DAY 1:
TOPIC: WHOLE NUMBER
GRADE 9
CONCEPTS & SKILLS TO BE ACHIEVED:
By the end of the lesson learners should know and be able to:
Describe the real number system by recognizing, defining and distinguishing properties of:
Natural number
Whole number
Integer
Rational number
Irrational number
RESOURCES: DBE Workbook, Sasol-Inzalo book, Textbooks
ONLINE RESOURCES: Page 5
LESSON DEVELOPMENT
Types of Numbers in Math
Just like different persons of the same family live in different homes, different numbers are of the
same family but have different characteristics or properties. These patterns of numbers are
different from each other due to different representations and properties.
The diagram of “stack of funnels” below will help us classify numbers easily.
But first, we need to describe what kind of elements are included in each group of numbers.
Each group or set of numbers is represented by a funnel.
GRADE 9 Page 1 of 41
WHOLE NUMBERS (draft)
, The numbers that we use to count are called natural numbers:
Natural numbers have the following properties:
Property 2:
Not closed under subtraction or division
When subtraction or division is done with
natural numbers, the answers are not
Property 1: always natural numbers
Closed under addition and multiplication Example 1:
When you add two or more natural numbers, There is no natural number
you get a natural number again. that provides the answer to 5 – 20.
Example 1: Example 2:
5 + 7 = 12, 12 → a natural number there is no natural
4 + 6 = 10, 10 → a natural number number that provides the answer to
10 ÷ 3.
When you multiply two or more natural
numbers, you get a natural number again.
Example 2:
5× 7 = 35, 35 → natural number
4 × 6 = 24, 24 → natural number
The whole numbers are a slight “upgrade” of the natural numbers because we simply add the
element zero to the current set of natural numbers. Think of whole numbers as natural numbers
together with zero.
0 When 0 is added to a
number the answer is just
the
number you start with:
24 + 0 = 24.
All whole numbers are also For this reason, 0 is called
integers. The set of whole the identity element for
numbers forms part of the set addition.
of integers.
For each whole number, there
is a negative number that
corresponds with it.
In the set of whole numbers, no answer is available when you subtract a
The negative number −5 number from a number smaller than itself.
corresponds to the whole For example, there is no whole number that is the answer for 5 – 8.
number 5. But there is an answer to this subtraction in the system of integers.
For example: 5 – 8 = −3. The number –3 is read as “negative 3”
Integers extend in both directions:
GRADE 9 Page 2 of 41
WHOLE NUMBERS (draft)
,The system of integers does not provide an answer for all possible division questions.
𝟐
For example, the answer for 12 ÷ 5 = 2.4 or 𝟐 𝟓 , is not an integer.
To have answers for all possible division questions, we must extend the number
𝒊𝒏𝒕𝒆𝒈𝒆𝒓
system to include fractions and negative fractions, in other words, numbers of the form
𝒊𝒏𝒕𝒆𝒈𝒆𝒓
Caution: The denominator cannot equal zero.
This system of numbers is called rational numbers.
We can represent rational numbers as common fractions or as decimal numbers.
Rational numbers do not provide for all situations that may occur in Mathematics.
For example, there is no rational number which will produce the answer 2 when it is
multiplied by itself : (𝒏𝒖𝒎𝒃𝒆𝒓) × (𝒔𝒂𝒎𝒆 𝒏𝒖𝒎𝒃𝒆𝒓) = 𝟐
2 × 2 = 4 and 1 × 1 = 1, so clearly, this number must be between 1 and 2.
But there is no number which can be expressed as a fraction, in either the common
fraction or the decimal notation, which will solve this problem.
Numbers like these are called irrational numbers.
Here are some more examples of irrational numbers as seen below:
Nos: mean
numbers
Rational and irrational numbers together, are called real numbers.
Remember:
Under the set of rational
numbers, we have the
subcategories or subsets of
integers, whole numbers,
and natural numbers.
GRADE 9 Page 3 of 41
WHOLE NUMBERS (draft)
, CLASSWORK:
1.
Classifying Real Numbers
Identify by writing a mark on the set or sets in which the given number belongs.
o Natural o Natural o Natural
0
o Whole o Whole o Whole
5
o Integers o Integers −6 o Integers
√9
o Rational o Rational 3 o Rational
o Irrational o Irrational o Irrational
o Real o Real o Real
o Natural o Natural o Natural
o Whole o Whole o Whole
−√9 o Integers 3 o Integers o Integers
√7 𝜋
−2 o Rational o Rational o Rational
o Irrational o Irrational o Irrational
o Real o Real o Real
o Natural o Natural o Natural
o Whole o Whole o Whole
o Integers o Integers o Integers
1.5 2 0.2222…
o Rational o Rational o Rational
o Irrational o Irrational o Irrational
o Real o Real o Real
2.
CREATE IT!
Directions:
Create an example of each of the types of numbers listed below.
If it is impossible to create a number that fits the classifications, write “Not possible”.
