rational numbers class 8 icse ml aggarwal

1. Rational numbers
Chapter 1
Class 8 - Math 1

2. CHAPTER 1
RATIONAL NUMBERS

3. INTRODUCTION
Why are there so many numbers?
Video Tutorial - https://www.youtube.com/watch?v=XN2Cj8GuVbY
Irrational - 22 trillion digits of pi (𝝅)
Visit the page to see the digits - https://pi2e.ch/blog/2017/03/10/pi-digits-download/
Why rational numbers? (Real life applications)
 http://explainingmath.blogspot.com/2011/04/real-world-examples-for-rational.html

4. Children will be able to:
Describe properties of rational numbers & express them in general form
Consolidate operations on rational numbers
Represent rational numbers on a number line
Understand that between any two rational numbers there lies another rational
number (unlike whole numbers)
Generalize & verify properties of rational numbers (including identities)
Apply the logic on word problems
Objectives

7. A rational number is any number that can be expressed as a ratio of two integers
It can be written as a fraction

8. Rational numbers

9. Standard Form of Rational Numbers
12/36 is a rational number.
But it can be simplified as 1/3;
common factors between the divisor and dividend is only one.
So we can say that rational number ⅓ is in standard form.

10. What are rational numbers? https://www.youtube.com/watch?v=SQ4cB9yXkHM
Examples of rational numbers & their decimal representation will be shown.
Decimal representation of irrational numbers will also be shown here to distinguish rational numbers from them.
Activity – Rational Maze
Description: A square maze will be given in a paper for each team. Each block in the maze contains numbers
(rational & irrational numbers)
Instructions to Students:
 Students should sit in four groups, each group will receive a maze
 The name of the group should be written on the paper given to them
 The activity should begin only after the teacher instructs them to start
 Students should identify only rational numbers and highlight (shade with highlighter) or circle with pen.
 Connect all the highlighted/circled blocks in order to form a route from start to end points in the maze
Instructions to Teacher: (File link : ..Chapter 1 Rational NumbersRational_Maze_HW_Half copies.docx
 Teacher should have the printout of all four different set of mazes in four different papers
 Each set will be distributed to each group & each group will be named/numbered
 Each group will use a different color to highlight or circle
 The instructions to solve the maze should be given clearly, only after which the students should begin
 The group which completes first gets a tally (to be updated in tally chart)
DELIVERY OF LESSON

11. Closure property of Rational numbers
What does it mean? https://www.youtube.com/watch?v=F0L2FENoJOo
Problems related to closure property verification will be given to students to solve as homework
Why a number divided by zero is UNDEFINED?
Watch to know: https://www.youtube.com/watch?v=J2z5uzqxJNU
Commutative, Associative & Distributive properties, Identity & Inverse (Additive & Multiplicative)
Students will be questioned for which operation these properties would fail
On the green board, teacher will be giving examples demonstrating each property over each operation
Rational numbers on a Number line
Watch to know how? - https://youtu.be/G9n8HbMdUIk?t=2300
On the green board, teacher will demonstrate how to plot rational numbers on the number line.
Word document (link) will displayed on smart board for the students to finish the exercise on plotting rational numbers.
Students completing all the questions will be rewarded a tally
DELIVERY OF LESSON

12. Compare & Order rational numbers
Watch the video to know how:
Worksheet will be given for the students to solve. (worksheet link - print to be taken)
Students completing the worksheet will be given a tally
Rational numbers between two rational numbers
Watch the video to know how: https://www.youtube.com/watch?v=lg04THe8wfY
On the green board, teacher will consolidate methods to find single or many rational number for same or different
denominators with examples
Word document (link) will be displayed for the students to solve. Students completing all the questions will be given a tally
Chapter Practice Worksheet
Worksheet with all the topics on rational numbers will be provided for the students to solve in class.
Teacher may help/clarify during this practice session.
Students completing the worksheet will be given a tally.
DELIVERY OF LESSON

13. ADDITIONAL LEARNING OPPORTUNITY
 Why is 𝜋 so irrational? Watch to know: https://www.youtube.com/watch?v=HSuqbqENIek.
 Why scientists keep computing digits of pi?
 History of 𝜋 - https://www.youtube.com/watch?v=1-JAx3nUwms
 What to do with the digits of 𝜋? - https://youtu.be/zhqdIhYvFxU
 Digits of 𝜋 & uses - What to do with the digits of π.docx

16. Subtracting Two Rational Numbers with different
denominators

17. Multiplication of Rational Numbers :-

19. Operations on rational numbers and their properties
There are some properties of operations on rational numbers.
They are closure,
commutative,
associative,
identity,
inverse
and distributive.

