Mathematical Literacy
Student’s Book
FET FIRST
NQF Level 4
Mathematical Literacy
Student’s Book
FET FIRST
NQF Level 4
PROTEC
FET First Mathematical Literacy NQF Level 4 Student's Book
FET First
© PROTEC 2008
© Illustrations and design Macmillan South Africa (Pty) Ltd, 2008
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claims for damages.
First published 2008
10
3 5 7 9 10 8 6 4
Published by
Macmillan South Africa (Pty) Ltd
Private Bag X19
2096 Northwold
Gauteng
South Africa
Text design by Resolution
Cover design by Deevine Design
Artwork by Geoff Walton
Typesetting by Resolution
The publishers have made every effort to trace the copyright holders.
If they have inadvertently overlooked any, they will be
pleased to make the necessary arrangements at the first opportunity.
e-ISBN: 978-1-43102-005-8
ISBN-13: 978-1-77030-476-5
WIP: 2121M000
It is illegal to photocopy any page of this book
without written permission from the publishers.
The publisher would like to thank the following for permission to
use photographs in this book:
Mike van der Wolk
Contents
Topic 1 Numbers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Module 1 Numbers and calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Unit 1.1 Types of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Unit 1.2 Working with decimal numbers, fractions and percentages . . . . . 11
Unit 1.3 Exponents and scientific notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Module 2 Measuring with accuracy and precision. . . . . . . . . . . . 20
Unit 2.1 Significant digits and measurement . . . . . . . . . . . . . . . . . . . . . . . . . 20
Unit 2.2 The basics of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Module 3 Calculating with measurement units . . . . . . . . . . . . . . 32
Unit 3.1 Measurement units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Unit 3.2 Ratio, rate and proportion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Unit 3.3 Time, duration and time zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Topic 2 Patterns and relationships
. . . . . . . . . . . . . . . . . . . . .50
Module 1 Identify and extend patterns . . . . . . . . . . . . . . . . . . . . . . . 52
Unit 1.1 Identifying and describing number patterns . . . . . . . . . . . . . . . . . . 52
Unit 1.2 Real-life number patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Unit 1.3 Direct proportion and inverse proportion . . . . . . . . . . . . . . . . . . . . 64
Module 2 Patterns, tables and graphs . . . . . . . . . . . . . . . . . . . . . . . . 68
Unit 2.1 Straight line graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Unit 2.2 Inverse proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Unit 2.3 Using graphs to show trends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Investigation: Protecting our environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Topic 3 Finance
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83
Module 1 Read and interpret financial information . . . . . . . . . . 85
Unit 1.1 Personal financial management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Unit 1.2 Being a wise consumer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Unit 1.3 Salaries and deductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Module assessment task: Budgets, deductions and borrowing . . . . . . . . . 111
Topic 4 Space, Shape and Orientation . . . . . . . . . . . . . . . . .115
Module 1 Calculations with space, shape and orientation . . 117
Unit 1.1 Area, perimeter and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Unit 1.2 Working with scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Assessment task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Module 2 Interpreting maps, plans and diagrams . . . . . . . . . . 132
Unit 2.1 Interpreting floor plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Unit 2.2 Assignment: Designing a restaurant . . . . . . . . . . . . . . . . . . . . . . . . 142
Unit 2.3 Working with maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Module 3 Work flow and processes . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Unit 3.1 Using symbols and diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Unit 3.2 Organising a factory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Topic 5 Information communicated through
numbers, graphs and tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
Module 1 Collecting, organising and representing
information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171
Unit 1.1 Summarising and organising information . . . . . . . . . . . . . . . . . . . . . . . . . .171
Unit 1.2 Representing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182
Unit 1.3 Collecting data by carrying out a survey . . . . . . . . . . . . . . . . . . . . . . . . . . .195
Research task: Investigating water resources in a community . . . . . . . . . . . . . . . .200
Module 2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206
Unit 2.1 The language of probability and calculating simple probabilities . . . . . .206
Unit 2.2 Dependent and independent events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213
Student’s Portfolio of Evidence
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218
Topic 1
Numbers
1
Topic 1
Numbers
Overview
This topic provides a foundation for all the other topics in
Mathematical Literacy. Many of the concepts in this topic are dealt
with in more depth in the other topics.
We begin by looking at different types of numbers and the
circumstances in which it is appropriate to use certain types of
numbers. In particular, we study decimal numbers and their use
in measurement and the need for measuring with accuracy and
precision. We look at different units of measurement and
converting between them. We also work with significant digits
and scientific notation.
This topic deals with calculator work, which you need to know
for all the topics.
We deal with the principles of measurement and how to measure
on scales.
The topic deals in some detail with selecting and using formulae
to solve measurement problems. Working with formulae prepares
you for Topic 2, in which you plot graphs from formulae and look
at relationships between numbers in more detail.
The topic also gives practice in solving problems involving
percentages, rate, ratio and proportion, which are useful for most
topics in the course.
The last part of this topic introduces notation used for writing
times and solving time problems related to duration and time
zones.
This topic is assessed in all of the assessment tasks throughout the
course.
2
Topic 1
Module 1
Numbers and
calculations
Overview
In this module you will:
•
•
•
•
•
•
•
•
•
•
•
learn about different types of numbers
understand that measurements are inexact and there is always
a part of a measurement value that is estimated
explore different units of measurement that are appropriate in
different contexts
learn why we round off and practise rounding off
work with the memory keys on a basic calculator
compare and order decimal numbers
convert between fractions, decimals and percentages
solve problems involving fractions, decimals and percentages
revise exponents
work with squaring and cubing numbers
learn about and use scientific notation.
