Recognize the properties of arithmetic operations
1. Exchange/commutative nature
The commutative property is the exchange property. For example, if you add or multiply two numbers. If the two numbers are exchanged the result remains the same.
Does the exchange apply to the reduction? To better understand the commutative property, consider the following example.
a. sum
Consider the following summation results.
1) 8 + 9 = 9 + 8
17 = 17
2) 20 + 30 = 30 + 20
50 = 50
b. Multiplication
Look at the following multiplication results.
1) 3 × 4 = 4 × 3
12 = 12
2) 7 × 5 = 5 × 7
35 = 35
12 – 5 = 7
5 – 12 = –7
So, the nature of exchange does not apply to subtraction
2. Grouping/associative nature
The associative property is a grouping property. For example, the addition operation
or multiplication of three numbers.
These operations are grouped differently. The result of the operation remains the same. To better understand the associative property, consider the following example.
a. sum
Example:
1) (3 + 4) + 5 = 3 + (4 + 5)
7 + 5 = 3 + 9
12 = 12
2) (15 + 20) + 25 = 15 + (20 + 25)
35 + 25 = 15 + 45
60 = 60
b. Multiplication
Example:
1) (2 × 3) × 4 = 2 × (3 × 4)
6 × 4 = 2 × 12
24 = 24
2) (4 × 5) × 7 = 4 × (5 × 7)
20 × 7 = 4 × 35
140 = 140
Does the associative property apply to subtraction? Consider the following example.
Example:
(15 – 4) – 6 = 5
15 – (4 –6) = 17
So (15 – 4) – 6 z 15 – (4–6)
So, the associative property does not apply to subtraction
3. The nature of the spread / distributive
The distributive property is the property of distribution. To better understand the distributive property,
consider the following example.
a. The distributive of multiplication over addition
Look at the distributive property of multiplication over addition.
1) 5 × (2 + 3) = (5 × 2) + (5 × 3)
= 10 + 15
= 25
2) (12 × 7) + (12 × 3) = 12 × (7 + 3)
= 12 × 10
= 120
b. The distributive of multiplication over subtraction
Now, consider the following distributive property of multiplication over subtraction.
1) 8 × (7 – 3) = (8 × 7) – (8 × 3)
= 56 – 24
= 32
2) (25 × 18) – (25 × 8) = 25 × (18 – 8)
= 25 × 10
= 250
The identity property is the property of the operation on a number whose result is that number
alone. Note the identity property of the following operations.
a. Addition identity
The identity in addition is 0. Consider the following example.
1) 8 + 0 = 8
2) 0 + 12 = 12
3) 23 + 0 = 23
4) 0 + 72 = 72
So, the additive identity is 0
b. Multiplication identity
The identity in multiplication is 1. Consider the following example.
1) 7 × 1 = 7
2) 1 × 12 = 12
3) 25 × 1 = 25
4) 1 × 36 = 36
So, the multiplicative identity is
1. Solving the problem of the nature of arithmetic operations
The following are problems related to the nature of arithmetic operations. Example:
Mother will help disaster victims. Mother bought 75 baskets of apples. Each basket
contains 55 pieces. Mother also bought 75 baskets of oranges. Each basket contains 45
fruit. How many fruits did mother buy?
Answer:
(75 × 55) + ( 75 × 45) = 75 × ( 55 + 45)
= 75 × (100)
= 7,500
So, the fruit that mother bought was 75,000 pieces.