20. Addition of rational numbers and their properties
 Closure: If a/b and c/d are any two rational numbers, then (a/b) + (c/d) is also a rational number.
Example :
2/9 + 4/9 = 6/9 = 2/3 is a rational number.
 Commutative: Addition of two rational numbers is commutative.
If a/b and c/d are any two rational numbers,
then (a/b) + (c/d) = (c/d) + (a/b)
Example :
2/9 + 4/9 = 6/9 = 2/3
4/9 + 2/9 = 6/9 = 2/3
Hence, 2/9 + 4/9 = 4/9 + 2/9
 Associative: Addition of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers,
then a/b + (c/d + e/f) = (a/b + c/d) + e/f
Example :
2/9 + (4/9 + 1/9) = 2/9 + 5/9 = 7/9
(2/9 + 4/9) + 1/9 = 6/9 + 1/9 = 7/9
Hence, 2/9 + (4/9 + 1/9) = (2/9 + 4/9) + 1/9

21. Addition of rational numbers and their properties
 Additive identity : The sum of any rational number and zero is the rational number itself.
If a/b is any rational number,
then a/b + 0 = 0 + a/b = a/b
Zero is the additive identity for rational numbers.
Example :
2/7 + 0 = 0 + 2/7 = 27
 Additive inverse : (- a/b) is the negative or additive inverse of (a/b)
If a/b is a rational number,then there exists a rational number (-a/b) such that
a/b + (-a/b) = (-a/b) + a/b = 0
Example :
Additive inverse of 3/5 is (-3/5)
Additive inverse of (-3/5) is 3/5
Additive inverse of 0 is 0 itself.

22. If a/b, c/d and e/f are any three rational numbers,
then a/b x (c/d + e/f) = a/b x c/d + a/b x e/f
Example :
1/3 x (2/5 + 1/5) = 1/3 x 3/5 = 1/5
1/3 x (2/5 + 1/5) = 1/3 x 2/5 + 1/3 x 1/5 = (2 + 1) / 15 = 1/5
Hence, 1/3 x (2/5 + 1/5) = 1/3 x 2/5 + 1/3 x 1/5
Therefore, Multiplication is distributive over addition.
Multiplication is distributive over addition.
 Multiplication is distributive over addition.

23. Subtraction of rational numbers and their properties
(i) Closure Property :
The difference between any two rational numbers is always a rational number.
Hence Q is closed under subtraction.
If a/b and c/d are any two rational numbers, then
(a/b) - (c/d) is also a rational number.
Example :
5/9 - 2/9 = 3/9 = 1/3 is a rational number.
(ii) Commutative Property :
Subtraction of two rational numbers is not commutative.
If a/b and c/d are any two rational numbers, then
(a/b) - (c/d) ≠ (c/d) - (a/b)
Example :
5/9 - 2/9 = 3/9 = 1/3
2/9 - 5/9 = -3/9 = -1/3
And,
5/9 - 2/9 ≠ 2/9 - 5/9
Therefore, Commutative property is not true for subtraction.

24. (iii) Associative Property :
Subtraction of rational numbers is not associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b - (c/d - e/f) ≠ (a/b - c/d) - e/f
Example :
2/9 - (4/9 - 1/9) = 2/9 - 3/9 = -1/9
(2/9 - 4/9) - 1/9 = -2/9 - 1/9 = -3/9
And,
2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9
Therefore, Associative property is not true for subtraction.
Subtraction of rational numbers and their properties

25. (ii) Distributive Property of Multiplication over Subtraction :
Multiplication of rational numbers is distributive over subtraction.
If a/b, c/d and e/f are any three rational numbers, then
a/b x (c/d - e/f) = a/b x c/d - a/b x e/f
Example :
1/3 x (2/5 - 1/5) = 1/3 x 1/5 = 1/15
1/3 x 2/5 - 1/3 x 1/5 :
= 2/15 - 1/15
= (2 - 1)/15
= 1/15 -----(2)
From (1) and (2),
1/3 x (2/5 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5
Therefore, Multiplication is distributive over subtraction.
Subtraction of rational numbers and their properties

26. Multiplication of rational numbers and their properties
(i) Closure Property :
The product of two rational numbers is always a rational number. Hence Q is closed under
multiplication.
If a/b and c/d are any two rational numbers, then
(a/b) x (c/d) = ac/bd is also a rational number.
Example :
5/9 x 2/9 = 10/81 is a rational number.
(ii) Commutative Property :
Multiplication of rational numbers is commutative.
If a/b and c/d are any two rational numbers, then
(a/b) x (c/d) = (c/d) x (a/b)
Example :
5/9 x 2/9 = 10/81
2/9 x 5/9 = 10/81
So,
5/9 x 2/9 = 2/9 x 5/9
Therefore, Commutative property is true for multiplication.