Unit 1.1 Types of numbers
Different types of numbers are used in different situations. We use
whole numbers for counting exact numbers of things, for example, a
person has exactly 10 toes. In this unit we deal with the kind of
numbers needed for measurements. Measurements are inexact, as we
round them off to a certain number of decimal places for convenience.
Learning activity 1
Words &
Terms
Accurate:
A measure
ment is
accurate
if it is clos
e to the tr
value.
ue
Decimals and measurement
1. a) Draw five straight lines of different lengths between about
5 and 10 cm long without measuring them.
b) Measure the lines and write down their lengths in
centimetres and millimetres.
c) Do any of the lines have a length that is a whole number of centimetres?
2. a) Now draw five straight lines that you estimate are less than 2 cm long.
b) Measure the lines and write down the lengths in millimetres.
c) Are the lengths a whole number of millimetres, or did you round off the numbers?
3. It is unlikely that you randomly drew lines that are a whole number of millimetres or
centimetres long. Discuss this and write down a short explanation.
4. How accurate do you think your measurements are? How could you describe the accuracy of
your measurements?
Module 1
3
In Learning activity 1 you probably found that the lengths of the lines
were decimal fractions and not whole centimetres or whole
millimetres. While we count numbers of things as whole numbers, we
need fractions or decimal numbers for measurement values. When we
give values for mass, length, volume, temperature, and many other
quantities, we need to be aware of the continuous nature of the values.
For example, if you draw a line between 1 and 2 cm long, it could have
any value you could think of between 1 and 2 cm, such as 1,2 cm;
1,02 cm; 1,578 cm; 1,99999 cm. There are an infinite number of fractions
between any two numbers.
We choose the number of decimal places to work with depending on
various factors, including how accurate we need to be for the context
and the accuracy of our measuring instrument.
For example, when we measure people’s height,
we are usually satisfied with measurements
correct to the nearest centimetre or half centimetre,
and so we ignore digits smaller than a centimetre
or half centimetre. So heights are usually
expressed as 1,82 m, 182 cm, 172,5 cm, 1,55 m,
162,5 cm, etc.
In a different context, such as doing a scientific
experiment to determine the effect of a substance
on children’s growth, it would be important to
measure height in smaller divisions, such as
millimetres, as small differences are more
important, and we need more accurate results in order to draw
conclusions. So we would use a method of measuring that would
allow us to measure more precisely.
The size of very small objects, such as
microorganisms, must be measured in very small
units to give meaningful results. Scientists use
units such as micrometres (one-thousandth of a
millimetre) to measure bacteria and even smaller
unit, such as nanometres, to measure the size of
virus particles. So these measurements need to be
much more precise.
Learning activity 1 should make you aware that
lengths can take on many different values between
the gradations on a ruler. When we measure, we
can only be as accurate (close to the real value) as
the measuring instrument allows us to be.
A certain part of a measurement is always an estimate.
You will learn more about this in the next module.
4
Module 1
Tobacco Mosaic Virus particles
This photograph is taken through
an electron microscope and is
called an electron micrograph. The
image is magnified about 150 000
times.
Some measurement values are rounded off, so you might mistake
them for discrete values. For example, you might read that the
distance between two towns is 300 km. This is not an exact value and
the distance between two towns is not a whole number of kilometres –
the rest of the measurement simply doesn’t matter in that context.
Learning activity 2
Words &
Terms
Context: T
he particu
lar
situation.
Continuou
s:
continuou A value is
s if it can
be
represente
d by a fra
ction. If a
set of num
be
means we rs is continuous, it
can take a
ny two
numbers
and find a
nother
number b
etween th
em.
Continuou
sn
obtained b umbers are usuall
y
y measuri
ng.
Correct to
: A numbe
r written
correct to
a certain
decimal
place is ro
unded off
to
decimal p
lace. It als that
o means
that we are
expressin
g
uncertain
ty about th
e decimal
places sm
aller than
this digit.
Discrete:
A value is
d
iscrete if
can take o
it
nc
Discrete n ertain values only.
umbers a
re obtaine
by countin
d
g.
Exact and inexact numbers
1. Is each of the following statements true or false? Explain your
answers.
a) Measurement values are usually rounded off to a certain
number of decimal places.
b) Measuring instruments do not limit how accurate a
measurement is.
c) Volume measurements are continuous numbers.
d) The distance of the earth from the sun is exactly
149 600 000 km – this number has not been rounded off.
2. Write down all the values in the list below that lie between
3 and 4 cm.
2 mm
3,2 mm
3,2 cm
4,0001 cm
4,6 cm
3,5 cm
2,99999999 cm 3,0000001 cm
3,99909 cm 2,66 cm
3,56 cm
35 mm
9,33 cm
1,34 cm
3,12 cm
304 mm
2,43 cm
3,00001 cm
0,4 cm
3,9 cm
3. State whether the value in each of the following situations is an exact number of units or
whether the number is inexact as it would need to be rounded off to a certain number of
decimal places.
a) the length of time it takes for a light bulb to burn out
b) the number of students attending a lecture
c) the number of leaves on a plant
d) the length of a blade of grass
e) the temperature of bathwater
f) a child’s height
g) the volume of rainwater in a tank
h) the number of jobs available in a company
i) the length of a matchstick
4. The picture below shows a pencil measured against a ruler with centimetre and millimetre
divisions.
a) Arrange the values in the box on the next page into two groups: A, the values which are
possible measurements of the pencil in the picture and B, values that are not possible
measurements of the pencil in the picture.
