LESSON 3.2 George Polya's Guidelines for Problem Solving

Question 1

What are George Polya’s four steps for problem solving?

Answer 1

1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look back.

Question 2

What does the first step ‘Understand the problem’ involve?

Answer 2

It involves asking questions, experimenting, or rephrasing the problem in your own words to grasp what is being asked.

Question 3

What is the second step ‘Devise a plan’?

Answer 3

It involves finding connections between the data and the unknown, looking for patterns, relating to previously solved problems, or simplifying the problem.

Question 4

What does ‘Carry out the plan’ mean in Polya’s model?

Answer 4

It means executing the plan step-by-step and checking each step as you go along.

Question 5

What does ‘Look back’ entail?

Answer 5

It means examining the solution and checking if the answer makes sense and meets the problem’s requirements.

Question 6

What are some strategies recommended by Polya for problem solving?

Answer 6

1. Drawing a diagram 2. Solving a simpler problem 3. Making a table 4. Working backwards 5. Guessing and checking 6. Finding a pattern 7. Using a formula or equation 8. Using logical reasoning.

Question 7

What is the strategy of ‘Drawing a diagram’ used for?

Answer 7

It helps visualize the problem, making it easier to understand and solve.

Question 8

What does ‘Solve a simpler problem’ mean as a strategy?

Answer 8

It means breaking down the complex problem into a simpler, more manageable version to find a pattern or understand the solution better.

Question 9

How does ‘Making a table’ help in problem solving?

Answer 9

It helps organize information and keep track of relationships between different variables.

Question 10

What is the strategy of ‘Working backwards’?

Answer 10

It involves starting from the desired outcome and figuring out the steps that lead to it.

Question 11

What is the strategy of ‘Guess and check’?

Answer 11

It involves making an educated guess and checking if it solves the problem, adjusting as needed.

Question 12

How can ‘Finding a pattern’ assist in solving problems?

Answer 12

Recognizing a pattern can provide insight into a general solution that applies to similar problems.

Question 13

What role does using ‘a formula or equation’ play in problem solving?

Answer 13

Using a formula or equation allows you to apply known mathematical rules to solve the problem.

Question 14

How does ‘Using logical reasoning’ help in solving problems?

Answer 14

Logical reasoning helps you make deductions and draw conclusions based on established facts or premises.

Question 15

In Example 25, how many handshakes occurred in a seminar with 30 attendees?

Answer 15

The total number of handshakes is 30(30 - 1) / 2 = 435.

Question 16

What is the formula used to find the number of handshakes in a group of k people?

Answer 16

The formula is k(k - 1) / 2, where k is the number of people in the group.

Question 17

In Example 26, how many magic cards does Andrew have?

Answer 17

Andrew has 4 magic cards.

Question 18

What strategy was used in Example 26 to solve the problem?

Answer 18

The strategy of ‘working backwards’ was used, starting from Patrick’s known number of magic cards.

Question 19

What was the process used to find how many cards Andrew had in Example 26?

Answer 19

1. Patrick has 8 cards.
2. Ken has 12 cards (since 2 × 8 - 4 = 12).
3. Ian has 10 cards (since 8 + 2 = 10).
4. Andrew has 4 cards (since (10 - 2) / 2 = 4).

Question 20

How do you check the solution in Example 26?

Answer 20

By verifying each step: If Andrew has 4 cards, then Ian has 10 cards, Patrick has 8 cards, and Ken has 12 cards, which all match the problem’s conditions.