Word Problems and Problem Solving

TL;DR

Word problems challenge you to translate everyday language into mathematical expressions and equations. They test your ability to understand a scenario, pick out key information, and then apply math skills to find a solution. Breaking down the problem into smaller, manageable steps is crucial for success.

1. The Mental Model

Think of a word problem as a puzzle where the pieces are words and sentences. Your job is to arrange those pieces into a mathematical picture (an equation or expression) that you can then solve to reveal the final answer. It's like being a detective for numbers.

2. The Core Material

Word problems can seem tricky because they don't immediately give you an equation to solve. You need to pull out the math from the story.

Understand the Goal

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Before you do anything else, figure out what the problem is asking you to find. What's the unknown? This will help you know when you've actually solved the problem.

Identify Key Information

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Read the problem carefully, and highlight or write down the numbers and keywords. Keywords often tell you what operation to use.

Formulate the Plan

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Once you know what you need to find and what information you have, think about how they relate. This is where you set up your equation or series of steps. Sometimes drawing a picture can help visualize the problem.

Execute the Plan

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Solve the equation or perform the calculations you planned. Be careful with your arithmetic!

Check Your Answer

Does your answer make sense in the context of the original problem? Is it a reasonable number? For instance, if you're calculating the number of people, a negative or fractional answer probably means something went wrong.

Here's a flowchart to help you visualize the problem-solving process:

graph TD
    A["Read & Understand Problem"] --> B{"What's the Question?"}
    B --> C["Identify Key Information/Numbers"]
    C --> D{"Keywords Indicate Operations?"}
    D --> E["Formulate Equation/Plan"]
    E --> F["Solve/Calculate"]
    F --> G{"Does Answer Make Sense?"}
    G -- No --> E
    G -- Yes --> H["State Final Answer Clearly"]

3. Worked Example

Let's try one:

"Sarah bought 3 notebooks for \$2.50 each and a pen for \$1.25. She paid with a \$10 bill. How much change did Sarah receive?"

1. Understand the Goal: We need to find out how much change Sarah received.
2. Identify Key Information:
   • 3 notebooks
   • \$2.50 each (cost of one notebook)
   • 1 pen
   • \$1.25 (cost of one pen)
   • \$10 (amount paid)
3. Formulate the Plan:
   • First, calculate the total cost of the notebooks: number of notebooks * cost per notebook.
   • Next, add the cost of the pen to find the total spending: notebook cost + pen cost.
   • Finally, subtract the total spending from the amount paid to find the change: amount paid - total spending.
4. Execute the Plan:
   • Notebook cost: 3 * $2.50 = $7.50
   • Total spending: $7.50 + $1.25 = $8.75
   • Change: $10.00 - $8.75 = $1.25
5. Check Your Answer: If she spent \$8.75 and paid with \$10, getting \$1.25 back makes sense. The numbers are reasonable.

Sarah received \$1.25 in change.

4. Key Takeaways

• Always start by understanding what the problem is asking you to find.
• Break down complex problems into smaller, more manageable steps.
• Highlight or write down all the relevant numbers and keywords.
• Use keywords (like "sum," "difference," "product") to determine the correct mathematical operations.
• Drawing diagrams or pictures can often help visualize the problem.
• Double-check your calculations to avoid simple arithmetic errors.
• Ask yourself if your final answer makes sense in the context of the problem.

5. Now Try It

A baker made 240 cookies. He sold half of them in the morning. In the afternoon, he sold 75 more cookies. How many cookies does the baker have left?

What to do: Follow the steps from section 2. Clearly state the goal, identify the numbers, plan your operations, calculate, and then check your answer.

What success looks like: You should arrive at a single number representing the remaining cookies and show the steps you took to get there.