Module 1
5
27 cm
14,903 cm
13,28 cm
13,99 cm
14,900 cm
14,995 cm
10 cm
3 cm
12,35 cm
14,89 cm
14,91 cm
12,5 cm
14 cm
15,5 cm
15 cm
14,5 cm
2 cm
6,5 cm
13,23 cm
147 mm
b) What is a reasonable level of accuracy with which you can state the measurement of the pencil
in this picture? Explain.
5.
The distance from the centre of town A to the centre of town B is measured to be 285,635 km.
Which of the following distances would you give to someone who is driving from town A to
town B, who wants an idea of how far it is? You can give more than one value, but explain your
choice clearly.
285 635 m
285,7 km
285,5 km
250 km
300 km
286 km
285,6 km
286 000 m
Rounding off
Numbers are rounded off to simplify them by reducing the number of
decimal places that they are written with. When we round off numbers
to a certain number of decimal places, we start by deciding which
decimal place we want to round off to. For example, the diameter of a
planet can be written correct to the nearest thousand kilometres and
the length of a pencil can be written correct to the nearest half
centimetre or millimetre. It depends on what the measurement is going
to be used for.
Example 1
diameter
If we have a value of 3 636 789,2 km for the diameter of a very large
planet, but only need the value to be written correct to a thousand
kilometres, we would not simply ‘drop’ the digits following the
6 thousands. This would give us an error of 789,2 km.
Millions
Hundred
thousands
Ten
thousands
Thousands
Hundreds
Tens
Units
tenths
3
6
3
6
7
8
9
2
Instead, we attempt to reduce the error by rounding off. We check
whether the digit that follows the 6 is closer to 6 or to 7. By
convention, if the digit following the 6 is smaller than 5, we round the
number down to 6 thousands, and if the digit is 5 or bigger, we round
the 6 up to 7. The remaining decimal places now have zero values, so
we omit them from the written number:
6
Module 1
Millions
Hundred
thousands
Ten
thousands
Thousands
Hundreds
Tens
Units
tenths
3
6
3
7
0
0
0
0
Example 2
The length of a piece of paper is measured as 217,349 mm. We need to
give the measurement to the nearest millimetre. The millimetre digit is
7. The digit following this is 3. So we round down to 7 and obtain the
number: 217 mm. (We do not look at the digits in the hundredths and
thousandths positions.)
Learning activity 3
Rounding off
1. In the South African currency, 1c and 2c coins have been phased
out. However, prices of goods in shops can still involve 1c and 2c
combinations, so shop tellers often round the prices down to the
nearest 5c.
a) Why do you think the cashiers do not simply round off the
total amount to the nearest 5c?
b) What would the customer pay for each of the following totals?
R12,99; R100,36; R100,31; R111,98; R21,02; 53c
2.
The following lengths of planetary orbits have been measured
with varying precision. To make them comparable, you need to
round off the measurements to the same place value digit.
a) What digit will this be? Explain.
b) Write down the rounded off values.
c) Arrange the values in ascending order.
A. 5 789 010 km
B. 6 278 393 km
C. 4 000 308 km
D. 10 201 000 km
E. 12 067 033,5 km
F. 8 098 900 km
?
??
Did you know?
The conventional method for
rounding off that we use here
results in a general error. If
you take the average of the
rounded off numbers, it will be
higher than the average of the
original numbers. If the next
digit ends in 1, 2, 3 or 4, we
round down, while if the next
digit ends in 5, 6, 7, 8 or 9,
then we round up. So there are
more cases where we round
up than cases where we round
down, resulting in an average
overestimation of the actual
number.
There are more complex
rounding off methods that can
be used to avoid this
overestimation error, but we
do not deal with these here.
Calculator work
Compare your calculator to the one in the photograph alongside. Note
any differences. Are there keys that your calculator does not have?
Some hints for using your calculator
•
Before you do a calculation on your calculator, it is good practice
to estimate the answer, even if you just do this mentally. This will
help you to ensure that the answer you get on the calculator is a
sensible one. If you pressed the incorrect key or left off a zero,
you will then notice the difference between your estimated
answer and the calculated answer immediately.
Module 1
7
•
•
We can easily make an error when adding up a long list of
numbers, as it often happens that we type in a number
incorrectly. There are a few strategies and good habits to get into
so that you pick up errors quickly.
o Do the whole calculation a second time. If your second answer
is different to the first, you have made a mistake. Do it again to
see which one is correct. If the answer is the same, you have
probably not made an error. This is a habit you should follow
whenever you use a calculator.
o You could also break up the calculation into parts and write
down the answer for each part.
When you are doing a calculation that involves a few steps, try
not to round off any of the numbers until the final answer. This
will make the final answer more accurate. Since the answer
would not be exact, we use the symbol ≈ (which means
“approximately equal to”) instead of =.
For example: 53,8789 … m ≈ 53,9 m
How to use the memory keys
The memory keys (M+, M–, and MRC) give your calculator some
special capabilities. They allow you to store a number such as the
answer to a calculation and do calculations in the memory of a
calculator.
The [M+] key is used to add a number to the memory, or to add it to a
number already in the memory.
The [M–] key is used to subtract a number from the number in the
memory.
The [MRC] key, pressed once, displays the number currently stored in
memory. If you press this key twice, the calculator’s memory is
cleared.
When you use the memory, the letter ‘M’ appears at the top of the
display, showing that the number on the display has been stored in the
calculator’s memory.
For example:
Follow these steps to do the calculation 100 + (2 × 80) – 60:
Enter 100 into the calculator and add it to the memory by pressing M+.
Calculate (2 × 80) and add it to the memory by pressing M+.
Then enter 60 and subtract it from the memory by pressing M–.
