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Mr. McDermott
Fractions & Decimals 31/8

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Recap:
Yesterday we looked at two types of decimals:
Recurring & Terminating
Look back at the definitions in your workbooks if
you can’t remember the difference!
These types of decimals represent a set of
numbers called RATIONAL numbers.
There is also a 3rd type of decimal: Non-
Terminating which represent a very special set of
numbers we call IRRATIONAL
This will be our focus for today :)

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Warm Up:
Make the target number
Use all four sets of numbers to find
the target
100 2
75 3
1 10
356
75 3
1 8
25 2
100 3
6 8
50 4
100 1
6 7
50 4
1. You can add, subtract, multiply and divide.
2. You can use as many or as little of the numbers as you want.
3. Each number can only be used once.

Types of numbers
Please take some notes in your workbooks as the definitions for
each of these number classifications appear (include a couple of
examples for each as well)
Natural numbers –
Real numbers -
Integers -
Rational numbers -
Irrational numbers -
Square numbers -
Cube numbers -
Prime numbers -

Types of numbers

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LI: WALT Investigate the concept
of irrational numbers, including
π
SC
3
SC
2
SC
1
Terms
Learn the
special
names given
to different
types of
numbers.
Calculate
Calculate
and
identify
square and
cube
numbers.
Advanced
Identify
prime
numbers.

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Learn the special names given to different types of
numbers.
Real
Integer
Rational
Natural
These are the counting
numbers
Includes all the
whole numbers and
zero
Includes all the
numbers you currently
know
A number that can be
expressed as a ratio
of two integers
(Recurring or
Terminating)
There are five number types (apart from whole numbers) that we
need to know at this level.
0
Irrational
Irrational numbers
cannot be expressed as
a ratio
of two integers.
(Always Non-
Terminating)
Natural Numbers
0, 1, 2, 3, 4, …
Real Numbers
3, 1 .62, -3.75,
4
5
, 𝜋
Integer
s
…, -4, -3, -2, -1, 0, 1, 2, …
Rational
Numbers
7 , 3.65, 0.0024, -0.24,
0.333333…
Irrational
Numbers
2 , 𝜋,
5
0
, 𝑒

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Where do the
numbers go?
B e f o r e c l i c k i n g , s e e i f y
g u e s s w h i c h c a t e g o r y e a c h
n u m b e r b e l o n g s i n .
2
𝜋
−7
1
2
−
5
23
−1 0.45 3
What does this mean?
2 𝑎𝑛𝑑 𝜋 are real and
irrational
−
5
23
,
1
2
𝑎𝑛𝑑 0.45 are
real and rational
−7 𝑎𝑛𝑑 − 1 are real,
rational and integers
3 is real, rational, an
integer and a natural
number

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Quiz – Question
1
Which of these sets of numbers
contains no rational numbers?
12,
1
5
, −15
A
C
D
B −3.401, 99, 0.2345 …
−8, − 169, 3
2
5
225, 0, 0.45
The … means that the numbers continue,
therefore this number is not rational
Attempt to answer all quiz
questions in your book before
clicking to reveal answer

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Quiz –
Question 2
What is the best classification
for - 6?
A
C
D
B
Irrational number, real number
Integer, rational number, real
number
Whole number, integer, real
number
Rational number, real number

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Quiz – Question
3
Which of the following square
roots is an irrational number?
A
C
D
B
− 18
81
12
1
169
Because, the root of 12 is 3.46410…
A, B and C have rational answers.

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Quiz – Question 4
Given the following numbers,
identify the irrational numbers.
A
C
D
B
34
5
3.64
0
E −3.444
F
5
9
G −4
H 2𝜋

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Quiz – Question 5
The number 0.7 belongs to which
of these sets? Name all that
apply.
A
C
D
B
Natural
Whole
Rational
E
Irrational
F
Integer
Real

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Calculate and identify square and cube numbers.
What are they? How do we calculate them?
When a number is multiplied by itself you get a
square number
The name "square
number" comes from
the fact
that these
particular numbers
of objects can be
arranged to fill a
perfect square.
The
symbol for squared is
². So, 3 squared can
be written as 32
12 = 1 × 1 = 1
22
= 2 × 2 = 4
52 = 5 × 5 = 25
This means that
1, 4 and 25 are
square numbers.
Can you think of
any more?

