Associative Property

The associative property in mathematics is applied to some binary operations, which states that changing the parenthesis in an expression or grouping the number differently won’t alter the outcome. That is, if a, b, c are three numbers in an expression.

    a × (b × c) = (a × b) × c 

    a + (b + c) = (a + b) + c

This is only applicable for addition and multiplication, as this property will not work for subtraction and division. We will learn more about it below.
 

Let’s look into the key difference between associative and commutative property in a table below: 
 

So firstly, the key difference is the number of numbers involved. Secondly, the associative property is the change in grouping of the numbers, and the commutative property is the change in order of the numbers. 

The associative property helps make calculations easier by allowing you to regroup numbers when adding or multiplying. It is useful in mental math and simplifying expressions. 

Step 1: Check that the given expression is multiplication or addition.
 

Step 2: If yes, then change the numbers into different groups.
 

Step 3: Then solve the expression. 
 

The associative property of addition is that when two or more numbers are grouped differently, their result remains the same.

For example, (2 + 3) + (5 + 6) = 16.

When you group it differently, (6 + 2) + (2 + 3), you still get 16. That is,

(A + B) + C = A + (B + C)
 

The associative property of multiplication is that when two or more numbers are grouped differently, their result remains the same.
 

For example, (2 × 3) × (5 × 6) = 180. 

When you group it differently, 

(6 × 5) × (3 × 2), you still get 180. 

That is, 

(A × B) × C = A × (B × C)

The associative property does not hold for subtraction. This means that rearranging the grouping of numbers in subtraction can lead to different results. Thus, subtraction does not follow associative property.

For example, (2 – 3) – (5 – 6) =  (-1 - (-1)) =  (-1 + 1) =  0

When you group it differently, 

(5 – 6) – (3 – 2)  = (-1 - (1)) = – 2. 

That is, 

(A – B) – C    A – (B – C)

The associative property does not apply for division as well. This means that altering the grouping of numbers in subtraction can result in different outcomes. Thus, division does not follow associative property.

For example, (2 ÷ 3) ÷ (5 ÷ 6) = 0.8. 
 

When you group it differently, 
 

(6 ÷ 2) ÷ (5 ÷ 2), you will get 1.2. 
 

That is,
 

(A ÷ B) ÷ C ≠ C ÷ (B ÷ A)

The associative property is one of the basic properties in mathematics. It helps in performing addition and multiplication easily by grouping the numbers without changing the results. Here are a few tips and tricks to master it: 

• Always memorize the formula for the associative property. The associative property of addition formula is (A + B) + C = A + (B + C) and associative property of multiplication is (A × B) × C = A × (B × C).
   
• Students can use visual aids like blocks, coins, or candies to demonstrate. For example, for addition, take 3 blocks, add 4 blocks, then add 2 blocks more, grouping them as (3 + 4) + 2 = 3 + (4 + 2) = 9. 
   
• Using number lines helps students to understand that the concept of addition works regardless of grouping. For example, to show 2 + (3 + 4), start from 0, move 2 steps, then move 7 steps (3 + 4), compare with (2 + 3) + 4, start from 0, moving 5 steps (2 + 3), then 4 more. Both results in 9. 
   
• To learning the associative property, students can use games that are interactive and encourage creative thinking. For example, students can form a group of numbers in different ways to find the sum.
   
• By regular practice, students can master the associative property, start with small numbers, and understand the pattern. Then try with bigger numbers or fractions.  
   

The associative property is useful in real-life situations where numbers are grouped differently to make calculations easier. Here are some examples:

• Shopping and Budgeting: If you buy three items and want to add their prices, you can group them in different ways to make mental math easier. For example, a shirt costs around $15, a hat $10, and shoes $25. This can be written and counted in whichever way you want. First group the shirt and shoes, $15 + $25, then add the hat $10. Or in any other way, (15 + 10) + 25, (25 + 10) + 15, etc. No matter how you group them and add, the resulting answer 50 remains the same. 
   
• Cooking and Baking: When measuring ingredients, you can group amounts differently for convenience. A recipe needs 1 cup of flour, \(1\over2 \) cup of sugar, and \(1\over2 \) cup of cocoa powder. Instead of adding in order, you can group them like (\(1\over2 \) + \(1\over2 \)) + 1 = 1 + 1 = 2 cups. This makes the measuring faster and easier. 
   
• Splitting a Bill at a Restaurant: If three friends are splitting a bill, they can add their shares in any order. The bill is $18 + $22 + $30. That can be grouped as (18 + 22) + 30 = 18 + (22 + 30) = 70. Grouping helps break the total into easier parts.
   
• Organizing Work Tasks: When planning your daily schedule, you can group tasks differently without changing the total time.
   
• Banking and Finances: When calculating total deposits or expenses, the order of grouping doesn’t matter.