Press MRC to recall what is in the memory – this gives you the answer
to the calculation. Try this out: your answer should be 200.
Compare this to the key sequence 100 + 2 × 80 – 60. The memory key
allows you to work with brackets.
Always clear the calculator’s memory by pressing [MRC] twice,
otherwise you will end up with unexpected answers.
8
Module 1
Positive and negative numbers (directed numbers)
Experiment with changing the sign of a number on your calculator, for
example, changing 10 to –10.
Some calculators have a
key, which changes the sign of a
number. (When we use the symbol ± (plus-minus) before a number in
other situations, we mean the amount is approximate or average, not
accurate. It is only on the calculator that this key refers to an operation
that changes the sign of the number.)
However, on calculators that do not have this key, we can usually
change the sign of a number by entering the number and then pressing
the minus key.
Examples
1.
2.
Sarah owes Lindi R400. She receives an amount of R1 000. How
much does she have?
Key sequence: 400 [–] [+] 1000 =
Answer: 600
If you move 3 m to the left of a point and then 5 m to the right,
how far have you moved away from the point? Let’s assume left is
negative distance and right is positive.
Key sequence: 3 [–] + 5 [=]
Answer: 2 to the right [or plus 2]
3. Susan earns R4 000 per month, and the bank allows her an
overdraft of R2 000. She begins the month with R50 in her
account. Her salary is deposited into her account. She then buys
groceries to the value of R850, pays her rent of R2 000, pays
accounts to the value of R500, spends R400 on clothing, R300 on
entertainment and she pays R300 for a medical bill. It is now the
15th of the month.
a) Show her income and expenditure on a number line.
b) How much money does she have available until the end of the
month?
c) Is she in credit (+) or debit (–) with the bank?
a)
b) She has R1 700 of her overdraft available.
c) She is in debit.
Module 1
9
Learning activity 4
Memory keys and directed numbers
1. For each of the following calculations, write the number sentences and give the answer.
a) 4 × 4 [M+] 10 × 20 [M+] [MRC]
b) 1000 [M+] 3 × 100 [M+] 550 [M–] [MRC]
Remember to clear the memory
c) 640 × 3 [M+] 2100 ÷ 7 [M–] [MRC]
between calculations!
d) 30 [M+] 2200 ÷ 11 [M+] 7 × 25 [M–] [MRC]
e) 10 240 × 3 [M+] 220 ÷ 11 [M–] [MRC]
f) 200 ÷ 2 [M+] 330 [M–] [MRC]
2.
Write down the calculator key sequence you could use to do the following calculations and give
the answer in each case:
a) 450 – (30 ÷ 2) + 42
b) 2 223 + (9 × 3) – (86 ÷ 2)
c) –20 + (300 ÷ 15)
d) (11 012 × 2) – (350 × 30)
e) (2 226 + 4 789) × 33
f) (59 678 × 43) – (40 000 ÷ 2)
3.
Write down the calculator key sequence and the answer for each of these problems.
a) Mbali has a debt of R7 000. She receives her monthly salary of R5 000. What is her balance?
b) A lift in a mine shaft moves 233 m downwards and then 150 m towards the surface. What is
the distance moved?
4.
The height of objects above and below sea level is called their altitude. Sea level is
regarded as 0 m. Here are the altitudes of some geographical features:
Table Mountain: 1 088 m
Johannesburg: 1 753 m
Lowest level of City Deep gold mine: – 2 134 m
Mount Everest: 8 848 m
Mount Fujiyama: 3 776 m
Surface of the Caspian Sea: 28 m
a) Show each of these features on a number line.
b) What is the difference between the altitude of the lowest level of City Deep gold mine
and the altitude of Johannesburg?
c) What is the difference between the altitude of the Caspian Sea and Mount Everest?
5.
The Mariana Trench in the North Pacific Ocean, near the Mariana Islands, is the deepest
part of the world’s oceans and the deepest location on the surface of the Earth’s crust. It
has a maximum depth of about 11 km.
a) Approximately how many times could Table Mountain be fitted into the trench, if
each mountain was stacked on top of the other?
b) If Mount Everest was moved into the deepest part of Mariana Trench, what would
the depth of the sea water over the highest point of Everest be?
10
Module 1
Assess yourself
To make sure that you understand this unit, answer the following questions:
Yes
I’m not sure
No
Can you explain why measurement values are usually rounded off?
Can you round off measurement values in a manner which is appropriate
to the context?
Can you work with place value to round off numbers to a given decimal
place?
Can you use the memory keys on your calculator to do calculations that
involve a few steps?
Unit summary
•
•
•
Measurement values are usually given to a convenient number of decimal places and so are
rounded off.
Different degrees of precision are needed in different contexts and we use measuring
instruments and units appropriate for the context.
The memory keys (M+, M–, and MRC) give your calculator some special capabilities. They
allow you to store a number such as the answer to a calculation and do calculations in the
memory of a calculator.
Unit 1.2 Working with decimal
numbers, fractions and
percentages
Comparing and ordering decimal numbers
We compare two or more numbers by stating that one number is less
than, equal to or greater than the value of another number.
We use the symbols <, = and > to compare two numbers.
For example, 2 < 3 and 3 > 2.
Using a place value table
Remember that the digit on the right of a number has the smallest
value, so it has the smallest effect on the size of the number. So, when
we compare two decimal numbers, we start from the left-hand side.
Let’s compare the numbers 4,679 and 4,668.
Module 1
11
It is useful to put numbers into a place value table to compare them:
Units
tenths
hundredths
thousandths
4
6
7
9
4
6
6
8
Start with the whole number
portion of the numbers. The
number with a larger whole
number portion is the larger
number.
