Lesson 21: Applications & Word Problems

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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

This lesson applies the concepts of factors, multiples, greatest common factor (GCF), and least common multiple (LCM) to practical word problems. Students will practice solving real-life scenarios such as scheduling, arranging objects, and simplifying fractions.

Learning Objectives

• Identify which method (GCF or LCM) is appropriate for different word problems.
• Solve problems involving grouping, sharing, and arranging objects.
• Apply GCF and LCM to simplify fractions and solve ratio problems.

Key Terms

• Word Problem: A math problem stated in a real-life context.
• GCF (Greatest Common Factor): The largest factor shared by two or more numbers, used to divide or group objects evenly.
• LCM (Least Common Multiple): The smallest multiple shared by two or more numbers, used to synchronize repeating events or schedules.

Example 1: Using GCF

1. A teacher has 24 red pencils and 36 blue pencils. She wants to make identical gift bags using all pencils.
2. Find the GCF of 24 and 36 → 12.
3. Number of pencils per bag = 12. Number of bags = 2 red pencils + 3 blue pencils per bag? Actually, divide 24 ÷ 12 = 2 red per bag, 36 ÷ 12 = 3 blue per bag → correct.

Example 2: Using LCM

1. Two traffic lights flash every 45 and 60 seconds. When will they flash together?
2. Find LCM of 45 and 60.
3. Prime factorization: 45 = 3² × 5, 60 = 2² × 3 × 5. LCM = 2² × 3² × 5 = 180 seconds → they flash together every 3 minutes.

Practice Problems

1. A baker has 18 chocolate muffins and 24 blueberry muffins. He wants to make identical boxes with all muffins. How many muffins in each box?
2. Three buses arrive every 12, 15, and 20 minutes. After how many minutes will they arrive together?
3. Simplify the fraction 36/60 using GCF.
4. Arrange 28 chairs and 42 tables in equal rows. How many chairs and tables per row?

Tips & Common Mistakes

• Always decide if the problem is asking for “even grouping” (GCF) or “synchronization” (LCM).
• Check calculations carefully; errors in factoring will lead to wrong GCF/LCM.
• Draw diagrams or lists for clarity in complex word problems.

Summary

Word problems help students connect factors, multiples, GCF, and LCM to real-world situations. Recognizing when to use GCF versus LCM is key to solving these problems efficiently.

Challenge Problems

1. A gardener has 60 tulips and 90 daffodils. She wants to plant them in identical rows with no leftover flowers. How many flowers per row?
2. Three alarms ring every 20, 30, and 45 minutes. When will they ring together in a 12-hour period?
3. Simplify 48/72 and 36/54 using GCF.