UNIT 1:
REVIEWING
NUMBERS
CONCEPTS
CORE AND EXTENDED
MATHEMATICS
QUARTER 1
MS TRIXIE
TUESDAY, AUGUST 15, 2023
WHAT ARE REAL NUMBERS?
Real numbers are numbers that can be found on the number line. These are the numbers
that we normally use and apply in real-world applications.
NUMBERS IN THE REAL NUMBER
SYSTEM
(Do a research on the types of numbers and then write a definition
for each in your words.)
 Natural numbers
 Whole numbers
 Integers
 Rational numbers
 Irrational numbers
MATCH THE CORRECT DEFINITION TO THE CORRECT TERM.
Definitions
1.
This is the set of negative and positive whole numbers including
zero.
2.
This is the set of counting numbers.
3.
This is the set of numbers that cannot be written as a fraction or a
decimal.
4.
This is the set of natural numbers including zero.
5.
This is the set of numbers that can be written as a fraction or a
decimal.
Terms
 Natural numbers
 Whole numbers
 Integers
 Rational numbers
 Irrational numbers
From the set of numbers below write all the numbers that belong to the different sets of real
numbers
2
-57, −2 , -1/2, -1, 0, 1/3, 1, 3.7, 5, 8 , 9, 13.333…, �, 3, 9
3
4
o Natural numbers:
3
o Whole numbers:
o Integers:
o Rational numbers:
o Irrational numbers:
What relationship(s) exist between the different set of numbers?
 All natural numbers are whole numbers.
 All natural numbers and whole numbers are integers.
SUMMARY OF
THE REAL
NUMBER
SYSTEM
 All natural numbers, whole numbers and integers are
rational numbers.
 A number is either rational or irrational (it cannot be both).
 Natural numbers, whole numbers, integers, rational and
irrational numbers are all real number.
VENN DIAGRAM OF THE REAL NUMBER SYSTEM
Even
numbers
Odd
numbers
OTHER TYPES
OF NUMBERS
Prime
numbers
Composite
numbers
THE SET OF EVEN NUMBERS
The set of even numbers consist of natural numbers that can be exactly
divided by 2.
So the set of even numbers consist of numbers that are multiples of two.
Hence even numbers are natural numbers ending with the digits 0, 2, 4, 6, or
8.
For example 30, 12, 24, 56, and 78.
THE SET OF ODD NUMBERS
The set of odd numbers consist of natural numbers that cannot be exactly divided by 2.
The set of odd numbers consist of counting numbers that are not even.
Hence, natural numbers ending with the digits 1, 3, 5, 7 or 9.
For example 21, 53, 65, 87, and 99.
THE SET OF PRIME NUMBERS
 A prime number is a number which can only be divided exactly by itself and 1.
 Prime numbers have only two factors.
THE SET OF COMPOSITE NUMBERS
 A composite number is a number which has other factors besides itself and 1.
 Composite numbers have more than 2 factors.
COURSEBOOK PRACTICE PAGE 2
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Starter
Using the numbers from the green hexagon, find
pairs that multiply together to give the totals below.
You must only use each number once.
a) 14
b) 24
c) 30
d) 16
4
7
2 3
8
6
4
5
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Multiple
A multiple is a number that appears in another number’s times table.
Q1 What are the first 5 multiples of 8?
8
16
24
32
40
Q2 Find the first 6 multiples of the following numbers:
9
6
7
9
6
7
18
12
14
27
18
21
36
24
28
45
30
35
54
36
42
Q3 Which of the following numbers are multiples of 4?
3
12
8
26
20
22
9
40
36
18
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q1
Is 20 a multiple of 4? Yes
Q2
Is 16 a multiple of 6? No
Q3
The next multiple of 5 after 20 is 30. True or False? False
Q4
20 is a multiple of both 5 and 10. True or False? True
Q5
7 is a multiple of 14. True or False? False
Q6
Write down the next two multiples of 4 after 32. 36
Q7
The multiples of 20 include 40 and 80. True or False? True
Q8
2000 is a multiple of both 10 and 100. True or False?
Q9
30 is a multiple of 2, 5, 6 and 30. True or False?
