Ace Maths Solutions Unit Five Reading: Exercises on Teaching Data Handling (word)

UNIT SOLUTIONS

Solutions Unit Five Reading: Exercises
on Teaching Data Handling
From the module:
Teaching and Learning Mathematics in Diverse Classrooms

South African Institute for Distance Education

Copyright
© South African Institute for Distance Education (SAIDE), 2009

The work is licensed under the Creative Commons Attribution-Non-commercial-Share Alike 2.5
License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/2.5/

Acknowledgements
The South African Institute for Distance Education wishes to thank those below:

For writing and editing the solutions guide to the module:

 Celeste Bortz – writer

 Ingrid Sapire - writer

 Commonwealth of Learning (COL) – for the OER instructional design template

 Tessa Welch – project manager

For permission to adapt the following study guide for the module :

UNISA (2006). Learning and teaching of Intermediate and Senior Mathematics (ACE ME1-C).
Pretoria: UNISA

For permission to use in Unit Five

 UNISA (2006). Study Units 7 to 10: Learning and Teaching of Intermediate and Senior Phase
Mathematics.

 MM French (1979). Tutorials for Teachers in Training Book 7

 RADMASTE Centre, University of the Witwatersrand (2005). Data Handling and Probability
(EDUC 187) Chapters 3, 8 and 9.

South African Institute for Distance Education

Box 31822
Braamfontein
2017
South Africa

Fax: +27 11 4032814
E-mail: info@saide.org.za
Website: www.saide.org.za

Solutions Unit Five Reading: Exercises on Teaching Data Handling

Contents
How the solutions for the module are presented...................................................................................1
Overview of content..........................................................................................................1
How the solutions unit is structured..................................................................................1
How to find alternative content material...........................................................................1

Unit Five Reading Solutions: Exercises on Teaching Data Handling..................................................3
Exercise 1: Collecting data................................................................................................3
Exercise 2: Representing data............................................................................................6
Exercise 3: Interpreting data............................................................................................10
Exercise 4a: Representing and interpreting data.............................................................11
Exercise 4b: Representing data – selecting the best representation................................15
Exercise 5: Interpreting data – impulse buyers...............................................................20
Unit mathematical content summary...............................................................................22


How the solutions for the module are presented

Overview of content
As an introduction to each activity notes about the teaching and learning
of the mathematical content in the activity are given. These notes are
intended to inform both lecturers using the materials in a teacher
education context and teachers who may wish to use the materials in their
own classes. The solutions to the activities are all given in full. Diagrams
are given to provide visual explanations where necessary.

How the solutions unit is
structured
The unit consists of the following:

 General points for discussion relating to the teaching of the mathematical
content in the activities.

 Step-by-step mathematical solutions to the activities.

 Annotations to the solutions to assist teachers in their understanding
the maths as well teaching issues relating to the mathematical content
represented in the activities.

 Suggestions of links to alternative activities for the teaching of the
mathematical content represented in the activities.

How to find alternative content
material
The internet gives you access to a large body of mathematical activities,
many of which are available for free downloading if you would like to
use them in your classroom. There are different ways of searching the
web for material, but a very easy way to do this is to use Google. Type in
the address http://www.google.co.za/ to get to the Google search page.
You can then search for documents by typing in the topic you are
thinking of in the space provided. You will be given many titles of
articles (and so on) which may be appropriate for you to use. You need to
open them in order to check which one actually suits your needs. To open
an article you click on the title (on the screen) and you will be taken to

How the solutions for the module are presented Error! No text
of specified style in document.

the correct web address for that article. You can then check the content
and see whether or not it suits your needs. When you do this search you
will see that there are many sites which offer worksheets open for use by
anyone who would like to use them. You need to check carefully that the
material is on the right level for your class and that there are no errors in
the text. Anyone can miss a typographical error and web material may
not be perfect, but you can easily correct small errors that you find on a
worksheet that you download.

You can also use a Google image search, to find images relating to the
topic you are thinking of. This usually saves you a lot of time, because
you will quickly see which images actually relate to what you are
thinking of and which do not. When you click on the image you like (on
the screen), this will take you to the full page in which this image is
actually found. In this way you can get to the worksheet of your choice.
You can then copy and download the material.