Whole and Integer: ____________ Integer and Natural: ____________
Irrational and Natural: ____________ Whole and Natural: ____________
GRADE 9 Page 4 of 41
WHOLE NUMBERS (draft)
TOPIC: WHOLE NUMBER
GRADE 9
CONCEPTS & SKILLS TO BE ACHIEVED:
By the end of the lesson learners should know and be able to:
Describe the real number system by recognizing, defining and distinguishing properties of:
Natural number
Whole number
Integer
Rational number
Irrational number
RESOURCES: DBE Workbook, Sasol-Inzalo book, Textbooks
ONLINE RESOURCES: Page 5
LESSON DEVELOPMENT
Types of Numbers in Math
Just like different persons of the same family live in different homes, different numbers are of the
same family but have different characteristics or properties. These patterns of numbers are
different from each other due to different representations and properties.
The diagram of “stack of funnels” below will help us classify numbers easily.
But first, we need to describe what kind of elements are included in each group of numbers.
Each group or set of numbers is represented by a funnel.
GRADE 9 Page 1 of 41
WHOLE NUMBERS (draft)
, The numbers that we use to count are called natural numbers:
Natural numbers have the following properties:
Property 2:
Not closed under subtraction or division
When subtraction or division is done with
natural numbers, the answers are not
Property 1: always natural numbers
Closed under addition and multiplication Example 1:
When you add two or more natural numbers, There is no natural number
you get a natural number again. that provides the answer to 5 – 20.
Example 1: Example 2:
5 + 7 = 12, 12 → a natural number there is no natural
4 + 6 = 10, 10 → a natural number number that provides the answer to
10 ÷ 3.
When you multiply two or more natural
numbers, you get a natural number again.
Example 2:
5× 7 = 35, 35 → natural number
4 × 6 = 24, 24 → natural number
The whole numbers are a slight “upgrade” of the natural numbers because we simply add the
element zero to the current set of natural numbers. Think of whole numbers as natural numbers
together with zero.
0 When 0 is added to a
number the answer is just
the
number you start with:
24 + 0 = 24.
All whole numbers are also For this reason, 0 is called
integers. The set of whole the identity element for
numbers forms part of the set addition.
of integers.
For each whole number, there
is a negative number that
corresponds with it.
In the set of whole numbers, no answer is available when you subtract a
The negative number −5 number from a number smaller than itself.
corresponds to the whole For example, there is no whole number that is the answer for 5 – 8.
number 5. But there is an answer to this subtraction in the system of integers.
For example: 5 – 8 = −3. The number –3 is read as “negative 3”
Integers extend in both directions:
GRADE 9 Page 2 of 41
WHOLE NUMBERS (draft)
,The system of integers does not provide an answer for all possible division questions.
𝟐
For example, the answer for 12 ÷ 5 = 2.4 or 𝟐 𝟓 , is not an integer.
To have answers for all possible division questions, we must extend the number
𝒊𝒏𝒕𝒆𝒈𝒆𝒓
system to include fractions and negative fractions, in other words, numbers of the form
𝒊𝒏𝒕𝒆𝒈𝒆𝒓
Caution: The denominator cannot equal zero.
This system of numbers is called rational numbers.
We can represent rational numbers as common fractions or as decimal numbers.
Rational numbers do not provide for all situations that may occur in Mathematics.
For example, there is no rational number which will produce the answer 2 when it is
multiplied by itself : (𝒏𝒖𝒎𝒃𝒆𝒓) × (𝒔𝒂𝒎𝒆 𝒏𝒖𝒎𝒃𝒆𝒓) = 𝟐
2 × 2 = 4 and 1 × 1 = 1, so clearly, this number must be between 1 and 2.
But there is no number which can be expressed as a fraction, in either the common
fraction or the decimal notation, which will solve this problem.
Numbers like these are called irrational numbers.
Here are some more examples of irrational numbers as seen below:
Nos: mean
numbers
Rational and irrational numbers together, are called real numbers.
Remember:
Under the set of rational
numbers, we have the
subcategories or subsets of
integers, whole numbers,
and natural numbers.
GRADE 9 Page 3 of 41
WHOLE NUMBERS (draft)
, CLASSWORK:
1.
Classifying Real Numbers
Identify by writing a mark on the set or sets in which the given number belongs.
o Natural o Natural o Natural
0
o Whole o Whole o Whole
5
o Integers o Integers −6 o Integers
√9
o Rational o Rational 3 o Rational
o Irrational o Irrational o Irrational
o Real o Real o Real
o Natural o Natural o Natural
o Whole o Whole o Whole
−√9 o Integers 3 o Integers o Integers
√7 𝜋
−2 o Rational o Rational o Rational
o Irrational o Irrational o Irrational
o Real o Real o Real
o Natural o Natural o Natural
o Whole o Whole o Whole
o Integers o Integers o Integers
1.5 2 0.2222…
o Rational o Rational o Rational
o Irrational o Irrational o Irrational
o Real o Real o Real
2.
CREATE IT!
Directions:
Create an example of each of the types of numbers listed below.
If it is impossible to create a number that fits the classifications, write “Not possible”.
Whole and Integer: ____________ Integer and Natural: ____________
Irrational and Natural: ____________ Whole and Natural: ____________
GRADE 9 Page 4 of 41
WHOLE NUMBERS (draft)