40
True
True
Q10 What is a multiple? A number that appears in another number’s times table.
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Factor
A factor is a number which divides exactly into another number.
Q1 List all the factors of 12.
1
2
3
4
6
12
Q2 Find the all of the factors of the following numbers:
9
24
15
1
1
1
3
2
3
9
3
5
4
15
6
8
12
24
Q3 Which of the following numbers are factors of 30?
30
12
5
4
6
20
10
2
8
1
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q1
Is 6 a factor of 12? Yes
Q2
Is 20 a factor of 30?
Q3
4 is a factor of 12, 16 and 20. True or False? True
Q4
11 is a factor of 55. True or False? True
Q5
List all of the factors of 11. 1
Q6
10 4 × ____
20 2 × ____
5
Complete these pairs of factors of 20. 1 × ____
Q7
Write a list of factors of 40. 1
Q8
Which of the following numbers are not factors of 36?
5
Q9
6
No
7
8
11
2
9
4
5
8
10
20
40
10
What is a factor? A number which divides exactly into another number.
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
The prime numbers that are multiplied to make
up another number are called prime factors.
Q1
We can find these using
a prime factor tree.
Write 30 as a product of its prime factors.
30
3
Remember to circle
the prime numbers
at every step!
10
2
5
30 = 2 × 3 × 5
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q2
Write 24 as a product of its prime factors.
24
2
Remember to circle
the prime numbers
at every step!
12
6
2
2
24 = 2 × 2 × 2 × 3
= 23 × 3
3
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q3
Write 36 as a product of its prime factors.
36
2
6
6
3
2
36 = 2 × 3 × 2 × 3
Remember to circle
the prime numbers
at every step!
3
= 22 × 32
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q5
Write the following numbers as products of their prime factors.
120
80
45
72
187
120 = 2 × 2 × 2 × 3 × 5
= 23 × 3 × 5
80 = 2 × 2 × 2 × 2 × 5
= 24 × 5
72 = 2 × 2 × 2 × 3 × 3
= 23 × 32
45 = 3 × 3 × 5
187 = 11 × 17
= 32 × 5
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Using the prime factors that you have worked out, can you find
the highest number that divides exactly into both 120 and 72?
120 = 2 × 2 × 2 × 3 × 5
72 = 2 × 2 × 2 × 3 × 3
2 × 2 × 2 × 3 = 24
120 ÷ 24 = 5
72 ÷ 24 = 3
24 is the highest common factor (HCF) of 120 and 72.
Highest common factor
The highest common factor of given numbers is the largest number that
exactly divides into them.
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q1
Find the highest common factor of 16 and 40.
16
2
40
8
4
2
2
16 = 2 × 2 × 2 × 2
2
2
4
10
2
2
5
40 = 2 × 2 × 2 × 5
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q1
Find the highest common factor of 16 and 40.
16 = 2 × 2 × 2 × 2
16 ÷ 8 = 2
40 = 2 × 2 × 2 × 5
2× 2× 2 =8
40 ÷ 8 = 5
8 is the highest common factor (HCF) of 16 and 40.
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q2
Find the HCF of the following numbers.
12 and 15
3
24 and 36
12
45 and 30
15
55 and 105
5
36 and 72 and 120
12
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Lowest common multiple
The lowest common multiple of given numbers is the smallest number that
appears in their times tables.
Q1
Find the lowest common multiple of 16 and 40.
16 = 2 × 2 × 2 × 2
40 = 2 × 2 × 2 × 5
2 × 2 × 2 × 2 × 5 = 80
80 ÷ 16 = 5
80 ÷ 40 = 2
80 is the lowest common multiple (LCM) of 16 and 40
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.
Key words: Common, divide, divisor, factor, highest, lowest, multiple, multiply, prime, product, power
Q2
Find the LCM of the following numbers using your previous answers.
12 and 15
60
24 and 36
72
45 and 30
90
55 and 105
1155
36 and 72 and 120
360
Objectives
• Developing learners will be able to explain the terms multiple, factor and prime number.
• Secure learners will be able to identify the prime factors of a given number.
• Excelling learners will be able to evaluate the highest common factor (HCF) and lowest common
multiple (LCM) of given numbers.