2


Unit Five Reading Solutions: Exercises on
Teaching Data Handling

Exercise 1: Collecting data
Collecting data for a project, for research or for Statistics S.A. can be
quite a challenge. We want our learners to learn how big our sample size
should be and what biases can occur which we need to factor in.

At school level, the main requirement is that we think

o before we choose what the sample should be

o and how big it should be.

The practical aspects of how to collect the data and the need to be as
accurate as possible are all factors which need to be taken into account .
These are ideas that learners will begin to learn about in school level data
handling exercises.

3

4 Error! No text of specified style in document.

Exercise 1

1 A survey was conducted to find the ten most spoken languages in the
world and the number of people speaking them. The results were
written out as follows:
Chinese: 700 million German: 119 million
English: 400 million Spanish: 240 million
Russian: 265 million Japanese: 116 million
Bengali: 144 million Arabic: 146 million
Hindustani: 230 million
Organise the information into an ordered list in two different ways.

2 In an experiment I toss a dice 50 times and keep a record of the
number that appears each time. The numbers are shown below:
2;4;3;3;1;5;6;3;2;2
2;2;6;1;5;5;3;3;4;2
2;3;4;3;6;5;1;1;2;1
3;5;6;3;1;2;2;5;5;1
6;2;2;4;1;6;2;3;3;5

Complete the tally table and then answer the questions.

Number

Notice:
Tally
The tally total is the same
as the frequency total
Frequen
The data can go across or
cy down the page.
The ‘items’ come first in
the table.
1 Do not confuse the
frequency with the number
on the dice

2

3

4


 How many threes were tossed?
 What number was tossed the most times?
 Why do you think more sixes were not tossed?
 How many more times was a two tossed compared to a five?
 Do you think this dice is a fair dice? What does fair mean in this
question?

Solutions to Exercise 1: Collecting data

1. Two ways of organizing the information into an ordered list:

ALPHABETICAL
Arabic: 146 million
Bengali: 144 million
Chinese: 700 million
English: 400 million
German: 119 million
Japanese: 116 million
Russian: 265 million
Spanish: 240 million
IN DESCENDING ORDER OF NUMBERS OF PEOPLE
Chinese: 700 million
English: 400 million
Russian: 265 million
Spanish: 240 million
Arabic: 146 million
Bengali: 144 million
German: 119 million
Japanese: 116 million
2.

Number Tally Frequency

1 |||| ||| 8

2 |||| |||| ||| 13

3 |||| |||| | 11

4 |||| 4

5 |||| ||| 8

6 |||| | 6

Total 50 50

5


 Eleven threes were tossed.
 Two was tossed the most times.
 More sixes were not tossed because the numbers appear randomly.
 A two was tossed five times more than a five.
 This dice seems to be fair. Fair means a relatively even spread of
the frequency of the numbers.

3. The survey of the learners' birthdays in your class will be an
individual solution.
Suggested links for other alternative activities:

• http://www.teachervision.fen.com/pro-dev/teaching-
methods/48944.html (information on collecting data)

• http://score.kings.k12.ca.us/standards/fourth.html#statistics (some
ideas that could be used in lesson planning)

Exercise 2: Representing data
The representation of data is one of the most important aspects of
data-handling. A well-organised representation of data can make any
task involving data much simpler. It is far easier to analyse the data
trends if the data is represented effectively by first tallying and then
using some sort of graphical representation such as bar charts,
pictograms, line graphs or pie charts.

Your learners need to find out about all of the different forms of
graphical representation of data. So you need to be sure to given the
activities in which they are called on to use each of the different
forms, once they know how to use them.

6


Exercise 2
1 The table below shows the estimated percentage HIV prevalence
per province in South Africa in 1998.

Province
%

Eastern Cape
16

Free State
22 This means that
16% of the
population of the
Eastern Cape is
Gauteng estimated to be
22 HIV positive.

