is the most popular after-school activity for this group of students? d How many students participated in the survey?

b How many students chose blue? c What is the most popular colour? d How many students participated in the class survey?

Favourite colours in 7B pink

Favourite colour

5 From a choice of pink, blue, yellow, green or red, each student of Year 7B chose their favourite colour.

The results are graphed here. a How many students chose yellow?

blue yellow green red

Now You Try questions in each section begin with accessible problems to support early success and build student confidence.

0 1 2 3 4 5 6 7 8 9 10 Number of students

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Essential Mathematics CORE

47

c In which year(s) are more people walking or cycling to work than driving? d Give a reason why the number of people driving to work has decreased. e What is one other trend that you can see from looking at this graph?

ESSENTIAL MATHEMATICS

7 a Draw a column graph to show the results of the following survey of the number of male and female puppies sold by a commercial dog breeder. Put time (years) on the horizontal axis.

b c d e

2010

2011

2012

2013

2014

2015

Number of male puppies born

40

42

58

45

30

42

Number of female puppies born

50

40

53

41

26

35

FOR THE AUSTRALIAN CURRICULUM FOURTH EDITION

During which year(s) were there more female puppies sold than male puppies?

Which year had the fewest number of puppies sold?

8C Column graphs and dot plots Which year had the greatest number of puppies sold?

During the entire period of the survey, were there more male or female puppies sold?

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Income (× $10 000)

9 A survey is conducted of students’ favourite subjects 8 8, 9 9, 10 30 PROBLEM-SOLVING from a choice of Art, Maths, English, History and Someone attempted to depict the results 8 Science.

The average (mean)has income of adults in a particular Average income in a town 7 in a column graph. town is graphed over a 6-year period.

6 25 aa What is wrong the scale thehappened vertical axis?

Describe in onewith sentence whatonhas 5 20 b Give least two other to theatincome over this problems period of with time.this graph.

4 5 cb Redraw graph an even scale Estimatethe what the with average income in and this appropriate town 3 0 labels. might have been in 2012.

2 dc The original graph makes Maths look twice Estimate what the average income might be as in popular 2028 as 1 Art, based on the column size. if this trend continues.

0 2013 2014 2015 2016 2017 2018 According to the survey, how many times more popular is Maths?

Year to the survey, e The original graph makes English look three times as popular as Maths. According how much more popular is English?

Assume that Music is now added to the survey’s choice of subjects. Five students who had previously chosen History now choose Music, and 16 students who had previously chosen English now choose Music. What is the most popular subject now?

10 Mr Martin and Mrs Stevensson are the two Year 3 teachers at a school. For the latest arithmetic quiz, they have plotted their students’ scores on a special dot plot called a parallel dot plot, shown opposite. a What is the median score for class 3M? b What is the median score for class 3S? c State the range of scores for each class. d Based on this test, which class has a greater spread of arithmetic abilities? e If the two classes competed in an arithmetic competition, where each class is allowed only one representative, which class is more likely to win? Justify your answer.

Arithmetic quiz scores Class 3M

0 1 2 3 4 5 6 7 8 9 10

Score

Class 3S REASONING

11

11, 12

12, 13

No. of passengers

11 At a central city train station, three types of services Train passengers at Urbanville Station 2500 run – local, country and interstate. The average 2000 number of passenger departures during each week is 1500 shown in the stacked column graph.

1000 a Approximately how many passenger departures 500 per week were there in 2013?

0 2013 2014 2015 2016 2017 2018 b Approximately how many passenger departures Year were there in total during 2018? interstate country local c Does this graph suggest that the total number of passenger departures has increased or decreased during the period 2013–2018? d Approximately how many passengers departed from this station in the period 2013–2018?

Explain your method clearly and try to get your answer within 10 000 of the actual number.

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533

CORE 476

8C

FOR THE AUSTRALIAN CURRICULUM SECOND EDITION

Chapter 8 Statistics and probability

Problem-solving and reasoning

10, 11

10 The results of a Year 4 spelling quiz are shown as a dot plot. a How many students got a score of 6? b What is the most common score in the class? c How many students participated in the quiz? d What is the range of scores achieved? e What is the median score? f Identify the outlier.

11–13

Q10 Hint: See Example 5 if you need help.

Year 4 students' spelling quiz results

Hints within the exercises point students to where they can find help.

The modified grading of problems allows lower and middle ability students to master the basics of the Australian Curriculum.

