Math with Mr. J: Order of Operations Lecture Notes

Introduction

• Topic: Order of Operations (PEMDAS)
• Purpose: Provide a comprehensive guide to understanding and applying the order of operations.
• Structure: Lecture includes examples, divided into chapters.

Why Use the Order of Operations?

• Consistency: Ensures everyone solves mathematical problems in the same way.
• Prevents Confusion: Provides a universal set of instructions for solving expressions.
• Example:
  • Without order of operations: 12 - 5 * 2 = 14
  • With order of operations (multiplication first): 12 - (5 * 2) = 2
• Analogy: Similar to traffic lights and measurement standards.

Order of Operations (PEMDAS)

• P: Parentheses
• E: Exponents
• MD: Multiplication and Division (left to right)
• AS: Addition and Subtraction (left to right)

Example Problems

Example 1

• Expression: 30 ÷ (13 - 8)
• Steps:
  1. Parentheses: 13 - 8 = 5
  2. Division: 30 ÷ 5 = 6
• Final Answer: 6

Example 2

• Expression: 16 - 5 * 3 + 12
• Steps:
  1. Multiplication: 5 * 3 = 15
  2. Subtraction and Addition (left to right): 16 - 15 + 12 = 13
• Final Answer: 13

Example 3

• Expression: 7² - 14 * 2
• Steps:
  1. Exponents: 7² = 49
  2. Multiplication: 14 * 2 = 28
  3. Subtraction: 49 - 28 = 21
• Final Answer: 21

Example 4

• Expression: 18 ÷ (6 + 3) * 15
• Steps:
  1. Parentheses: 6 + 3 = 9
  2. Division: 18 ÷ 9 = 2
  3. Multiplication: 2 * 15 = 30
• Final Answer: 30

Examples with Parentheses and Exponents

Example 1

• Expression: 13 + 5(2³ + 4) ÷ 2
• Steps:
  1. Parentheses: 2³ = 8, then 8 + 4 = 12
  2. Multiplication: 5 * 12 = 60
  3. Division: 60 ÷ 2 = 30
  4. Addition: 13 + 30 = 43
• Final Answer: 43

Example 2

• Expression: (15 - 7)² - (36 ÷ 9) * 10
• Steps:
  1. Parentheses: 15 - 7 = 8, then 8² = 64
  2. Division in brackets: 36 ÷ 9 = 4
  3. Multiplication: 4 * 10 = 40
  4. Subtraction: 64 - 40 = 24
• Final Answer: 24

Examples without Parentheses or Exponents

Example 1

• Expression: 20 - 4 * 4 + 15 - 6
• Steps:
  1. Multiplication: 4 * 4 = 16
  2. Subtraction and Addition (left to right): 20 - 16 + 15 - 6 = 13
• Final Answer: 13

Example 2

• Expression: 7 * 2 + 9 ÷ 3 - 10 - 7
• Steps:
  1. Multiplication: 7 * 2 = 14
  2. Division: 9 ÷ 3 = 3
  3. Subtraction and Addition (left to right): 14 + 3 - 10 - 7 = 0
• Final Answer: 0

Examples with Multiple Grouping Symbols

Example 1

• Expression: [28 - (6 + 4)] ÷ 2
• Steps:
  1. Innermost Parentheses: 6 + 4 = 10
  2. Brackets: 28 - 10 = 18
  3. Division: 18 ÷ 2 = 9
• Final Answer: 9

Example 2

• Expression: 50 - { [5(7+1)] + 3² }
• Steps:
  1. Parentheses: 7 + 1 = 8
  2. Multiplication in brackets: 5 * 8 = 40
  3. Exponents: 3² = 9
  4. Addition in braces: 40 + 9 = 49
  5. Subtraction: 50 - 49 = 1
• Final Answer: 1

Examples Involving Fraction Bars

Example 1

• Expression: (10 + 5 * 6) / (12 ÷ 3)
• Steps:
  1. Numerator: 5 * 6 = 30, then 10 + 30 = 40
  2. Denominator: 12 ÷ 3 = 4
  3. Division: 40 ÷ 4 = 10
• Final Answer: 10

Example 2

• Expression: (28 - 22 + 15) / [(27 - 9) ÷ (3 * 2)]
• Steps:
  1. Numerator: 28 - 22 = 6, then 6 + 15 = 21
  2. Denominator: 27 - 9 = 18, 3 * 2 = 6, then 18 ÷ 6 = 3
  3. Division: 21 ÷ 3 = 7
• Final Answer: 7

Complex Fractions

• Example: (6² - 9) / 3
  • Steps: 6² = 36, 36 - 9 = 27
  • Final Result: 27 / 3 = 9

Fraction Result Example

• Example: (5² - (10 + 9)) / ((11 - 8) * (4 + 2))
  • Steps:
    1. Numerator: 5² = 25, 10 + 9 = 19, 25 - 19 = 6
    2. Denominator: 11 - 8 = 3, 4 + 2 = 6, 3 * 6 = 18
  • Simplification: 6/18 = 1/3
  • Final Answer: 1/3

Examples with Positive and Negative Integers

Example 1

• Expression: -18 ÷ -2 * (-11 + 6)
• Steps:
  1. Parentheses: -11 + 6 = -5
  2. Division: -18 ÷ -2 = 9
  3. Multiplication: 9 * -5 = -45
• Final Answer: -45

Example 2

• Expression: (-4 - 4)² + (-3)³ ÷ 9
• Steps:
  1. Parentheses: -4 - 4 = -8
  2. Exponents: (-8)² = 64, (-3)³ = -27
  3. Division: -27 ÷ 9 = -3
  4. Addition: 64 + (-3) = 61
• Final Answer: 61

Conclusion

• A comprehensive guide covering various scenarios using order of operations.
• Emphasized the importance of following PEMDAS to ensure consistency and accuracy.