KwaZulu-Natal
33

Mpumalanga
30

Northern Cape
10

Limpopo
12

North West
21

Western Cape
5

South Africa
22



7


Solutions to Exercise 2: Representing data

1. A PICTOGRAM

Province Estimated Percentage of HIV prevalence
in SA (1998)

Eastern Cape

Free State

Gauteng

KwaZulu-Natal

Mpumalanga

Northern Cape

Limpopo

North West

Western Cape

South Africa

Key: = 5%; = 1%

A VERTICAL BAR CHART

Estimated Percentage of HIV prevalence in SA
(1998)

35
30
25
Percentage

20
15
10
5
0
e
ga
l

po
ng

a
e
pe

pe
ta

t
es
ap
at

ric
Na

an
Ca

te

po

Ca
St

W
C

Af
au

al
u-

m
n
ee
n

rth

rn

h
m
G

ul

Li
er
er

ut
te
pu
Fr

No
aZ

rth
st

So
es
M
Ea

Kw

No

W

Province

The graph would not look the same eleven years later. Much
misunderstanding and a dire lack of anti-retrovirals have hindered the

8


solving of the Aids pandemic. Unfortunately the vertical bars would be
much longer on a bar chart today.

2. A BAR CHART OF SOUTH AFRICAN HOUSEHOLDS

Households in SA with 2 (or less) rooms (1996)

45
40
35
30
Percentage

25
20
15
10
5
0

e
ga
l

po
ng

a
e
pe

pe
ta

t
es
ap
at

ric
Na

an
Ca

te

po

Ca
St

W
C

Af
au

al
u-

m
n
ee
n

rth

rn

h
m
G

ul

Li
er
er

ut
te
pu
Fr

No
aZ

rth
st

So
es
M
Ea

Kw

No

W
Province

The percentage of households that have 2 or fewer rooms may be lower
in the Western Cape for a number of possible reasons. One is that the
statistics may not be accurate. Another reason may be that there are far
fewer people in the Western Cape to accommodate compared to other
provinces, so it may be easier to organize better housing. Also it may be
possible that the Western Cape Housing Department is an efficient
corrupt-free department which is organizing housing at a rapid rate!


• http://www.ict.oxon-lea.gov.uk/best_practice/1e_pictograms/unit
%201e.html (introduction to the use of prictograms)

• http://www.technologystudent.com/struct1/model4.htm (pictograms as
bar charts)

• http://www.coolschool.ca/lor/AMA11/unit1/U01L02.htm (some
misleading graphs)

• http://www.blueclaw-db.com/download/barchart_access_form.htm
(how to draw a bar graph on Microsoft Access)

• http://www.mrbartonmaths.com/resources/software%20tutorials/Excel
%20-%20Drawing%20Charts.doc (drawing a bar chart using
Microsoft Excel)

9


Exercise 3: Interpreting data
The pie chart is an effective way of representing data if handled properly.
The diagram is divided into slices like a pie - each slice represents a part
of the whole. The whole circle represents the whole population.

Exercise 3

1 Copy and complete this pie chart to represent the data about eye
colour given above.
2 Draw a radius in the circle. This is where you start measuring the
angles.
3 Measure the angles at the centre.
4 Give a title and a key for the pie chart.
5 Use Excel to draw the pie chart if you are able to.
6 Compare your hand drawn graph with the computer generated graph
if possible.

Solutions to Exercise 3: Pie charts

Colour Number
This is the table of
Brown 32 data on eye-colour
that was given on
Grey 6 page 10 of the reading
for Unit Five of the
Blue 22 SAIDE ACEMaths
materials.
Total 60

10


Eye colour of learners in a class.

Brow n
Grey
Blue

The angles (if you worked them out to draw them by hand) are:

• Brown eyes – 192°

• Blue eyes – 132°

• Grey eyes – 36°

(The sum of all of the angles must come to 360° you should check this
when you work them out using your calculator. 192°+132°+36°=360°)


• http://en.wikipedia.org/wiki/Pie_chart (information on Wikipedia)

• http://www.fsmq.org/data//files/ihausdrawpsi-9345.doc (a very clear
guide to drawing pie charts using excel)

• http://pubs.logicalexpressions.com/Pub0009/LPMArticle.asp?ID=390
(another guide to drawing a pie chart using Microsoft Excel with
information about alternative graphs and their uses)

Exercise 4a: Representing and
interpreting data
The learners will have fun collecting the data required by the teacher
but it is much more fun and meaningful once it is correctly
represented by means of a bar graph, a pictogram, a line graph or a pie
graph. When the data is well-represented visually, the message is
communicated so much more effectively. There are times when one
representation is more effective than another. This depends on the
situation presented.

11


Exercise 4 a
Thembi kept a record of the hours she spent on different activities during
the day. This information is shown below.
1 Complete the table to show the degrees needed for each activity when
drawing a pie chart.
2 Draw the pie chart.
3 Represent the information using a line graph.

12


Activity
Number of
hours
Number of
degrees

School
5
……. x 360°
= ……

Meals
1

Homework
3

TV
2

Travel
1

Sleep
8

Other
4

Total
24

13


Solutions to Exercise 4a: Representing and Interpreting Data

1.

Activity Number of hours Number of degrees

School 5 5
24 × 360° = 75°

Meals 1 1
24 × 360° = 15°

Homework 3 3
24 × 360° = 45°

TV 2 2
24 × 360° = 30°

Travel 1 1
24 × 360° = 15°

Sleep 8 8
24 × 360° = 120°

Other 4 4
24 × 360° = 60°

Total 24 360°

2. Pie chart to represent the data about Thembi’s daily activities.

Thembi's daily activities

School
Meals
Homew ork
TV
Travel
Sleep
Other

14


3. Line graph to represent the data about Thembi’s daily activities.

Thembi's daily activities

9
8

Number of hours
7
6
5
4
3
2
1
0

el
k

TV

p
ol

er
ls

or

av

ee
ea
ho

th
ew

Tr

Sl

O
Sc

M

m
Ho
Activity


• http://www.fsmq.org/data//files/ihausdrawlsi-9880.pdf (how to draw
line graphs using Microsoft Excel)

• http://www.ixl.com/math/practice/grade-4-line-graphs (online
activities on interpreting line graphs)

• http://members.westnet.com.au/molinasantos/strands/C_D/piegraphs.p
df (interpreting pie graphs/charts)

• http://members.westnet.com.au/molinasantos/strands/C_D/piegraphs.p
df (interpreting pie graphs/charts)

• http://www.lmpc.edu.au/resources/Science/research_projects/4%20Dr
awing&interpreting%20grap.pdf (information on interpreting line (and
other) graphs)

Exercise 4b: Representing data –
selecting the best representation

15


Exercise 4 b

Joe did a survey of the colours of cars parked at the local sports club. The
observation sheet is given below.
1 Complete the frequency table.
2 Display the data as a pictogram, a bar graph and a pie chart.
3 Which representation do you think is best? Explain your response.

16


Joe’s observation sheet

Colours

Tallies

Frequency

Red

|||| ||||

Blue

||

Green

|||| |||| ||||

Black

||||

Orange

|||| |||| ||

17


Solutions to Exercise 4 b: Representing data - selecting the best
representation

1. tallies and frequencies below:

Joe’s observation sheet

Colours Tallies Frequency

Red |||| |||| 10

Blue || 2

Green |||| |||| |||| 15

Black |||| 5

Orange |||| |||| || 12

2. Pictogram, Bar graph and Pie chart graphs follow below.

Pictogram:
Joe’s car survey

Colours

Red

Blue

Green

Black

Orange

Key: = 5 cars and =
=2 cars

Bar graph:

18


Joe's car survey

16
14
12
10

Number
8
6
4
2
0
Red Blue Green Black Orange
Colour of car

Pie chart

Joe's car survey

Red
Blue
Green
Black
Orange

3. In the example above, the pictogram, the bar chart and the pie chart
are good representations of the data. Perhaps a pictogram using cars
for representation would be fun here, especially if the colour of the car
icons could be varied to show the particular colours of the cars he
counted!


• http://wsgfl.westsussex.gov.uk/ccm/cms-
service/stream/asset/;jsessionid=aE7nigzVl2Sf?asset_id=2554092
(information on selecting the best type of graph)

• http://42explore.com/graphs.htm (online information on selection of
graphs for representing data. Also gives information on misleading
data. Also a child friendly online graphing activity.)

19


Exercise 5: Interpreting data –
impulse buyers
It is common in our media for data to be misrepresented and
misinterpreted. Sometimes this can be deliberate to shock or delight us,
or actually to mislead us. Other times it can be a genuine
misunderstanding of the data by the journalist or reported involved. The
more we as ordinary civilians know about the representation and
interpretation of data, the more analytical and informed we become. If
we can help our learners to confidently represent and interpret data at
school level, we will be helping them to develop a tool for everyday life
long after formal schooling is complete!