27. (iii) Associative Property :
Multiplication of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b x (c/d x e/f) = (a/b x c/d) x e/f
Example :
2/9 x (4/9 x 1/9) = 2/9 x 4/81 = 8/729
(2/9 x 4/9) x 1/9 = 8/81 x 1/9 = 8/729
So,
2/9 x (4/9 x 1/9) = (2/9 x 4/9) x 1/9
Therefore, Associative property is true for multiplication.
(iv) Multiplicative Identity :
The product of any rational number and 1 is the rational number itself. ‘One’ is
the multiplicative identity for rational numbers.
If a/b is any rational number, then
a/b x 1 = 1 x a/b = a/b
Example :
5/7 x 1 = 1 x 5/7 = 5/7
Multiplication of rational numbers and their properties

28. (v) Multiplication by 0 :
Every rational number multiplied with 0 gives 0.
If a/b is any rational number, then
a/b x 0 = 0 x a/b = 0
Example :
5/7 x 0 = 0 x 5/7 = 0
(vi) Multiplicative Inverse or Reciprocal :
For every rational number a/b, b ≠ 0, there exists a rational number c/d
such that a/b x c/d = 1. Then,
c/d is the multiplicative inverse of a/b.
If b/a is a rational number, then
a/b is the multiplicative inverse or reciprocal of it.
Example :
The multiplicative inverse of 2/3 is 3/2.
The multiplicative inverse of 1/3 is 3.
The multiplicative inverse of 3 is 1/3.
The multiplicative inverse of 1 is 1.
The multiplicative inverse of 0 is undefined.

29. Division of rational numbers and their properties
(i) Closure Property :
The collection of non-zero rational numbers is closed under division.
If a/b and c/d are two rational numbers, such that c/d ≠ 0, then
a/b ÷ c/d is always a rational number.
Example :
2/3 ÷ 1/3 = 2/3 x 3/1 = 2 is a rational number.
(ii) Commutative Property :
Division of rational numbers is not commutative.
If a/b and c/d are two rational numbers, then
a/b ÷ c/d ≠ c/d ÷ a/b
Example :
2/3 ÷ 1/3 = 2/3 x 3/1 = 2
1/3 ÷ 2/3 = 1/3 x 3/2 = 1/2
And,
2/3 ÷ 1/3 ≠ 1/3 ÷ 2/3
Therefore, Commutative property is not true for division.

30. (iii) Associative Property :
Division of rational numbers is not associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b ÷ (c/d ÷ e/f) ≠ (a/b ÷ c/d) ÷ e/f
Example :
2/9 ÷ (4/9 ÷ 1/9) = 2/9 ÷ 4 = 1/18
(2/9 ÷ 4/9) ÷ 1/9 = 1/2 - 1/9 = 7/18
And,
2/9 ÷ (4/9 ÷ 1/9) ≠ (2/9 ÷ 4/9) ÷ 1/9
Therefore, Associative property is not true for division.
Division of rational numbers and their properties

31. Add the following
1)
4
7
and
5
7
2)
7
−13
and
4
−13
3)
5
11
𝑎𝑛𝑑 4
3
9
−4
9
2
12
13
𝑎𝑛𝑑
4)

32. SUMMARY
 Rational number general form:
𝑝
𝑞
where 𝑝, 𝑞 are integers and 𝑝 ≠ 0
 All integers and fractions are rational numbers
 To find rational number between two rational numbers, calculate their average
 There are infinitely many rational numbers between any two distinct rational numbers
 Comparing and ordering of rational numbers are done by comparing their numerators after making their
denominators common (by taking LCM)
 Decimal representation of rational number can either be terminating or non-terminating recurring
Operation
Property
Addition Subtraction Multiplication Division
Closure property ✓ ✓ ✓ ❌
Commutative ✓ ❌ ✓ ❌
Associative ✓ ❌ ✓ ❌
Existence of Inverse ✓ ✓
Existence of Identity ✓ ✓

33. Rational numbers on number line
Watch to know how? - https://youtu.be/G9n8HbMdUIk?t=2300
Compare & Order rational numbers
Watch the video to know how: https://www.youtube.com/watch?v=CbrfJPv2qP8
Rational numbers between two rational numbers
Watch the video to know how: https://www.youtube.com/watch?v=lg04THe8wfY

38. Chapter test