Name: ______________________________________ QUARTER 1

Section: _______________________ MATHEMATICS 7

Module 2

Content Standards: The learner demonstrates understanding of key concepts of sets and the real
number system.
Performance Standards: The learner in able to formulate challenging situations involving sets and real
numbers and solve these in a variety of strategies.
Learning Competencies: The learner
i. Represents the absolute value of a number on a number line as the distance of a number from 0;
ii. Performs fundamental operations on integers;
iii. Illustrates the different properties of operations on the set integers; and

Lesson 3: Absolute Value of a Number

The absolute value of a number is its distance from zero, without considering direction. You
write the absolute value of a number by using 2 vertical bars “ (||)”. The absolute value of a number
is never negative.
Take a look at|−5|, it is read as “the absolute value of negative 5”. Use a number
line to find its absolute value.
 The absolute value of −¿ 5 is 5. The distance from 5 to 0 is 5 units.
  The absolute value of +5 is 5. The distance from –5 to 0 is 5 units.

Example 1: Find the absolute value and compare using >, <, or =.
1. |+8|∧|−9|
2. |−12|∧|+3|
3. |−10|∧|+10|
4. −|−24|∧|−4|
Answer:
1. 8< 9 , hence |+8|<|−9|
2. 12>3 , hence |−12|>|+3|
3. 10=10 ,hence |−10|=|+10|
4. −( 24 ) <4 ,hence −|−24|<|−4|

Example 2: Find two numbers with the following as their absolute value.
1. 15
2. 28
3. 89
Answer:
1. 15∧−15
2. 28∧−28
3. 89∧−89

Lesson 4: Fundamental Operations on Integers

ADDING OF INTEGERS
An integer is a whole number that is either positive or negative.
Addition of integers can be done in different ways.
1.Integers with the same sign
 Using the number line
To add a positive integer, move to the right.
To add a negative integer, move to the left.
Example 1: 3+ 4=7
First, from 0, we move 3 units to the right and from where it stopped, we move
another 4 units to the right.

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 Example 2: −2+ (−5 )=−7
From 0, we move 2 units to the left and from where it stopped, we move another 5
units to the left.

 Using Absolute
Values
Add their absolute value and copy the common sign.
1) |3|+|4|=7
2) |−2|+|−5|=7

2.Integers with unlike or different signs

 Using the number line

From 0, we move 6 units to the right, and from where it stopped, we move 2 units
Example 3: 6+ (−2 )=4

to the left.

From 0, we move 5 units to the left and from where it stopped, we move 4 units the
Example 4: −5+ 4=−1

right.

 Using absolute value

1) |6|+|−2|=4
Subtract their absolute values and copy the sign of the number with greater absolute value.

2) |−5|+|4|=−1
Rules for Adding Integers
To add integers with like signs, add their absolute values and copy the common
signs.
To add integers with unlike signs, subtract their absolute values and copy the sign
of the number with greater absolute value.
Example 5: Find the sum
1) 11+38
2) −12+(−72)
3) −65+ 84
4) 15+ (−11 )+ (−8 )
Answer:
1) 11+38
¿|11|+|38|

2) −12+ (−72 )
¿ 49

¿|−12|+|−72|
¿−84
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 3) −65+ 84
¿|−65|+|84|

For number 4, there are other options.

¿ 19

4) 15+ (−11 )+ (−8 )

OPTION 1:
|15|+|−11|=4
|4|+|−8|=−4
OPTION 2:
|−11|+|−8|=−19
|−19|+|15|=−4

Example 6: Problem Solving

1. Yesterday the temperature was 65 degrees and today it dropped
by 8 degrees. What was the temperature today?
Answer:
65+ (−8 )=57 degrees

0 1 2 3 4 5 5 6 7 8 9 100

SUBTRACTION OF INTEGERS
Recall that the sign – can indicate the sign of a number, as in –3 (negative 3), or
can indicate the operation of subtraction, as in 9 – 3 (nine minus three).
Rule for Subtracting Two Integers
To subtract two integers, add the opposite of the second integer to the first
integer. This can be written symbolically as a - b = a + (-b).

