Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Number System → Whole Numbers ↓


Properties of Whole Numbers (Associative, Commutative, Distributive)


Whole numbers are the set of numbers that includes all non-negative integers. This means that they include zero and all positive integers. One of the fascinating aspects of whole numbers is that they follow certain rules, or properties, that make arithmetic operations easier and more predictable. In this guide, we will explore three fundamental properties of whole numbers: the associative property, the commutative property, and the distributive property.

Associative property

The associative property refers to the way in which numbers are grouped in an operation. It states that the way the numbers are grouped does not change the result of the operation. This property applies to both addition and multiplication.

Associative property of addition

For the action of addition, the associative property can be expressed as:

(a + b) + c = a + (b + c)

This means that when you're adding numbers it doesn't matter what group you put them in; the sum will be the same.

For example:

(2 + 3) + 4 = 2 + (3 + 4)

Calculating both sides, we get:

5 + 4 = 2 + 7

Therefore, both sides are equal to 9, which shows the associative property.

2 + 3 + 4 (2 + 3) + 4 = 9 2 + (3 + 4) = 9

Associative property of multiplication

For multiplication, the associative property is expressed as:

(a × b) × c = a × (b × c)

Again, this means that it doesn't matter how you group numbers when you multiply them; the product will be the same.

For example:

(2 × 3) × 4 = 2 × (3 × 4)

Calculating both sides, we get:

6 × 4 = 2 × 12

Both sides are equal to 24, which shows the associative property for multiplication.

2 × 3 × 4 (2 × 3) × 4 = 24 2 × (3 × 4) = 24

Commutative property

The commutative property refers to the order of numbers in an operation. It states that changing the order of the numbers does not change the result. This property is true for both addition and multiplication.

Commutative property of addition

The commutative property of addition is expressed as follows:

a + b = b + a

This means you can add the numbers in any order.

For example:

4 + 5 = 5 + 4

Both sums are 9, which shows the commutative property in action.

4 + 5 = 95 + 4 = 9

Commutative property of multiplication

The commutative property of multiplication is expressed as follows:

a × b = b × a

This means that you can multiply the numbers in any order.

For example:

6 × 3 = 3 × 6

The two products are 18, which verifies the commutative property.

6 × 3 = 183 × 6 = 18

Distributive property

The distributive property combines the operations of addition and multiplication. It states that multiplying a number by the sum of two other numbers is the same as doing each multiplication separately. In mathematical terms, this is expressed as:

a × (b + c) = (a × b) + (a × c)

This property provides an easy way to simplify expressions and is often used in mental arithmetic and algebra.

For example, using the numbers 2, 3, and 4:

2 × (3 + 4) = (2 × 3) + (2 × 4)

Simplifying both sides:

2 × 7 = 6 + 8

Both sides are equal to 14, thus the distributive property is verified.

2 × (3 + 4) (2 × 3) + (2 × 4)

Benefits of distributive property

The distributive property is particularly useful because it allows complex expressions to be simplified, especially in algebra. It also plays an important role in arithmetic, helping to break down more challenging multiplication problems into more manageable parts.

Bringing assets together

These properties are fundamental because they provide simple rules that work for whole numbers. By understanding and applying these properties, you can perform arithmetic operations more easily and develop better number sense. Let's look at some examples where we use more than one property at a time:

Example: Applying properties

Let's calculate the expression: (5 + 3) × 2 + 8

  1. Use the associative property to change the grouping:
    2. (5 + 3) = 5 + 3
  2. Use the commutative property to rearrange:
    3. 5 + 3 = 3 + 5
  3. Use the distributive property for multiplication:
    4. (3 + 5) × 2 = (3 × 2) + (5 × 2)
  4. Calculate each part:
    5. 6 + 10 + 8
  5. Simplify to get the final sum:
    6. 24

Understanding these properties helps simplify many mathematical problems and develop logical reasoning skills. Keep practicing these properties with new examples to get more comfortable.

The more we learn about the properties of numbers, the easier it becomes to perform mathematical operations efficiently. By mastering these properties, we create a strong foundation for further mathematical studies.

Practical applications in daily life

These properties are not just theoretical; they also have practical applications. Whether you are calculating prices while shopping or creating your budget, these properties make the math faster and simpler. For example, when summing prices, you can apply the commutative property to add items in a more convenient order or use the distributive property to simplify tax calculations.

In conclusion, the properties of whole numbers - associative, commutative and distributive - form the basis for many mathematical operations that are part of daily life as well as more complex mathematical challenges. Practicing these properties consistently will increase both speed and accuracy in arithmetic calculations.


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Properties of Whole Numbers (Associative, Commutative, Distributive)

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