0 1 2 3 4 5 6 7 8 9 10 Score out of 10

11 Every five years, a company in the city conducts a transport survey of the way people get to work in the mornings. The results are graphed below.

Number of employees

6

ESSENTIAL MATHEMATICS

70 60 50 40 30 20 10 0

Types of transport used to get to work

public transport car walk/bicycle

2000

2005

2010 2015 Year of survey

a Copy and complete this table to show the data in the graph. b In which year(s) was public transport the most popular option?

2020

Use public transport Drive a car Walk or cycle

2000 30 60 10

2005

2010

2015

2020

c In which year(s) were more people walking or cycling to work than driving? d Suggest one reason why the number of people driving to work has decreased. e What is one other trend that you can see from looking at this graph?

Annotated sample pages • © Cambridge University Press & Assessment

Essential Mathematics CORE

ESSENTIAL MATHEMATICS

FOR THE AUSTRALIAN CURRICULUM FOURTH EDITION

Chapter 8 Statistics and probability

12 The mean value of data represented in a dot plot can be found by adding the value represented by each dot, then dividing by the number of dots. a Use this method to find the mean number of children per household, shown in this dot plot of 10 households. b A faster method is to calculate (2 × 1) + (3 × 2) + (4 × 3) + (1 × 4) and divide by 10. Explain why this also works to find the mean.

0 1 2 3 4 5 Number of children

13 Classify the following dot plots as representing symmetrical, positively skewed or negatively skewed data. Find the mean number in each case and compare it to the median. a b

0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 c

d

0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 ENRICHMENT: How many ways?

–

–

14

Animal sightings

Animal sightings

ha ep

0

2 4 6 Number of sightings

Annotated sample pages • © Cambridge University Press & Assessment

ar

lo te

Animal

be

o ng

an

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nt ep el

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Animal sightings

8 6 4 2 0 ha

How many different column graphs could be used to represent the results of this survey? (Assume that you can only change the order of the columns, and the horizontal or vertical layout.) Try to list the options systematically to help with your count.

Number of sightings

Animal

di

nt

o

dingo cougar antelope bear elephant

el

ng di

ar ug co

te

be ar

pe

Animal

8 6 4 2 0 lo

Number of sightings

14 As well as being able to draw a graph horizontally or vertically, the order of the categories can be changed. For instance, the following three graphs all represent the same data.

an

534

8

ESSENTIAL MATHEMATICS

CORE

FOR THE AUSTRALIAN CURRICULUM SECOND EDITION

8C Dot plots and column graphs

477

12 a Draw a column graph to show the results of the following survey of the number of males and females born at a certain hospital. Put the years on the horizontal axis.

Number of males born Number of females born

b c d e

2012 40 50

2013 42 40

2014 58 53

2015 45 41

2016 30 26

2017 42 35

During which year(s) were more females born than males?

Which year had the smallest number of births?

Which year had the greatest number of births?

During the time of the survey, were more males or females born?

13 Mr Martin and Mrs Stevensson are the two Year 3 teachers at a school. For the latest arithmetic test, they plotted their students’ scores on dot plots.

Class 3M

0 1 2 3 4 5 6 7 8 9 10 Score

Class 3S

0 1 2 3 4 5 6 7 8 9 10 Score

a b c d

What is the median score for class 3M? What is the median score for class 3S? State the range of scores for each class.

Based on this test, which class has a greater spread of arithmetic abilities? e If the two classes competed in an arithmetic competition where each class is allowed only one representative, which class is more likely to win? Why?

Misleading graphs

30

Annotated sample pages • © Cambridge University Press & Assessment

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25 20 5 0

M

Gold star sections provide an extra challenge for students at the higher end of this cohort, while remaining within the scope of the curriculum.

14

Q14 Hint: Check the Key ideas to see what features graphs should show.

A

14 A survey is conducted of students’ favourite subjects. Someone has tried to show the results in a column graph. a What is wrong with the scale on the vertical axis? b Give at least two other problems with this graph. c Redraw the graph with an even scale and appropriate labels. d The original graph makes Maths look twice as popular as Art, based on the column size. According to the survey, how many times more popular is Maths? e The original graph makes English look three times more popular than Maths. From the survey, how many times more popular is English? f Look in a newspaper or magazine for a graph with an uneven scale that makes the graph misleading.

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Essential Mathematics CORE

47