1. Mathematics
Class VII
Chapter 1
Integers

2. Module Objectives By the end of this chapter, you will be able to
Learn the basic concepts of Integers
Learn integer addition and subtraction
Multiply positive integer by positive integer as well as negative integer
Multiply negative integer by positive integer as well as negative integer
Compare integers and identify smaller and greater among them
Follow proper method in multiplication as well as relate multiplication and division of numbers

3. Module Objectives By the end of this chapter, you will be able to
Divide an integer by another integer
Understand why an integer cannot be divisible by 0
Understand the basic properties of integers with respect to fundamental operations.

4. Welcome to Module 1

5. Definition
Positive number – a number greater than zero. 1 2 3 4 5 6

6. Definition
Negative number – a number less than zero. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

7. Definition
Opposite Numbers – numbers that are the same distance from zero in the opposite direction -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

8. Definition
Integers – Integers are all the whole numbers and all of their opposites on the negative number line including zero. 7
opposite -7

9. The absolute value of 9 or of –9 is 9.
Definition
Absolute Value – The size of a number with or without the negative sign.
The absolute value of
9 or of –9 is 9.

10. Negative Numbers Are Used to Measure Temperature

11. Negative Numbers Are Used to Measure Under Sea Level30 20 10
-10
-20
-30
-40
-50

12. Negative Numbers Are Used to Show Debt
Let’s say your parents bought a car but
had to get a loan from the bank for $5,000.
When counting all their money they add
in -$5,000 to show they still owe the bank.

13. Hint
If you don’t see a negative or positive sign in front of a number it is positive. 9 +

14. Integer Addition Rules
Rule #1 – If the signs are the same, pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.
9 + 5 = 14
= -14

15. Solve the Problems -3 + -5 = 4 + 7 = (+3) + (+4) = -6 + -7 = 5 + 9 == -8 11 7
-13 14
-18

16. Check Your Answers
= 21
2. – = -33
= 72
4. – = -49

17. Integer Addition Rules
Rule #2 – If the signs are different pretend the signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer. =
Larger abs. value
Answer = - 4
9 - 5 = 4

18. Solve These Problems 3 + -5 = -4 + 7 = (+3) + (-4) = -6 + 7 = 5 + -9 =-2 = =
(+3) + (-4) = = = =
5 – 3 = 2
7 – 4 = 3 3
4 – 3 = 1 -1
7 – 6 = 1 1
9 – 5 = 4 -4
9 – 9 = 0

19. Some more of them…
1. – = 10
2. – = -15
(-7) = 7
4. – = -55

20. One Way to Add Integers Is With a Number Line
When the number is positive, count
to the right.
When the number is negative, count
to the left. - + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

21. One Way to Add Integers Is With a Number Line= -2 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

22. One Way to Add Integers Is With a Number Line= +2 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

23. One Way to Add Integers Is With a Number Line= -4 + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -

24. One Way to Add Integers Is With a Number Line= +4 - 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 +

25. Integer Subtraction Rule
Subtracting a negative number is the same as adding a positive. Change the signs and add.
2 – (-7)
is the same as
2 + (+7)
2 + 7 = 9!

26. Here are some more examples.
12 – (-8)
12 + (+8)
= 20
-3 – (-11)
-3 + (+11)
= 8

27. Try these out 1. 8 – (-12) = 8 + 12 = 20 2. 22 – (-30) = 22 + 30 = 52
1. 8 – (-12) = = 20
2. 22 – (-30) = = 52
3. – 17 – (-3) = = -14
4. –52 – 5 = (-5) = -57

28. How do we know that “Subtracting a negative number is the same as adding a positive” is true?
We can use the same method we use to check our answers when we subtract.

29. Suppose you subtract a – b and it equals c:
a – b = c
5 – 2 = 3
To check if your answer is correct, add b and c:
a = b + c
5 = 2 + 3

30. Here are some examples:
a – b = c a = b + c
9 – 5 = 4 9 = 5 + 4
20 – 3 = =

31. If the method for checking
subtraction works, it should
also work for subtracting
negative numbers.

32. If a – b = c, and….
2 – (-5) is the same as
2 + (+5), which equals 7,
Then let’s check with the negative numbers to see if it’s true…

33. a – b = c a = b + c 2 – (-5) = 7 2 = -5 + 7 It works!
2 – (-5) = =
It works!
a – b = c a = b + c
-11 – (-3) = =
YES!

34. Check Your Answers
1. Solve: 3 – 10 = 7
Check: 3 = 10 + (-7)
2. Solve: 17 – ( 12) = 29
Check: 17 =
Continued on next slide

35. Check Your Answers
1. Solve: 20 – ( 5) = 25
Check: 20 =
1. Solve: -7 – ( 2) = -5
Check: -7 =

36. You have learned lots of things
About adding and subtracting
Integers. Let’s review!

37. Integer Addition Rules
Rule #1 – If the signs are the same, pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.
9 + 5 = 14
= -14

38. Integer Addition Rules
Rule #2 – If the signs are different pretend the signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer. =
Larger abs. value
Answer = - 4
9 - 5 = 4

39. One Way to Add Integers Is With a Number Line
When the number is positive, count
to the right.
When the number is negative, count
to the left. - + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

40. Integer Subtraction Rule
Subtracting a negative number is the same as adding a positive. Change the signs and add.
2 – (-7)
is the same as
2 + (+7)
2 + 7 = 9!