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Calculate and identify square and cube numbers.
What are they? How do we calculate them?
A cube number is a number multiplied by itself 3
times
The name “cube
number" comes from
the fact
that these
particular numbers
of objects can be
arranged to fill a
perfect cube.
The
symbol for cubed is 3.
So, 3 cubed can be
written as 33
13 = 1 × 1 × 1 = 1
33 = 3 × 3 × 3 = 27
This means that
1 and 27 are
cube numbers.
Can you think of
any more?

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Quiz – Question
1
What is 52?
10
A
C
D
B 125
25
50

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Quiz – Question
2
What is 43?
16
A
C
D
B 64
12
512

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Quiz – Question
3
Which of these numbers are
square numbers?
9
A
C
D
B 36
18
86
E 110
F 81
G 256
H 75

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Quiz – Question
4
Which of these numbers are cube
numbers?
100
A
C
D
B 144
125
27
E 343
F 81
G 2150
H 8000

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Quiz – Question
5
Which of these numbers are
neither square or cube numbers?
169
A
C
D
B 85
729
196
E 156
F 1
G 8
H 4913

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Identify prime numbers.
What are they? How do we find them?
A prime number is a natural number greater than 1
that is not a product of two smaller natural
numbers. I.e. it can only be divided by 1 and itself
without leaving a remainder.
Is the number 25 a prime
number?
Is there any number that divides into 25 without
leaving a remainder?
25 ÷ 5 = 5
No, 25 cannot be a prime
number.

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Identify prime numbers.
What are they? How do we find them?
A prime number is a natural number greater than 1
that is not a product of two smaller natural
numbers. I.e. it can only be divided by 1 and itself
without leaving a remainder.
Is the number 9 a prime
number?
Is there any number that divides into 9 without
leaving a remainder?
9 ÷ 3 = 3
No, 9 cannot be a prime
number.

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Identify prime numbers.
What are they? How do we find them?
A prime number is a natural number greater than 1
that is not a product of two smaller natural
numbers. I.e. it can only be divided by 1 and itself
without leaving a remainder.
Is the number 7 a prime
number?
Is there any number that divides into 7 without
leaving a remainder?
𝐼 𝑐𝑎𝑛′
𝑡 𝑡ℎ𝑖𝑛𝑘 𝑜𝑓 𝑎𝑛𝑦
Yes, 7 must be a prime number.

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Sieve of Eratosthenes
276 BC
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Eratosthenes was a Greek born in
Cyrene in modern-day Libya. As a
mathematician, poet, athlete,
geographer, astronomer, and music
theorist.
He invented the Sieve of
Eratosthenes, an algorithm for
finding prime numbers still used
today. He sketched the course of the
Nile River and correctly predicted
its source. He also may have
accurately calculated the distance
from the Earth to the Sun and
calculated the circumference of the
Earth with stunning accuracy and
Steps:
1. Shade in 1 because 1 is not a
prime.
2. Leave 2 and shade in all
multiples of 2. (2, 4, 6, 8, 10,
...)
3. Leave 3 and shade in all
multiples of 3. ...
4. Leave 5 and shade in all
multiples of 5. ...
5. Continue this process until
finished.
Prime numbers to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, and 97.

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Test Your Knowled
Before clicking see if
you can Identify which
numbers go where.
(Numbers can go in more than
one place)
Platinum
15 – 18 correct
Gold
Silver
Bronze
Natural Numbers
Integer
s
Prime Numbers
Square Numbers
Cube Numbers
Rational
Numbers
−6 7
144 216
−0.32
5
9
5
−6
−6
7 does not go with any of these as it is
irrational.
−0.32
−0.32
144 144
144
144
216
216
216
216 5
9
5
5
5
5
10 – 14 correct
5 – 9 correct
1 – 4 correct

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Early Finishers
Be mindful of those around you
but feel free to watch this
awesome video which
demonstrates my personal
favourite irrational number:
The Golden Ratio also known as
the divine proportion.
https://www.youtube.com/watch?v
=sj8Sg8qnjOg