Download Maths basics - NSW Department of Education

Maths basics
Introduction to mathematical language
3
The basic operations of mathematics
3
Another operation: Squaring
5
Words that indicate a number
7
Some other common words
9
Order of operations
10
A look at the number line
12
Negative numbers
13
Suggested answers to activities
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Introduction to mathematical language
Once the early mathematicians had developed their Number System,
they were able to efficiently calculate. The next step was to develop
mathematical language to describe the type of calculation being performed.
Today, a combination of both words and symbols are used to describe a
calculation and, as long as any given word or symbol means the same thing
each time it is used, there can be no confusion.
Symbols are a shorthand way of writing a mathematical instruction.
For example, most people know the four basic operations in mathematics
are addition, subtraction, multiplication and division. Every hand-held
calculator shows these basic operations using the symbols +, –,  and .
The basic operations of mathematics
The symbols +, –, ,  are the building blocks for arithmetic. Let’s look at
their meanings.

–
+

add
subtract
multiply
divide
sum of
take away
times
share
increase
decrease
product
quotient
total
difference
between
groups of
plus
lots of
Addition
‘Add 20 and 12’ means ‘increase 20 by 12’ and is written 20 + 12 = 32.
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Subtraction
Subtraction reverses the addition process. So if we start with the answer
(32) and subtract 12, we arrive back at our original starting point (20).
Thus, ‘subtract 12 from 32’ means ‘decrease 32 by 12’ and is written
32 – 12 = 20.
Multiplication
Mutiplication is just a shortcut to performing many additions. For example,
Daniel needs 8 lengths of timber, each measuring 40 cm, for his wood
project. In order to calculate how much timber he needs to buy, he worked
out the following.
40 + 40 + 40 + 40 + 40 + 40 + 40 +40 = 320.
So Daniel should buy 320 cm of timber.
Obviously, Daniel did his calculation the hard way. It would have been
much more efficient for him to perform a multiplication.
40 cm  8 = 320 cm
Division
As you might expect, division and multiplication are opposite operations as
well. In other words, division reverses the multiplication process. So if we
start with the answer (320) and divide by 8, we again arrive back at our
original starting point (40).
Thus ‘divide 320 by 8’ means the same as ‘share 320 between 8’ or ‘divide
8 into 320’. It is written as follows.
4

Subtraction is the opposite of addition.

Multiplication is continuous addition.

Division is the opposite of multiplication.
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Activity 2A
Translate each of these word sentences into a number sentence, then use your calculator to
find the answer. The first one has been done for you.
Word sentence
Number sentence
Answer
(a) add 16, 85 and 9
16 + 85 + 9 =
110
(b) subtract 28 from 72
(c) multiply 12 by 7
(d) find the sum of 36 and 66
(e) find the quotient of 45 and 9
(f) total 101, 1001 and 10 001
(g) share 87 among 3
(h) increase 265 by 58
(i) find the product of 15 and 5
(j) find the difference between 92 and 29
(k) what are 12 groups of 8?
(l) by how much does 36 exceed 9?
Check your answers against those at the end of the topic.
Another operation: Squaring
We have seen that the four basic operations in mathematics are +, –,  and .
A fifth operation, which is really just an extension of ‘multiply’, is so
important that we should mention it here.
This operation is the squaring operation, which means ‘multiply a number
by itself’.
So 3 squared = 3  3 = 9
7 squared = 7  7 = 49
The symbol for the squaring operation makes use of index notation, which
will be studied in more detail in Topic 4.
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Because squaring a number means two identical numbers are multiplied
together, it is written as follows.
The opposite operation to squaring
We have seen that:
The opposite of + is –
The opposite of – is +
The opposite of  is 
The opposite of  is 
And
the opposite of ‘square’ is ‘square root’.
The symbol that we use for ‘square root’ is
.

Squaring a number means multiplying it by itself.

The opposite of square is square root.