These numbers have the same
whole number portion, 4.
If they have the same value,
compare tenths and then
hundredths, and so on. If one
decimal has a higher number in
the tenths place then it is larger
and the decimal with fewer
tenths is smaller.
The tenths digit is equal to 6 in
both these numbers.
If the tenths are equal compare
the hundredths, then the
thousandths, and so on, until one
decimal is larger or there are no
more places to compare.
4,679 has 7 in the hundredths
position, so it is bigger than 4,668,
which has a 6 in the hundredths
position.
We don’t need to go any further,
as we have enough information
to compare the two numbers.
For example: Use the correct symbol to compare: 1,90 _ 1,09
Both numbers have the same units value, but the first number has nine
tenths while the second number has no tenths, so 1,90 > 1,09.
Arranging a list of numbers in ascending order means that we write
the list of numbers starting with the smallest number and ending with
the largest number. Descending order means from the largest number
to the smallest number.
When we arrange a list of numbers in order, we are doing the same
sort of comparison, but with more numbers:
For example: arrange the following numbers in ascending order:
1,63; 0,9; 1,627; 2,06; 1,653; 1,27; 1,06.
Of the seven numbers in the list above, only one of them has a units
value of 0. So 0,9 is the smallest number.
Five of the numbers have a units value of 1.
Only one of the numbers has a units value of 2. So 2,06 is the largest
number.
Then we look at the next highest digit, the tenths, to arrange the rest of
the numbers in order: 1,63; 1,627; 1,653; 1,27; 1,06
The correct order is: 0,9; 1,06; 1,27; 1,627; 1,63; 1,653; 2,06
12
Module 1
?
??
Did you know?
You may have noticed that in
many contexts a decimal point
is used instead of a decimal
comma, for example, in the
media and on bank statements
and till slips. Also, sometimes
people use commas to
separate groups of three digits
in very large numbers, e.g. one
million rand can be written as
R1,000,000.00. These are all
methods that make numbers
easier to read. In this course,
we will use a decimal comma
to separate wholes from
decimal fractions, and we use
spaces to separate groups of
three digits. So we write one
million rand as R1 000 000,00.
Calculators use decimal
points, so when we give
calculator key sequences, we
use the decimal point in this
book.
It doesn’t matter which
convention you use, but you
must be consistent in the
convention you follow and
don’t mix them as far as
possible.
Converting fractions to decimals using a
calculator
Any fraction can be written in a decimal form. To write a common
fraction as a decimal, we use mental arithmetic or a calculator to
divide the numerator by the denominator.
Examples
1 = 1 ÷ 4 = 0,25.
4
1 = 1 ÷ 6 = 0,16666…
6
2 = 2 ÷ 10 = 0,2
10
3 = 3 ÷ 100 = 0,03
100
You can recognise some fractions in their decimal form easily, such
as 1 = 0,5; 1 = 0,25, without needing to use your calculator.
2
4
It would be helpful to remember and use some other common
fractions, such as 0,375 and 0,125.
Mixed numbers as decimals
To convert a mixed number to a decimal number, write the whole
number first and then convert the fraction to decimal form and add it
to the whole number.
Examples
13 1 = 13 + 1 = 13 + (1 ÷ 4) = 13 + 0,25 = 13,25
4
4
60 3 = 60 + 3 = 60 + (3 ÷ 8) = 60 + 0,375 = 60,375
8
8
When comparing fractional numbers to whole numbers, convert the
fraction to a decimal number by division and compare the numbers.
Percentages
A percentage is a fraction that has a denominator of 100. The word
‘percentage’ means ‘of 100’ and the symbol % shows that the
denominator is 100.
For example: 10 = 10%.
100
Whenever you encounter a percentage, find out what the whole or the
100% of the amount is. A percentage is not meaningful on its own.
Think about these examples:
• 50% of the students in your class travel by bus. We cannot put a
value to the 50% until we know that there are 32 students in the
class. Then we can easily work out that this means that 16
students travel by bus.
Module 1
13
•
A large company retrenches 12% of its workforce. If 200 workers
are retrenched, then we know that 12% = 200. We can work out
the number of staff members in the whole company as follows:
200 × 100 = 1 667.
12
•
A newspaper report tells us that crime in an area has increased by
200% over one year. What does 100% represent in this situation?
100% is the crime incidence from the previous year. So 200%
simply means that there has been double the number of incidents
in the year.
Any fraction can be written as a percentage, because the denominator
can be converted to 100. So for example, 1 = 50%.
2
Converting fractions and decimals to percentages
A percentage is a fraction with a denominator of 100. So 20% = 20 ,
100
and so on.
To convert a fraction into a percentage, we need to make the
denominator 100, and then turn it into a % symbol. We do this by
multiplying by 100 %. (This is the same as multiplying by 100 ,
1
100
because the % symbol means a denominator of 100.)
Example: 2 × 100 % = 200 % = 13,3%
15
1
15
1 × 100 % = 100 = 16,7%
6
1
6
Working out percentages, percentage increases and discounts
The calculator key sequences for calculating percentages, discounts
and increases can save you a lot of time in everyday calculations as
you can do these in one step.
Example 1: To find 45% of R320:
Press the following keys: 320 × 45%
Do not press the = key. The answer comes up on the calculator display
as 144. So R144 is 45% of R320.
To work out a percentage increase or discount, you don’t need to work
out the percentage and then add it or subtract it – you can do it all in
one step.
Example 2: The price of an item is now R56,70 and it will increase by
12%. What is the new price?
Press the following keys: 56.7 + 12%
Do not press the = key. The answer is R63.504. So the new price is
R63,50.
