• 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
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).
Discrete numerical data can be counted and presented as a dot plot, with the number of dots representing the frequency.
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.
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.)
Key vocabulary identified at the beginning of each lesson.
Thedatacouldalsobeshowninadotplot(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.
■ 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
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.
■ Data represented as a dot plot could be described as symmetrical or skewed (or neither).
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8C Column graphs and dot plots
ESSENTIAL MATHEMATICS CORE
FOR THE VICTORIAN CURRICULUM SECOND EDITION
Keyideas
A dotplot isagraphthatcanbeusedtodisplaydata,whereeachdotrepresentsone datum
A columngraph isagoodwaytoshowdataindifferentcategories,andisusefulwhenmorethan afewitemsofdataarepresent.Columngraphscanbedrawnvertically(goingup)orhorizontally (goingacross).
Graphsshouldhavethefollowingfeatures:
A title explaining what the graph is about
labels for any non-numerical data
An outlier isavaluethatisnoticeablysmallerorlargerthanthemainclusterofpoints.
Fewer Key ideas are covered, to focus on helping students meet the minimum requirements of the Achievement Standards.
1 Fillintheblanksinthefollowingsentences.
a A isagraphwhichusesdotstorepresentdata.
b Agraphshowingdataindifferentcategoriesasrectanglesiscalleda
c An isavaluethatisnoticeablysmallerorlargerthanthemainclusterofpoints.
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.
ESSENTIAL MATHEMATICS
FOR THE VICTORIAN CURRICULUM THIRD EDITION
First decide which scale goes on the vertical axis.
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.
Now you try
Draw a column graph to represent the following people’s arm spans.
Name MattPatCarlyKimTristan
Exercise 8C
1 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.
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.
Explanation
Thereare 4 dotsinthe‘2 pets’category, so 4 childrenhave 2 pets.
Thetotalnumberofdotsis 15
Range = highest lowest Inthiscase,highest = 8,lowest = 0
ESSENTIAL MATHEMATICS
FOR THE VICTORIAN CURRICULUM THIRD EDITION
SOLUTION
a 4 children
b 22 children
c 8 0 = 8
d 1 pet
e the child with 8 pets
f 1 pet
Now you try
EXPLANATION
There are 4 dots in the ‘2 pets’ category, so 4 children have 2 pets.
The total number of dots is 22
Range = highest lowest In this case, highest = 8, lowest = 0
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.
The main cluster of children has between 0 and 3 pets, but the person with 8 pets is significantly outside this cluster.
The most common number of pets is 1
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
Constructing a column graph
Draw a column graph to represent the following people’s heights.
Howmanystudentschoseblue? b Whatisthemostpopularcolour? c
Howmanystudentsparticipatedintheclasssurvey? d
Now You Try questions in each section begin with accessible problems to support early success and build student confidence.
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.
FOR THE VICTORIAN CURRICULUM THIRD EDITION
b During which year(s) were there more female puppies sold than male puppies?
c Which year had the fewest number of puppies sold?
d Which year had the greatest number of puppies sold?
e During the entire period of the survey, were there more male or female puppies sold?
PROBLEM-SOLVING
9 A survey is conducted of students’ favourite subjects from a choice of Art, Maths, English, History and Science. Someone has attempted to depict the results in a column graph.
8 The average (mean) income of adults in a particular town is graphed over a 6-year period.
a What is wrong with the scale on the vertical axis?
b Give at least two other problems with this graph.
a Describe in one sentence what has happened to the income over this period of time.
b Estimate what the average income in this town might have been in 2012
c Redraw the graph with an even scale and appropriate labels.
c Estimate what the average income might be in 2028 if this trend continues.
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 as popular as Maths. According to the survey, how much more popular is English?
f 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. REASONING
11 At a central city train station, three types of services run – local, country and interstate. The average number of passenger departures during each week is shown in the stacked column graph.
a Approximately how many passenger departures per week were there in 2013?
b Approximately how many passenger departures were there in total during 2018?
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.
a Copyandcompletethistableto showthedatainthegraph.
b Inwhichyear(s)waspublictransport themostpopularoption?
c Inwhichyear(s)weremorepeoplewalkingor cyclingtoworkthandriving?
d Suggestonereasonwhythenumberofpeople drivingtoworkhasdecreased.
e Whatisoneothertrendthatyoucanseefrom lookingatthisgraph?
ESSENTIAL MATHEMATICS
FOR THE VICTORIAN CURRICULUM THIRD EDITION
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.
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.
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.
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.
ESSENTIAL MATHEMATICS CORE
FOR THE VICTORIAN CURRICULUM
12 a Drawacolumngraphtoshowtheresultsofthefollowingsurveyofthenumberofmalesand femalesbornatacertainhospital.Puttheyearsonthehorizontalaxis.
b Duringwhichyear(s)weremorefemalesbornthanmales?
c Whichyearhadthesmallestnumberofbirths?
d Whichyearhadthegreatestnumberofbirths?
e Duringthetimeofthesurvey,weremoremalesorfemalesborn?