To find the square root, think ‘what number do I multiply by itself to get this?’
Activity 2B
Without the use of a calculator find:
1
62
2
22
3
92
4
202
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1002
6
7
8
9
10
Check your answers against those at the end of the topic.
Words that indicate a number
Our everyday language includes many words which indicate a certain
operation or number. For example:
1

double and twice mean ‘multiply by 2’

triple and treble mean ‘multiply by 3’

a score is 20

a dozen is 12.
It is said that most humans live for ‘three score and ten’. What does this mean?
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2
What does ‘century’ mean? Think of two examples where it is used.
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3
What is a baker’s dozen?
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3  20 + 10 = 60 + 10 = 70 years.
2
A century means one hundred (100). We use ‘century’ when referring to
100 years and in cricket ‘a century’ is 100 runs.
3
A baker’s dozen refers to 12 + 1 = 13. For example, you may want to
buy a dozen bread rolls, but the baker gives you an extra one for the
price of 12.
Our everyday language also includes many words that use prefixes to
indicate a certain number. For example, the prefix ‘bi-’ means two (so a
bicycle is a two-wheeled machine) and ‘deca-’ means ten (so a decade is a
period of 10 years) and so on.
Note: A ‘prefix’ is something that comes before a word.
Use your dictionary to find the difference between ‘biannual’ and ‘biennial.
Activity 2C
Rule a line between each prefix and its number meaning. You may find a dictionary
helpful.
Check your answers against those at the end of the topic.
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Some other common words
Some words are used so commonly in mathematics that we should not
progress too much further without mentioning them.
Average
To find the average of a group of numbers, add them all up
and divide by how many numbers there are in the group.
Example: The average of 4, 5, 7 and 8 is:
Evaluate
Evaluate means (just as it sounds) ‘find the value of”.
Simplify
Simplify means that your answer should end up looking
simpler than the question.
Some hidden meanings
Every language has its hidden messages and meanings which may not be
obvious at first sight. In the English language, for example, there are certain
rules about where to use full stops, capital letters, and apostrophes, and there
are customs about how to say different words. We gradually learn the rules
and customs by speaking, reading and writing.
The language of mathematics is no different. Certain rules must always be
followed by mathematicians all around the world so that there can be no
confusion about what is meant.
Here is an example.
Evaluate 2 + 3  5.
This calculation could be approached in two ways.

Add 2 + 3 first (to get 5), then multiply by 5, giving a final answer
of 25.

Mutiply 3  5 first (to get 15), then add 2, giving a final answer of 17.
As it happens, only one of these is right, much to the disappointment of
mathematics students over the years. Let’s look at the question of order.
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Order of operations
Calculations with several operations to be performed have to be handled in
the correct order. That order is shown in the flowchart below.
1
Do operations in grouping symbols.
Grouping symbols can be ( ) or [ ] or { }.

2
Do multiplications and divisions from left to right.

3
Do additions and subtractions from left to right.
Example 1
20  (2 + 3)
= 20  5
= 100
(first, do the operation in grouping symbols)
(next do the multiplication)
Example 2
30  2 – 3  4
= 15 – 12
= 3
(do the multiplication and division from left to right)
(do the subtraction)
Example 3
80  10  4  2
= 842
= 22
= 1
(all the operations are divisions so work from left to right)
Example 4
Note: Questions like the last one use the fraction bar as a grouping symbol.
Sometimes, more than one set of grouping symbols may occur in the same
calculation. In this case, the inner grouping symbols are worked first.
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Example 5
3 + [10 – (4 + 3)]
= 3 + [10 – 7]
= 3+3
= 6
(inner grouping symbols worked first)
(other grouping symbols then worked)
Activity 2D
Questions 1–19
Simplify each of the following:
1
15 – 4 – 5 – 3
2
120  6  2  2
3
20 – (29 – 17)
4
(230 + 170)  10  5
5
6
6  (9 + 6)  3
7
100 – 40  2
8
32 + 42
9
10  4 + 10  5
10 10 – 6  3 + 3
11 10 – 4  2  2
12 10 – (4+ 2)  2
13 (10 – 4) + 2  2
14 3  42
15
16 (8 + 3)  (8 – 4)
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17 (5 + 2)  (8  8)
18 4  [12 – (6 + 1)]
19 [20 – (4 + 2)]  2
Questions 20–24
Put in grouping symbols where necessary to make each of the following statements true:
20 15 – 3  5 = 60
21 8 + 4 – 3  2 = 10
22 8 + 4 – 3  2 = 6
23 8 + 4 – 3  2 = 18
24 24  4  3  2 = 1
Challenge
Using four 4’s and any mathematical operations we can get an answer of 1.
For example: 4  4  4  4 = 1
Now can you use four 4’s and any mathematical operations to get an answer of 7?
Check your answers against those at the end of the topic.
A look at the number line
If we were to try to map the types of numbers we have met so far, we would
end up with a number line like this.
Notice that:

the numbers are spaced at equal distances.

