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Lesson 1-4: Complex Operations with Parentheses

Scenario: Temperature Fluctuations - Learn to calculate temperature differences and solve complex expressions!

Duration: 55-70 minutesScenario: Temperature Fluctuations

Learning Objectives

  • Apply order of operations to complex expressions with parentheses
  • Understand and calculate absolute values
  • Solve temperature difference and elevation problems
  • Verify solutions using step-by-step calculations

Temperature Difference Scenario

The Problem

During one day, the highest temperature was 5°C and the lowest temperature was -3°C. What is the temperature difference?

Calculation: |5 - (-3)| = |8| = 8°C

The temperature difference is 8 degrees Celsius.

Understanding Absolute Value

Absolute Value Definition: |a| represents the distance from a to 0 on the number line.

Distance is always positive, so absolute value is always positive or zero.

Examples

|5| = 5
|-5| = 5
|0| = 0
|-3.2| = 3.2

Key Properties

|a| ≥ 0 (always positive or zero)
|a| = |-a|
|a × b| = |a| × |b|

Order of Operations with Parentheses

Complex Expression

Calculate: -2 + 3 × (-4 - 1)

Solution: -2 + 3 × (-4 - 1) = -2 + 3 × (-5) = -2 + (-15) = -17

Step-by-Step Process

Step 1: Parentheses first: (-4 - 1) = -5

Step 2: Multiplication: 3 × (-5) = -15

Step 3: Addition: -2 + (-15) = -17

PEMDAS Reminder

Order of Operations:

  • 1. Parentheses (and absolute value)
  • 2. Exponents
  • 3. Multiplication and Division (left to right)
  • 4. Addition and Subtraction (left to right)

Absolute Value in Complex Expressions

Priority Problem

Calculate: 10 ÷ (-2) + |-3| × 2

Solution: 10 ÷ (-2) + |-3| × 2 = -5 + 3 × 2 = -5 + 6 = 1

Calculation Steps

Step 1: Division: 10 ÷ (-2) = -5

Step 2: Absolute value: |-3| = 3

Step 3: Multiplication: 3 × 2 = 6

Step 4: Addition: -5 + 6 = 1

Practice Problems

Problem 1: Absolute Value

Calculate: |7 - 12| + |-5|

Your solution:

|7 - 12| = || =
Final answer: 5 + 5 =

Problem 2: Complex Expression

Calculate: 4 × (3 - 7) + |-2| × 3

Your solution:

Step 1: (3 - 7) =
Step 2: 4 × (-4) =
Step 3: |-2| × 3 =
Final: -16 + 6 =

Problem 3: Real-World Application

A weather station recorded temperatures of -8°C, 3°C, and -2°C on three consecutive days. What is the largest temperature difference between any two days?

Your calculation: