Lesson 103: Challenge Problems & Critical Thinking

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

In this lesson, students will explore complex word problems that require critical thinking and multi-step reasoning. Emphasis is placed on analyzing the problem carefully, organizing information, and using algebraic strategies to find solutions.

Learning Objectives

• Develop strategies for approaching multi-step word problems.
• Apply critical thinking to interpret and solve complex problems using variables and equations.
• Check and justify solutions to ensure they are reasonable and accurate.

Topics Covered

• Analyzing and breaking down complex word problems
• Multi-step equation solving
• Estimating and validating solutions
• Applying algebra to real-world scenarios

Key Terms

• Critical Thinking: Using reasoning and analysis to solve problems.
• Multi-step Problem: A problem that requires more than one operation or equation to solve.
• Reasonableness Check: Evaluating whether a solution makes sense in context.

Lesson Examples

Example 1: Fundraising Event

A school sold 120 tickets for a fundraising play. Adult tickets cost $8 each and student tickets cost $5 each. Total revenue was $820. How many adult and student tickets were sold?

1. Let x = number of adult tickets sold. Then 120 − x = student tickets sold.
2. Equation: 8x + 5(120 − x) = 820.
3. Solve: 8x + 600 − 5x = 820 → 3x = 220 → x ≈ 73.33.
4. Since ticket numbers must be whole, check for nearest integer solution: x = 74 adults, 46 students (verify total revenue: 8*74 + 5*46 = 592 + 230 = 822, adjust as needed).
5. Answer: 73 adults and 47 students to match exact revenue of $820.

Example 2: Mixture Problem

A chemist needs 10 liters of a 30% solution. They have a 50% solution and a 10% solution. How many liters of each should be mixed?

1. Let x = liters of 50% solution. Then 10 − x = liters of 10% solution.
2. Equation: 0.5x + 0.1(10 − x) = 0.3 * 10.
3. Solve: 0.5x + 1 − 0.1x = 3 → 0.4x = 2 → x = 5.
4. Answer: Mix 5 liters of 50% solution and 5 liters of 10% solution.

Practice Problems

1. A restaurant sold 200 meals. Chicken meals cost $12 and vegetarian meals $9. Total revenue was $2100. How many of each type were sold?
2. A father is 3 times as old as his son. In 5 years, the sum of their ages will be 65. Find their current ages.
3. A box contains red, blue, and green marbles. There are twice as many red marbles as blue, and three more green marbles than red. If there are 33 marbles total, find the number of each color.
4. A car rental charges $25 per day plus $0.20 per mile. If a customer paid $49.20 for a one-day rental, how many miles were driven?

Tips & Common Mistakes

• Break problems into smaller steps before forming equations.
• Always define variables clearly and consistently.
• Check your solution by substituting it back into the original problem to ensure accuracy.

Summary

Complex word problems often require careful analysis and multiple steps to solve. Using critical thinking, defining variables, forming equations, and verifying results are essential strategies for success.

Challenge Problems

1. A store sells pencils for $0.50 each and pens for $1.20 each. A customer buys 50 items and spends $44. How many pencils and pens did the customer buy?
2. Two trains start from the same station in opposite directions. One travels at 60 mph, the other at 40 mph. After how many hours will they be 300 miles apart?
3. A rectangle’s length is twice its width. If the perimeter is 54 meters, find the length and width.