the arrow on the end of the number line is used to show that it continues
forever.
The number line above shows ‘the set of counting numbers’, in other words,
the numbers we would use if we were counting.
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There are many more numbers than these. In fact, as you will soon see, there
are an infinite number of numbers! Let’s explore some of them, now.
Negative numbers
Consider these questions.
1
Adam’s cheque account had a balance of only $230 when he wrote a cheque for $250
to pay for two new tyres. What is the balance of his account now?
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2
At 5 am one winter’s day at Cowra the temperature was 2ºC. By 6 am, the temperature
had fallen 3º. What was the temperature then?
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$20 overdrawn ($ – 20)
2
1ºC below zero ( – 1ºC).
These situations show the need for negative numbers, or numbers less than
zero. On the number line, negative numbers are shown to the left of zero.
When we travel around the number line we move to the right for positive
numbers and to the left for negatives. This is as easy as moving east and
west across a map or up and down on a thermometer.
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Activity 2E
1
(a) At 4 am the temperature was –2ºC. Mark this on the thermometer below.
(b) By 6 am the temperature had fallen another degree. What’s the temperature now?
(c) Over the next two hours, the temperature rose 2 degrees. What was the
temperature at 8 am?
(d) By midday, the temperature was 9ºC. By how many degrees did the temperature
rise between 8 am and midday?
2
Look at this diagram of a department store.
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If ground level is floor 0, then the third floor would be + 3 (you would go up three
floors, or add three floors to your height above ground level). Parking level P2,
two floors below ground level, would be – 2.
(a) Mary parks her car on parking level P1. She then takes the lift up three floors.
What will she be shopping for on this level?
(b) Quan works in this department store. He parks on parking level P2, then takes the
lift up two floors. What floor does Quan work on?
(c) Rhett enters the store on the ground level. He takes the lift up four floors then
down two floors, which departments does he visit?
(d) On Monday Anelise met her friend for lunch in the Skyline Café. After lunch she
took the lift down seven floors to go home. Where did she park her car?
3 Sven has an overdraft account with Starbank. His bank statement for the month of
October is shown below. Complete the balance column.
Check your answers against those at the end of the topic.
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Suggested answers to activities
Activity 2A
(b) 72 – 28 = 44
(c) 12 × 7 = 84
(d) 36 + 66 = 102
(e) 45  9 = 5
(f) 101 + 1001 + 10 001 = 11 103
(g) 87  3 = 29
(h) 265 + 58 = 323
(i) 15 × 5 = 75
(j) 92 – 29 = 63
(k) 12 × 8 = 96
(l) 36 – 9 = 27
Activity 2B
1
36
2
4
3
81
4
400
5
10 000
6
2
7
10
8
4
9
9
10 12
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Activity 2C
1
= uni
7
= hept or sept
2
= bi
8
= oct
3
= tri
9
= nona
4
= quad
10 = deca
5
= pent
12 = dodec
6
= hex or sex
Note: Did you notice that there is no prefix to indicate 11?
Activity 2D
1
3
13
7
2
5
14
48
3
8
15
6
4
800
16
44
5
13
17
7
6
30
18
20
7
20
19
7
8
25
20
(15 – 3) × 5 = 60
9
90
21
8 + (4 – 3) × 2 = 10
10
11
22
(8 + 4) – 3 × 2 = 6
11
6
23
(8 + 4 –3) × 2 = 18
12
7
24
24  (4 × 3)  2 = 1
Challenge
4+4–44=7
Note: It is, in fact, possible to get all the numbers from 0 to 10 using four
4’s in this manner. Try it!
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Activity 2E
1
(a)
(b) –3C
(c) –1C
2
(d) 10
(a) Ladies’ fashions
(b) Ground floor – shoes
(c) Manchester then Ladies Fashions
(d) P2
3
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