Exercise 5
Below are a graph and a newspaper article taken from The Star newspaper
(1999) about impulse buying. Read the article carefully. Think about things like
the assumptions that are made by the writer/researcher and whether the article
and the graph tell the same story.

Durban
29,6%
Cape Town
42,6%
Pretoria
31,8%
Johannesburg
38,5%

Who are the impulse buyers?

South Africa is a nation of shoppers with increasing numbers defined as
impulse buyers who respond to glossy adverts and come-ons such as
‘never to be beaten bargains’ and ‘buy one and get one free’.
This, in part, has emerged from one of the most comprehensive surveys
of consumer shopping behaviour which has just been compiled by Media
& Marketing Research (MMR).
MMR’s research provides answers to a host of questions about South
Africa and how its people buy.
Capetonians were the most likely to respond to bargains, good buys as
well as advertising come-ons on TV and in newspapers (42,6%).
Johannesburg shoppers came in second at 38,5%, Pretoria notched up
31,8% and Durbanites were rated at 29,6%.
The ranks of impulse shoppers were most likely to come from these
‘bargain hunting’ groups.
1 What is wrong with the pie graph?
2 Who do you think ‘impulse buyers’ are?

20


3 Who do you think ‘bargain hunter’ shoppers are?
4 Redraw the given data in a more suitable and correct graph.
5 According to the researchers, what does the given data represent?
6 What does the newspaper headline suggest the data represents?
7 What assumptions are made about the meaning of results of research
on ‘bargain-hunting’ shoppers?
8 What does the data not tell us about impulse buying?

Solutions to Exercise 5: Interpreting data - Impulse Buyers

1. The percentages in the pie graph do not add up to 100%! The
percentages add up to 142,5%. The pie graph should represent the
spread over the total population of 100%, so these percentages are
misleading.

2. 'Impulse buyers' are people who buy things on the spur of the moment.
They probably do not analyse whether they really have a need for the
item beforehand.

3. 'Bargain hunter' shoppers are those shoppers who look for good deals.
Bargain hunters are not necessarily impulse buyers. In fact, it is quite
probable that the impulse buyer does not look for a bargain and
similarly, it is quite probable that the bargain hunter would not be an
impulse buyer.

4. A more suitable graph would be a bar graph because the percentages
represent those who are impulse buyers out of all the buyers in a
particular city.

The bar graph is shown below.

Impulse buyers in SA cities (The Star, 1999)

50
Percentage of city

40
population

30

20

10

0
Durban Cape Town Pretoria Johannesburg
SA Cities

21


5. According to the researchers, the given data represents the percentage
of people 'most likely to respond to bargains, good buys, as well as
advertising come-ons on TV and in newspapers': in other words
bargain-hunting shoppers. The percentages given are those of the
bargain hunters out of the total shoppers in that city.

6. The headline suggests that the data represents the percentages relative
to the South African population, of impulse buyers from a selection of
cities in the country.

7. The assumption made is that the bargain hunters are impulse shoppers.
This is not necessarily true as mentioned previously.

8. Actually the data tells us nothing concrete about impulse-buying.


• http://hospitality.hud.ac.uk/studyskills/usingData/InterpretingData/mis
leadingData.htm (info on misleading data but also other info about
data representation and interpretation)

• http://www.blackgold.ab.ca/ict/Division4/Math/Div.
%204/analyzingdata/misleading.htm (activity on misleading data)

Unit mathematical content
summary
Summary of content covered in this unit:

• Exercise 1 involved mathematical concepts relating to the collecting
and tallying of data

• Exercise 2 involved mathematical concepts relating to drawing
pictograms and bar charts.

• Exercise 3 involved mathematical concepts relating to the drawing of
a pie chart, using a pair of compasses and Excel on the computer.

• Exercise 4a involved mathematical concepts relating to the working
out of degrees and the drawing of a pie chart/ line graph.

• Exercise 4b involved mathematical concepts relating to the displaying
of data using the pictogram, a bar graph and a pie chart.

• Exercise 5 involved mathematical concepts relating to the
misinterpretation of data - how data can sometimes be misrepresented
by reporters in media.

22

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