41. Verbal Problem

44. Thank you 

45. Welcome to Module 2

46. Verbal Problem

51. Remember

54. Multiplication and Division Rules
When multiplying and dividing integers there are two rules:
When multiplying or dividing integers that have the SAME sign, the answer is POSITIVE.
Positive x positive = positive
Negative x negative = positive
+7 x +12 = +84
-6 x –5 = +30

55. Rules continued… Rule 2: Positive x negative = negative
When multiplying or dividing integers whose signs are DIFFERENT, the answer will always be NEGATIVE.
Positive x negative = negative
Negative x positive = negative
+8 x –4 = -32
-5 x +7 = -35

56. Activity: Let’s Sing a Song!
Multiplying Integers Song
(sing to “Dead Bones” tune)
Verse 1 
A negative times a negative is a, (um) positive 
A negative times a positive is a, (um) negative
A negative times a negative is a, (um) positive
These are the rules for signs.
Verse 2 
If you have the same signs, you get a positive, 
But if they’re different, you get a negative,
If you have the same signs, you get a positive,
Yes, these are the rules for signs. 

57. Let’s Practice!
What sign would the answer have in front of the integer???
-5 x -7 = ???

58. POSITIVE!!!

59. What sign will be in front of the answer??
How about…
-6 x 9 = ???
What sign will be in front of the answer??

60. NEGATIVE!!!

61. Try this one!
What sign will be in front of the answer???
-81 ÷ +9 = ???

62. NEGATIVE!!!

63. Let’s solve some problems!
-8 x –9 =

64. +72

65. +4 x +12 =

66. +48

67. -45 ÷ +5 =

68. -9

69. -63 ÷ +7 =

70. -9

71. -8 x –11 =

72. +88

73. When multiplying and dividing integers:
Remember…
When multiplying and dividing integers:
Same signs = a positive
Different signs = a negative

74. Verbal Problem

76. Activity

77. Activity

78. Why is it not possible to divide by zero?

80. Thank you 

81. Welcome to Module 3

82. Properties of integers
Commutative Property
Addition
For any two integers a and b , if
a + b = b+a , then integer addition is said to be commutative.
e.g = 7
4 + 3 = 7
Therefore, the integers satisfy commutative property under addition
ORDER DOES NOT MATTER

83. Properties of integers (contd.)
Commutative Property
2. Multiplication
For any two integers a and b , if
a x b = b x a , then integer multiplication is said to be commutative.
e.g. 3 x 4= 12
4 x 3 = 12
Therefore, the integers satisfy commutative property under multiplication.
ORDER DOES NOT MATTER

84. Properties of integers (contd.)
Commutative Property
3. Subtraction
For any two integers a and b , if
a - b = b - a , then integer subtraction is said to be commutative.
e.g = -1
4 - 3 = 1
Therefore, the integers do not satisfy commutative property under subtraction
ORDER DOES MATTER

85. Properties of integers (contd.)
Commutative Property
4. Division
For any two integers a and b , if
a / b = b / a , then integer subtraction is said to be commutative.
e.g. 12/ 4= 3
4 /12 = 1/3
Therefore, the integers do not satisfy commutative property under division
ORDER DOES MATTER

86. Properties of integers (contd.)
Associative Property
1. Addition
For any two integers a and b , if
a + (b + c) = (a + b) + c , then integer subtraction is said to be associative.
e.g. 2+ (4 + 3) = 9
(2 + 4) + 3 = 9
Therefore, the integers do not satisfy associative property under addition
ORDER DOES NOT MATTER

87. Properties of integers (contd.)
Associative Property
2. Multiplication
For any two integers a and b , if
a x (b x c) = (a x b) x c , then integer subtraction is said to be associative.
e.g. 2x (4 x 3) = 24
(2 x 4) x 3 = 24
Therefore, the integers do not satisfy associative property under multiplication.
ORDER DOES NOT MATTER

88. Properties of integers (contd.)
Associative Property
3. Subtraction
For any two integers a and b , if
a - (b x c) = (a x b) x c , then integer subtraction is said to be associative.
e.g. 2x (4 x 3) = 24
(2 x 4) x 3 = 24
Therefore, the integers do not satisfy associative property under subtraction.
ORDER DOES NOT MATTER

89. Properties of integers (contd.)
Associative Property
4. Division
For any two integers a and b , if
a / (b / c) = (a / b) / c , then integer subtraction is said to be associative.
e.g. 12/ (6 / 3) = 6
(12 / 6) / 3 = 2/3
Therefore, the integers do not satisfy associative property under division.
ORDER DOES MATTER

90. Properties of integers (contd.)
Additive Identity
Zero is the identity element on either side. By adding zero on either side number will not change
5 + 0 = 5
0 + 5 = 5

91. Properties of integers (contd.)
Multiplicative Identity
One is the identity element on either side. By multiplying with 1 on either side number will not change
5 x 1 = 5
1 x 5 = 5

92. Properties of integers (contd.)
Distributive Property
For any three integers a, b, c,
a x (b + c) = (a x b) + (a x c) . This is called Distributive property
e.g. 2x (4 + 3) = (2 x 4 ) + (2 x 3) = 14
2 X (4 – 3) = (2 x 4) – (2 x 3) = 2
Therefore, the integers satisfy distributive property

93. Verbal Problems

97. Thank you ! 