Subtraction of integers can be written as the addition of the opposite number. To

subtract two integers, rewrite the subtraction expression as the first number plus the
opposite of the second number. Example 7:
1) 8 – 15 = 8 + (-15) = -7
2) 8 - (-15) = 8 + 15 = 23
3) -8 - 15 = -8 + (-15) = -23
4) -8 - (-15) = -8 + 15 = 7

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MULTIPLICATION AND DIVISION OF INTEGERS
Multiplication of integers is the process of repetitive addition including positive
and negative numbers or can simply say integers. When we come to the case of
integers, the following cases must be taken into account:

When you multiply integers with two positive signs, the product will be positive.
Positive x Positive = Positive
When you multiply integers with two negative signs, the product will be positive.
Negative x Negative = Positive
When you multiply integers with one negative sign and one positive sign, the
product is negative.
Negative x Positive = Negative

Types of Integers Product Example

Both Positive Positive 2 x 5 = 10

Both Negative Positive –2 x –3 = 6

1 Positive and 1 Negative Negative –2 x 5 = –10

Division of integers involves the grouping of items. It includes both positive numbers
and negative numbers. Just like multiplication, the division of integers also involves the
same cases.

When you divide integers with two positive signs, the quotient is positive.
Positive ÷ Positive = Positive

When you divide integers with two negative signs, the quotient is positive.
Negative ÷ Negative = Positive

When you divide integers with one negative sign and one positive sign, the quotient is
negative.
Negative ÷ Positive = Negative

Types of Integers Quotient Example

Both Positive Positive 16 ÷ 8 = 2

Both Negative Positive –16 ÷ –8 = 2

1 Positive and 1 Negative Negative –16 ÷ 8 = –2

Example 8: Perform the indicated operations.

1) (−6 )(−4 )
2) ( 14 )( 7 )
3) (−32 ) ( 10 )
4) (−65 ) ÷ (−13 )
5) 72 ÷ 4
6) (−225 ) ÷ ( 25 )

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Answer:
1) (−6 )(−4 ) =24 5) 72 ÷ 4=18
2) ( 14 )( 7 )=98 6) (−225 ) ÷ ( 25 )=−9
3) (−32 ) ( 10 )=−320
4) (−65 ) ÷ (−13 )=5
LESSON 5: PROPERTIES OF INTEGERS
There are properties that are applied in performing operations in integers.

CLOSURE PROPERTY OF INTEGERS

Addition :
Observe the following examples:
1) 19 + 23 = 42
2) - 10 + 4 = - 6
3) 18 + (- 47) = - 29
In general, for any two integers a and b, a + b is an integer.
Therefore the set of integers is closed under addition.
Subtraction :
Observe the following examples:
1) 12 - 5 = 7
2) 5 - 12 = -7
3) 18 - (-13) = 18 + 13 = 31
In general, for any two integers a and b, a - b is an integer.
Therefore, the set of integers is closed under subtraction.
Multiplication :
Observe the following:
1) 10 × (– 5) = 50
2) 40 × (– 15) = – 600
In general, a × b is an integer, for all integers a and b.
Therefore, integers are closed under multiplication.

COMMUTATIVE PROPERTY OF INTEGERS

Addition :
Two integers can be added in any order. In other words, addition is commutative for
integers.
Example:
8 + (- 3) = 5 and (- 3) + 8 = 5
Therefore, 8 + (- 3) = (- 3) + 8
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In general, for any two integers a and b we can say,
a+b=b+a
Multiplication :
Observe the following :
5 × (– 6) = – 30 and (– 6) × 5 = – 30
Therefore, 5 × (– 6) = (– 6) × 5
In general, for any two integers a and b, a × b = b × a.
ASSOCIATIVE PROPERTY OF INTEGERS
Addition :
Observe the following example :
Consider the integers 5, – 4 and 7.
Look at
5 + [(– 4) + 7] = 5 + 3 = 8
and
[5 + (– 4)] + 7 = 1 + 7 = 8
Therefore, 5 + [(– 4) + 7] = [5 + (– 4)] + 7
In general, for any integers a, b and c, we can say,
a + (b + c) = (a + b) + c.
Therefore addition of integers is associative.
Multiplication :
Consider the integers 2, – 5, 6.
Look at
[2 x (-5)] x 6 = -10 x 6 = -60
2 x [(- 5) x 6] = 2 x (-30) = -60
Thus,
[2 x (-5)] x 6 = 2 x [(- 5) x 6]
In general, for any integers a, b, c,
(a × b) × c = a × (b × c)

IDENTITY PROPERTY OF INTEGERS

Additive Identity :
When we add zero to any integer, we get the same integer.
Observe the example: 5 + 0 = 5.
In general, for any integer a, a + 0 = a.
Therefore, zero is the additive identity for integers.
Multiplicative Identity :
Observe the following:
1) 5 x 1 = 5
2) 1 x (- 7) = -7
This shows that ‘1’ is the multiplicative identity for integers. In general a x 1 = 1 x a =
a.

INVERSE PROPERTY
Addition
It says that any number added to its opposite will equal zero. What is the opposite
you might ask? All you have to do is change the sign from positive to negative or
negative to positive.

Let's see what that looks like.

Example 1: 5 + (-5) = 0 -5 is the opposite of 5
Example 2: -4 + (4) = 0 -4 is the opposite of 4
Multiplication
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 It says that any number multiplied by its reciprocal is equal to one.
Let's start by defining a reciprocal. To find the reciprocal of any number write it as a
fraction and then flip it.

Example 1: find the reciprocal of . Flip it → .

The reciprocal of is .

Example 2: find the reciprocal of 5. → Write it as a fraction →flip it

The reciprocal of 5 is

Example 3: find the reciprocal of . → flip it

The reciprocal of is 2.

Example 4: find the reciprocal of - . → flip it -

The reciprocal of - is -

DISTRIBUTIVE PROPERTY OF INTEGERS

Multiplication is Distributive Over Addition :
Consider the integers 12, 9, 7.
Look at
(12) (9 + 7) = 12 x 16 = 192
(12) (9 + 7) = 12 x 9 + 12 x 7 = 108 + 84 = 192
Thus,
(12)(9 + 7) = (12 x 9) + (12 x 7)
In general, for any integers a, b, c.
a x (b + c) = (a x b) + (a x c)
Multiplication is Distributive Over Subtraction :
Consider the integers 12, 9, 7.
Look at
(12) (9 - 7) = 12 x 2 = 24
(12) (9 - 7) = (12 x 9) - (12 x 7) = 108 - 84 = 24
Thus,
(12) (9 - 7) = (12 x 9) - (12 x 7)
In general, for any integers a, b, c.
a x (b - c) = (a x b) - (a x c)

Example: Name the property illustrated in each mathematical sentence.

1) 4 +7=7+ 4 4) 5 ×1=5

2) 9+ 0=9 5) (−12 ) ( )
−1
=1
12
3) 3+ ( 2+5 )=( 3+2 ) +5 6) 8 ( 7+−3 )=( 8 ×7 )( 8×−3 )
Answer:
1) Commutative Property of Addition
2) Identity Property of Addition
3) Associative Property of Addition
4) Identity Property of Multiplication
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5) Inverse Property of Multiplication
6) Commutative Property of Multiplication

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Name:______________________________________
_ Module 2 Activity Score:________________
Section:________________________

A. Operation with Integers

MATH DECODER!
Direction: Solve each problem and write the matching letter on the blank above the
answer to unlock the code/message.

A 20+ (−10 )−2 K (−6)(−19)

O −50−50 L (−8)×(−15)

C −18 ÷ 6 M 75 ×2

S 15 ×(−12) N (−33 ) −(21)

E (−5 ) +(−4 ) B 335 ÷ 5+42

F 20+(−62) W (−43 )−(71)

G −125 ÷ 5 Q −30 ÷−3

H 8 ×3 R (−12)× 5

I 42+(−21) D −108 ÷(−3)

U (−27 )−(−14) T 100 ÷ 10

_____ _____ _____ _____ _____ _____ _____ _____ _____ __________
11 21 - 36 - -9 - - 21 - 8
4 54 54 180 180 180
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
12 8 - _____- _____
- _____
8 - -9 - 144 24 21 -3 24 10 24 - 9
0 54 25 13 25

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
_____
36 - _____
9 8 - -3 8 - 24 - 9 8 -60 8 - 36
42 54 54

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
10 24_____- 9_____ 10
_____.
12 21 - 36 -3 8 - - -9 -9
9 0 54 54 180

- Mark Twain

10 | P a g e
11 | P a g e
B. Number Line
Solve the following using a number line. Label/ Number your number line.
1) 10+ (−3 )

2) 6+3

3) −9+(−6)

12 | P a g e