Integer Operations Properties

Overview

• Topic: Properties of integers (properties of operations on integers).
• Lesson objective: Simplify numerical expressions involving integers using number properties.
• Five properties covered: Commutative, Associative, Distributive, Identity, Inverse.
• Scope: Applies to addition and multiplication; subtraction and division generally do not follow commutative/associative rules shown here.

Commutative Property

• Principle: Order of numbers does not affect result for addition or multiplication.
• Addition form: A + B = B + A.
• Multiplication form: A × B = B × A.
• Examples:
  • 5 + (−3) = (−3) + 5 = 2.
  • 5 × (−3) = (−3) × 5 = −15.

Associative Property

• Principle: Grouping (parentheses) does not affect result for addition or multiplication.
• Addition form: (A + B) + C = A + (B + C).
• Multiplication form: (A × B) × C = A × (B × C).
• Examples:
  • (−2 + 3) + 4 = −2 + (3 + 4) = 5.
  • (−2 × 3) × 4 = −2 × (3 × 4) = −24.
• Note: Associative property applies only to addition and multiplication in this lesson.

Distributive Property

• Principle: A number outside parentheses multiplies each term inside.
• Form (addition): A × (B + C) = A×B + A×C.
• Form (subtraction): A × (B − C) = A×B − A×C.
• Examples:
  • 3 × (4 + 2) = 3×4 + 3×2 = 12 + 6 = 18.
  • 3 × (4 − 2) = 3×4 − 3×2 = 12 − 6 = 6.

Identity Property

• Principle: Adding zero or multiplying by one does not change the value.
• Addition identity: A + 0 = A.
• Multiplication identity: A × 1 = A.
• Examples:
  • (−7) + 0 = −7.
  • (−7) × 1 = −7.

Inverse Property

• Principle: Adding the opposite gives zero; multiplying by reciprocal gives one.
• Additive inverse: A + (−A) = 0.
• Multiplicative inverse (reciprocal): A × (1/A) = 1, provided A ≠ 0.
• Examples:
  • 10 + (−10) = 0.
  • (−10) × (1/(−10)) = 1.

Key Terms and Definitions

| Term | Definition | | Commutative Property | Reordering operands does not change sum or product. | | Associative Property | Changing grouping (parentheses) does not change sum or product. | | Distributive Property | Multiply outside term across each term inside parentheses. | | Identity Property | 0 is additive identity; 1 is multiplicative identity. | | Inverse Property | Opposite (additive) yields 0; reciprocal (multiplicative) yields 1. | | Reciprocal | The multiplicative inverse of A is 1/A (A ≠ 0). |

Worked Examples (Summary)

• Commutative addition: 5 + (−3) = (−3) + 5 = 2.
• Commutative multiplication: 5 × (−3) = (−3) × 5 = −15.
• Associative addition: (−2 + 3) + 4 = −2 + (3 + 4) = 5.
• Associative multiplication: (−2 × 3) × 4 = −2 × (3 × 4) = −24.
• Distributive (addition): 3 × (4 + 2) = 3×4 + 3×2 = 18.
• Distributive (subtraction): 3 × (4 − 2) = 3×4 − 3×2 = 6.
• Identity: (−7) + 0 = −7; (−7) × 1 = −7.
• Inverse: 10 + (−10) = 0; (−10) × (1/(−10)) = 1.

Tips and Restrictions

• Use these properties to simplify expressions mentally or on paper.
• Distributive property is helpful when parentheses include addition or subtraction with multiplication outside.
• Do not apply commutative or associative properties blindly to subtraction or division.
• For reciprocals, never use 0 (reciprocal of 0 is undefined).

Action Items / Next Steps

• Practice matching expressions with the correct property (exercise suggested in lesson).
• Review any unclear parts by revisiting respective property examples.
• Prepare for the next lesson: order of operations (GEMDAS/GEM DUST).