Question

Exercises 37-62 are short-answer questions. List the four steps in Polya's How to Solve It list.

Short answer

Understand the problem, devise a plan, carry out the plan, and review and reflect.

Step-by-step solution

Understand the Problem

The first step in Polya's problem-solving process is to understand the problem. You should begin by clarifying the problem statement by asking yourself what is being asked and identifying what information is given and what is unknown.

Devise a Plan

The second step is to devise a plan for solving the problem. This involves determining how to approach the problem and considering strategies you can use such as drawing diagrams, working backwards, looking for patterns, or creating equations.

Carry Out the Plan

The third step is to carry out the plan you have devised. Carefully implement your chosen strategy step by step, ensuring all calculations and reasoning are correct.

Review and Reflect

The last step is to review and reflect on your solution. Check the answer for correctness and think about what worked or didn’t work in your problem-solving process to improve future attempts.

Key concepts

Understanding the Problem

Grasping the core of the problem is the first essential step in Polya's problem-solving process. At this stage, it's crucial to make sure you completely understand what is being asked. Start by carefully reading the problem statement. Like a detective, identify the given information and list what is known and unknown.
Ask questions like:

• What are we trying to find or prove?
• What do the terms mean in this context?
• Are there any constraints or conditions?

Breaking down the problem into parts makes it easier to tackle. A strong understanding anchors all subsequent steps and avoids confusion later on. To solidify your understanding, consider paraphrasing the problem in your own words or discussing it with peers.

Devise a Plan

Once you have a solid understanding of the problem, it's time to devise a plan. This involves brainstorming and selecting a strategy or a combination of strategies that best fits the problem. Here’s where creativity and logic blend to form a path forward.
Consider options such as:

• Drawing diagrams to visualize the problem
• Looking for patterns or relationships
• Using algebraic equations
• Breaking the problem into smaller, manageable parts
• Reflecting on similar problems you've solved before

While planning, anticipate potential challenges and contemplate multiple approaches. This planning phase could significantly impact the ease and success of solving the problem. Flexibility is key—be prepared to adjust your plan if it doesn't yield the desired results.

Carrying Out the Plan

After devising your plan, the next step is to carry it out meticulously. This stage is where you put your strategy into action through logical reasoning and computation. Ensure that each step is carefully executed and take your time to avoid errors.
As you work through:

• Closely follow your strategy step by step
• Check calculations for accuracy
• Maintain clear and organized work
• Be methodical and patient

If you encounter difficulties, revisit your plan and consider alternate strategies. Persistence and careful execution are vital here. It's normal for your initial attempts to require adjustments; this is part of the process.

Review and Reflect

The final step in Polya's process is reviewing and reflecting on your solution. Even if you arrive at an answer, this step is crucial for deep learning and future improvement.
Take measures by:

• Verifying that your solution answers the question posed
• Checking each step of your work for correctness
• Evaluating alternative ways to solve the problem
• Understanding what strategies were effective or not
• Reflecting on what could be done differently next time

Reflection helps solidify your understanding and builds problem-solving skills. Consider discussing your solution with others to gain new insights or perspectives. This stage is not just about correcting mistakes, but about enhancing your overall approach to tackling future problems.