Example 3: What is the discounted price if there is 20% taken off an
original price of R66,90?
Press the following keys: 66.9 – 20%
Do not press the = key. The answer comes up on the calculator display
as 53.52. So the discounted price is R53,52.
14
Module 1
Working with VAT
Remember to read the question carefully when working with VAT. Are
you given the price including or excluding VAT? Do you need to work
out the VAT itself, the price including VAT, or the price excluding VAT?
To find the VAT of a price simply work out 14% of the value, using the
method given on the previous page.
Examples
1.
What VAT will be charged on an item of clothing which costs
R55,99 (excl.)? Calculator key sequence: 55.99 × 14 % = 7.8386. So
the VAT will be R7,84.
You can add VAT to a price in one step rather than working out the
VAT and then adding it on.
2.
What is the VAT-inclusive price of R64,69 (excl.)?
64.69 + 14% = 73.7466. So the price will be R73,75.
When you are given a price that includes VAT, you cannot simply
subtract 14% of the amount to find the VAT exclusive price! This is
because the VAT is 14% of the VAT-exclusive price, not of the inclusive
price.
3.
Find the VAT exclusive price if the VAT-inclusive price is R85,50.
To work this out, divide by 114% or 1,14. Calculator key sequence:
85.5 ÷ 1.14 = 75. So the original price is R75,00.
Assessment activity
Working with decimals and percentages
Read the questions carefully!
1. Increase the following prices by the given percentages. Round off the answers to the nearest cent.
a) R225,50 by 33%
b) R350 by 16%
c) R408 by 120%
2. Work out the discounted prices if each price is decreased by the given percentage. Round off the
answers to the nearest cent.
a) 30% off R187,50
b) 5% off R1 899
c) 28% off R6 402
d) 15% off R15 863
e) 8% off R250,50
f) 5% off R1 160,65
3. The following prices do not include VAT. Give the VAT-inclusive prices, rounded off to the
nearest cent.
a) R15,85
b) R67,50
c) R106,45
4. The following prices are VAT-inclusive. Work out the amount of VAT charged in each case.
a) R19,85
b) R69,13
c) R356,45
5. The following prices are VAT-inclusive. Work out the VAT-exclusive prices.
a) R25,75
b) R75,50
c) R136,65
6. The government charges taxes on some goods, which is included in the cost of the goods. These
taxes are calculated as a percentage of the selling price. In a particular year, the tax on sparkling
wine rises from 10% to 20% of the cost of the wine and the tax on unfortified wine from 10%
to 12,5%.
a) If the average price of a bottle of sparkling wine was R55 the previous year, calculate the new
average price after the tax increase.
b) The average cost of unfortified wine was R47,50 the previous year. What will the average
price be this year?
Module 1
15
7.
The following is an extract from a newspaper article which appeared in the New York Times,
27 February 2008:
The economic fallout from
South Africa’s electricity crisis
continued to reverberate as
Gold Fields, the second largest
gold producer, announced that
it might cut up to 6,900 jobs,
13 per cent of its work force,
because of a 10 per cent power
reduction by the state utility,
Eskom. It said its production
would decline by 20 to 25 per
cent this quarter because of the
power cuts. The National
Union of Mineworkers, which
represents 320,000 workers,
said its members would “take
to the streets” if there were
major job cuts.
a) How big is the work force of Gold Fields?
b) If quarterly production usually amounts to 25 million rand, what is the predicted decline?
c) What percentage of the NUM members would lose their jobs, assuming that Gold Fields did
cut all the jobs and that all are members of NUM?
Assess yourself
To make sure that you understand this unit, answer the following questions:
Yes
I’m not sure
No
Can you order and compare decimal numbers correctly?
Can you convert between decimal numbers, fractions and percentages?
Can you convert fractions to percentages?
Can you calculate percentages of amounts and percentage increases and
decreases?
Can you work with percentages in real contexts?
Unit summary
•
•
•
•
•
•
16
Compare decimal numbers by starting with the digits on the left, which have a higher value.
To write a common fraction as a decimal, we use mental arithmetic or a calculator to divide
the numerator by the denominator.
To convert a mixed number to a decimal number, write the whole number first and then
convert the fraction to decimal form and add it to the whole number.
A percentage is a fraction with a denominator of 100. The % symbol means ‘of a hundred’.
To convert a fraction to a percentage, multiply by 100 %.
1
Using the % key on a calculator allows you to work out percentages of amounts, and increase
or decrease amounts by a given percentage in one step.
Module 1
Unit 1.3 Exponents and scientific
notation
Many complex formulae involve working with exponents. When a
number is raised to a power, it means that it is multiplied by itself a
certain number of times. For example, 32 or “three squared” means
3 × 3 = 9.
The number that is being raised to the power is called the base. The
number that the base is raised to is called the exponent.
Examples
1.
2.
The formula for the area of a square is (side)2, so a square with a
side of 12 m has an area of (12 m)2 = 144 m2.
The volume of a cube is equal to (side)3. The volume of a cube
with a side of 12 m is equal to (12 m)3 = 1 728 m3.
Powers of 10
A power is a number raised to a
certain number. The exponent is the
number of times that the number is
multiplied by itself.
We can use exponents to write powers of 10 in a convenient way. This
is illustrated in this decimal place table:
Ten
Thousands
thousands
Hundreds Tens
Units
tenths
hundredths thousandths tenthousandths
10 000
1 000
100
10
1
0,1
0,01
0,001
0,0001
104
103
102
101
100
10–1
10–2
10–3
10–4
Notice that raising a number to a negative exponent means inverting
the number. 10–2 = 1 2 = 0,01. You may need to remember this if you
10
encounter a formula that uses negative exponents. You also need to
know this for scientific notation. Keeping the table above in mind will
help you with this.
