MathIsimple
Back to Unit 1

Lesson 1-5: Comprehensive Applications

Scenario: Hiking & Financial Planning - Apply all rational number operations to solve complex real-world problems!

Duration: 65-80 minutesScenario: Hiking & Financial Planning

Learning Objectives

  • Integrate all rational number operations in complex problems
  • Solve multi-step real-world problems systematically
  • Apply problem-solving strategies to financial and travel scenarios
  • Verify solutions and check reasonableness of answers

Hiking Trip Scenario

The Problem

A hiker starts at sea level (0 meters). During the day, they:

  • • Morning: Climb up 250 meters
  • • Afternoon: Descend 180 meters
  • • Evening: Descend another 75 meters

What is their final position relative to the starting point?

Calculation: 0 + 250 + (-180) + (-75) = -5 meters

The hiker ends up 5 meters below the starting point.

Step-by-Step Solution

Step 1: Start at sea level: 0 meters

Step 2: Morning climb: 0 + 250 = 250 meters

Step 3: Afternoon descent: 250 + (-180) = 70 meters

Step 4: Evening descent: 70 + (-75) = -5 meters

Financial Planning Scenario

The Problem

Sarah has $120 in her savings account. She spends $150 on a purchase, then deposits $80 from her paycheck. What is her final account balance?

Calculation: 120 + (-150) + 80 = 50 dollars

Sarah's final account balance is $50.

Alternative Calculation Method

Method 1: Sequential calculation

120 - 150 + 80 = -30 + 80 = 50

Method 2: Group positive and negative amounts

(120 + 80) + (-150) = 200 - 150 = 50

Problem-Solving Strategies

Efficient Calculation Methods

Grouping Strategy

Group positive and negative numbers separately, then combine.

Example: 5 + (-3) + 8 + (-2) = (5 + 8) + (-3 + -2) = 13 + (-5) = 8

Complementary Numbers

Look for numbers that add to convenient values (like 10, 100).

Example: 7 + 3 + 5 = (7 + 3) + 5 = 10 + 5 = 15

Verification Strategies

Always check your answer:

  • • Does the answer make sense in the context?
  • • Can you work backwards to verify?
  • • Are the units correct?
  • • Is the magnitude reasonable?

Comprehensive Practice Problems

Problem 1: Elevator Journey

An elevator starts at the 5th floor, goes up 8 floors, down 12 floors, up 3 floors, and down 7 floors. What floor does it end on?

Your calculation:

Problem 2: Stock Market

A stock starts at $50. It gains $3, loses $8, gains $5, and loses $2. What is the final price? What is the total change from the starting price?

Your calculation:

Problem 3: Temperature Variation

The temperature at 6 AM was -5°C. It rose 8°C by noon, dropped 3°C by 6 PM, and dropped another 6°C by midnight. What was the temperature at midnight? What was the total change from 6 AM to midnight?

Your calculation: