Lesson 7.1: Problem-solving and Modelling Worded Problems

Introduction

In this lesson, we will focus on a fundamental aspect of mathematics: solving worded problems. Worded problems are a common format for mathematical questions, requiring students to interpret language and translate it into mathematical operations. By the end of this lesson, students will:

• Understand the four-step problem-solving strategy: understand, plan, solve, and check.
• Be able to select appropriate methods from different areas of mathematics such as arithmetic, ratio, algebra, and graphing.
• Effectively translate between real-world contexts and mathematical expressions.
• Apply structured strategies to tackle unfamiliar problems.

To engage with this topic effectively, we will use practical examples that relate to everyday situations.

Understanding Worded Problems

The Four-Step Problem-Solving Strategy

When faced with a worded problem, it can be beneficial to approach it using a structured strategy. This strategy consists of four steps:

1. Understand: Read the problem carefully. Identify what is being asked and what information is given.
2. Plan: Decide on a strategy for solving the problem, such as which mathematical operations to apply.
3. Solve: Carry out the plan and perform the necessary calculations.
4. Check: Review the solution to ensure it makes sense in the context of the problem.

Example 1: Basic Worded Problem

Problem: A baker made 24 muffins. He packed them into boxes with 6 muffins each. How many boxes did he use?

1. Understand: The problem asks for the number of boxes used, given the total muffins and the capacity of each box.
2. Plan: We can use division to find out how many boxes are needed: total muffins divided by muffins per box.
3. Solve: $ \text{Number of boxes} = \frac{\text{Total muffins}}{\text{Muffins per box}} = \frac{24}{6} = 4 $
4. Check: If 4 boxes are used and each contains 6 muffins, that gives $ 4 \times 6 = 24 $ muffins, confirming our solution is correct.

Choosing the Right Mathematical Tool

As we move forward, it's important to identify which mathematical tools to use in different situations. The context of the problem often dictates the method.

Types of Tools

1. Arithmetic: Basic calculations such as addition, subtraction, multiplication, and division.
2. Ratios: Useful for comparing quantities, especially in proportional reasoning.
3. Algebra: Employed for problems involving unknowns or variables.
4. Graphing: Effective for visualizing relationships, particularly in problems that involve change, like distance or cost over time.

Example 2: Ratio Problem

Problem: A recipe requires a ratio of 2 cups of flour to 3 cups of sugar. If you have 8 cups of flour, how much sugar do you need?

1. Understand: We need to maintain the same ratio of flour to sugar.
2. Plan: Set up a proportional relationship. If 2 cups of flour correspond to 3 cups of sugar, we can set up the equation: $ \frac{2}{3} = \frac{8}{x} $
3. Solve: Cross-multiply to find $ x $: $ 2x = 3 \times 8 $ $ 2x = 24 $ $ x = \frac{24}{2} = 12 $
4. Check: Using 12 cups of sugar with 8 cups of flour maintains the ratio $ \frac{8}{12} = \frac{2}{3} $, which is correct.

Translating Between Context and Mathematics

Another critical skill is translating worded problems into mathematical expressions and vice versa. This involves using keywords and phrases as cues for the operations needed.

Example 3: Translation Problem

Problem: John has three times as many marbles as David. If David has 5 marbles, how many does John have?

1. Understand: The problem states a relationship between John’s and David’s marbles.
2. Plan: We interpret "three times" as multiplication. If David has 5 marbles, John has $ 3 \times 5 $.
3. Solve: $ \text{John's marbles} = 3 \times 5 = 15 $
4. Check: If David has 5 marbles and John has 15, then John indeed has three times as many.

Applying Structured Strategies to Unfamiliar Problems

As we encounter more complex problems, it is essential to remain calm and break them down systematically using the aforementioned strategy.

Example 4: Complex Worded Problem

Problem: A train travels 60 miles per hour. How long will it take to travel 180 miles?

1. Understand: We need to determine the time required to cover a distance at a specific speed.
2. Plan: Use the formula $ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $.
3. Solve: $ \text{Time} = \frac{180 \text{ miles}}{60 \text{ miles per hour}} = 3 \text{ hours} $
4. Check: At 60 miles per hour, the train covers 60 miles in 1 hour; thus, in 3 hours, it would cover $ 3 \times 60 = 180 $ miles, confirming our solution.

Conclusion

Problem-solving in mathematics can seem daunting at first, especially when confronted with worded problems. However, by adopting a structured approach—understanding the problem, planning a solution, solving it, and checking the work—students can significantly improve their problem-solving skills. Additionally, knowing which mathematical tools to use and how to translate real-world scenarios into mathematical terms will prepare students for a variety of problems they will encounter in their studies.

Study Notes

• Remember the four steps: Understand, Plan, Solve, Check.
• Choose the right tool: Arithmetic, Ratio, Algebra, or Graphing based on the problem.
• Translate word problems into mathematical expressions using context clues.
• Break down unfamiliar problems using a structured strategy.