Scientific notation
Scientific notation is a method of writing numbers so that there is
always only one significant digit in front of the decimal comma, and
the number is multiplied by the appropriate power of 10. (You will
learn more about significant digits in the next module.)
Scientific notation is a very convenient way to write large or small
numbers and do calculations with them. It also quickly conveys two
properties of a measurement that are useful to scientists, significant
digits and size of the number.
To convert a number to scientific notation, write the number with the
decimal comma after the first significant digit (non-zero digit) and
multiply it by a power of 10 so that it is equal to the original.
Module 1
17
Examples
1.
To write 3 300 in scientific notation: first write 3,3 and then work
out what power of 10 to multiply it by. 3,3 must be multiplied by
1 000 to equal 3 300, so 3 300 = 3,3 × 103.
2.
The mass of an electron is approximately
0,00000000000000000000000000000091093826 kg.
In scientific notation, this is written 9,1093826 × 10–31 kg.
3.
The Earth’s mass is approximately
5 973 600 000 000 000 000 000 000 kg.
In scientific notation, this is written 5,9736 × 1024 kg.
Note that the number 0,8 is not written in scientific notation. The digit
0 is not a significant number when it is written before the first nonzero digit. So to write 0,8 in scientific notation, it should be 8 × 10–1.
Learning activity 5
Exponents and scientific notation
1. Write the following numbers as powers of 10:
a) 100 000
b) 10
c) 0,01
d) 0,00001
e) 1 000 000
f) 0,000001
g) 0,0001
2. The circumference of the Earth is approximately 40 000 000 m.
a) To what digit has this measurement been rounded off to?
b) Write this number in centimetres.
c) Write the centimetre number in scientific notation.
3. The distance between the centre of the Earth and the centre of the Moon is approximately
384 403 km.
a) Write this number in metres.
b) Write the distance in metres in scientific notation.
c) Round off the original number to the nearest thousand kilometres.
d) Now write the rounded off number in metres and in scientific notation.
4. The length of a certain type of virus is 0,000000004 m. Write this number in scientific notation.
5. Write the following measurements in ordinary notation:
b) 6,5 × 105 g
c) 1,08 × 10–2 kg
a) 2,5 × 10–1 m
–4
5
e) 9,99 × 10 kg
f) 1,00005 × 106 mm
d) 3,008 × 10 g
6. Are the following numbers written in scientific notation?
a) 10,8 × 102
b) 1,3
c) 1 000,8 × 10–1
–2
3
d) 1,0003 × 10
e) 0,9 × 10
f) 100 × 10–2
18
Module 1
Assess yourself
To make sure that you understand this unit, answer the following questions:
Yes
I’m not sure
No
Can you use exponents to write squares and cubes?
Can you identify numbers written in scientific notation?
Can you convert between numbers written in scientific notation and
ordinary notation?
Unit summary
•
•
•
When a number is raised to a power, it is multiplied by itself a certain number of times.
Powers of 10 are a convenient way of writing numbers to indicate decimal place.
In scientific notation, numbers are written with one significant digit in front of the decimal
comma, multiplied by the appropriate power of 10.
Module 1
19
Module 2
Measuring with accuracy
and precision
Overview
In this module you will:
•
•
•
•
•
find out about principles of measurement
learn about significant digits
learn about reading scales
learn about using the trundle wheel and Vernier callipers and
practise reading Vernier scales
learn about measuring volume and temperature using different
instruments.
Unit 2.1 Significant digits and
measurement
Every measuring instrument has limits. Whenever we work with a
particular instrument, we know that part of our measurement, the
larger digits, are definitely correct, but the last digit is an estimate. This
is true even if the measurement is a whole number or if the
measurement appears to be on an exact division.
20
980
10
30
970
mm
40
960
0
The example opposite demonstrates what the uncertainty limit of an
instrument is. Look at the grey line and the part of a ruler below it. It
is easy to see that the grey line is more than 20 mm and less than
30 mm long. So we can say with confidence that 2 is the first digit in
our measurement.
But what about the next digit? We can see that the line is more than 7
and less than 9 mm. But is it 7 or 8? The line might line up exactly
with the 28 mm mark, in which case you would report the
measurement as 28 mm. However, bear in mind that you have some
confidence that the measurement is 28 mm, but you are not completely
certain. The way in which you report this measurement should also
indicate your degree of confidence in the measurement (usually taking
into account the uncertainty limit of the instrument).
990
1000
How confident are you? If the line were 27,8 mm or 28,2 mm long,
then you would see quite easily that the line does not line up exactly
with the 28 mm mark. In fact, the uncertainty limit with a ruler like
this is 0,02 cm (0,2 mm).
20
Module 2
The number of digits believed to be correct and the last (estimated)
digit are the significant digits of a measurement. A measurement
reading has one more significant digit than the smallest gradation on a
scale. So on a metre ruler, the smallest division is 1 mm or 0,1 cm, so
we should read to 0,01 cm.
Example: Kim measures the length of a book, using a ruler with
millimetre divisions. She finds that the end of the book lines up exactly
with the 240 mm division on the ruler. However, the ruler is accurate
to two-tenths of a millimetre (0,2 mm), so she estimates that the last
digit should be zero.
A metre ruler is divided into a thousand millimetre divisions with no
gradations between the millimetre gradations. We have to estimate the
measurement value between the two millimetre marks. A metre ruler
is accurate to half a millimetre.
Example: Dineo measures the length of a block of wood using a metre
ruler and finds that it is 25,57 cm or 255,7 mm. The number seven, the
last digit in the measurement, is the number that she estimated
between two adjacent millimetre divisions. The estimated value is a
significant digit in this measured quantity for the wood’s length. Thus,
the measurement has four significant digits.
When machines give us measurement readings, the same idea applies.
We know what the precision of a machine is, and so we know which
digits in a measurement are definitely correct and which are estimates.
The last digit given for any measurement is the uncertain or estimated
digit. It is uncertain because it is estimated. The last digit given for a
measurement is always assumed to be estimated and it is significant.
All non-zero digits in a measurement are significant (1, 2, 3, 4, 5, 6, 7, 8
and 9). Zeros are only sometimes significant when reported in a
measurement.
?
??
Did you know?
Benchmark means a standard
measurement that others are
compared to. It is often used in
non-mathematical contexts
now as well, for example,
education specialists may
refer to a benchmark for
children’s reading levels.
The word “benchmark”, when
used in the context of
measurement, means a point
of reference for a
measurement. The term
comes from the practice of
making dimensional height
measurements of an object on
a workbench and using the
surface of the workbench as
the origin for the
measurements.
Learning activity 1
Significant digits
1. How many significant digits are there in each of the following measurements:
a) 303,0 m
b) 2,9801 × 102 km
c) 0,008320
d) 5 000
e) 0,023
f) 40,08
Unit 2.2 The basics of measurement
Measuring is about comparing. It is not absolute – the only ways we
have of measuring are comparing objects to standard measurements.
Even machines that measure do so by using an internal system of
comparison.
Principles of measurement
Two concepts are important here:
• accuracy: how close the measured value is to the true value
• precision: how closely the measured values agree with each other.
Module 2
21
We need to ensure that our measurements are both accurate and
precise. Errors in measurement are frequently caused by lack of skills
and practice in measuring, not only by the limits of the instrument.
General rules for measuring
•
•
•
•
•
Record your results with all the digits that you can measure, to
the limit of the uncertainty of the equipment. Make sure to
include zeros when they are actually being measured.
On a digital instrument, read and record all the numbers,
including zeros after the decimal point, exactly as displayed.
On a scaled instrument, estimate one more digit than you can
actually read from the scale.
Usually the manufacturer of the equipment indicates how
accurately or precisely it can measure. This tells you how many
digits you should record when using this device. For example,
suppose that the manufacturer of a measuring cylinder indicates
that it is accurate to within 0,5 ml. If you use such a graduated
cylinder to measure the volume of a substance, then you should
only report it to the nearest half millilitre – 95,0; 95,5; 96,0, etc.
When talking a reading on any ruled scale, avoid the error of
parallax by making sure that you are not looking at the scale
from an angle. Your eyes should be in line with the place you are
taking the measurement.
Words &
Terms
Error of parallax:
The error in a
measurement re
ading that
comes from look
ing at the scale
from an angle su
ch that the
item being measu
red does not
line up with the co
rrect scale
lines.
Calibration
This is a principle that you should be aware of even when you are
using instruments that you can’t calibrate, such as a simple
thermometer. Instruments such as these are calibrated by the
manufacturer.
Calibration is the process of setting a measuring instrument so that it
is correct for the current conditions. Precision instruments such as a
micrometer are calibrated before you purchase them, and this
calibration can be checked at regular intervals.
Reading scales
Calibrate: To set
a measuring
instrument so th
at it measures
accurately for th
e present
conditions. This
is usually done
using a set of stan
dards with
known measurem
ents.
Gradation: A mar
k indicating a
measurement on
a scaled
measuring instru
ment.
Many measuring instruments work with scales, a series of gradations
that are calibrated. Usually some of the gradations or graduations are
labelled, with smaller unlabelled gradations between them.
10
20
980
990
1000
Module 2
mm
30
40
970
22
0
960
There are three important steps to take whenever reading a scale
measurement:
1. Determine the interval of the scale (sometimes called the scale
increment). This means determining how far apart, or how many
units apart each adjacent gradation is. On the ruler alongside, the
labelled gradations are 10 mm apart and there are 10 intervals
between the labelled gradations. So the interval is 1 mm per
gradation.
2. Read all the certain digits of your measurement, using the
gradations on the instrument.
A ruler is an example of a scaled
measuring instrument.
3.
Estimate the uncertain digit: look at where the item being
measured is relative to the nearest gradation and write this as the
last digit in the measurement.
Learning activity 2
Significant digits and scale measurement
1. Three students are doing an investigation on the volume of cold drink in a can. They each take
measurements. Their measurements are:
333 ml; 325 ml; 345 ml.
a) Discuss why their results are different.
b) What should they do to make sure they are following a valid method for measuring the
volume?
c) How should they report their results?
2. Explain the difference between considerations in significant digits and in rounding off.
3. a) The tape measure in the photograph below has gradations in inches and eighths of an inch.
Use the three steps of scale measurement to give the measurement of the drill bit to three
significant digits.
b) The tape measure below has gradations in centimetres and millimetres. Use the three steps of
scale measurement to give the measurement of the drill bit to three significant digits.
4.
An imaginary scale measuring instrument measures units called torons. Each toron is composed
of 10 units between them called bit-torons, shown as smaller gradations on the scale.
a) If you read 23 torons and 4 bit-torons using the gradations on the scale, and estimate that the
measurement is three-fifths of the way between 4 bit-torons and 5 bit-torons, what will your
reading be and how many significant digits do you have?
b) If you read 10 torons on the scale and estimate that the measurement lines up exactly with
the 5 bit-toron gradation, what are your certain digits and what is your estimated digit?
Module 2
23