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Here are 7 Pólya problem-solving steps with answers:
1. Understand the Problem:
Problem: Find the area of a rectangle with a length of 8 cm and a width of 5 cm.
Answer: The area of the rectangle is 8 cm x 5 cm = 40 square cm.
2. Devise a Plan:
Problem: How many ways can you arrange 4 different books on a shelf?
Answer: There are 4! = 4 x 3 x 2 x 1 = 24 ways to arrange the books.
3. Carry Out the Plan:
Problem: Solve the equation 2x + 5 = 11.
Answer: Subtract 5 from both sides to get 2x = 6, then divide by 2 to find x = 3.
4. Look Back:
Problem: A car travels at 60 km/h for 2 hours. How far did it travel?
Answer: Distance = speed x time = 60 km/h x 2 h = 120 km. Confirming the answer against the original problem ensures accuracy.
5. Generalize:
Problem: Find the 10th term of the sequence: 2, 4, 8, 16, ...
Answer: Each term doubles the previous one, so the 10th term is 2^10 = 1024.
6. Make an Extension:
Problem: If a triangle has sides of length 4 cm, 5 cm, and 6 cm, is it a right triangle?
Answer: Check if the Pythagorean theorem holds: 4^2 + 5^2 = 16 + 25 = 41 which is not equal to 6^2, so it is not a right triangle.
7. Solve a Similar Problem:
Problem: If a basketball player makes 70% of their free throws and attempts 20 free throws, how many can we expect them to make?
Answer: 70% of 20 is 0.7 x 20 = 14, so we can expect the player to make 14 free throws.
These 7 Pólya problem-solving steps illustrate a structured approach to solving a variety of mathematical problems, emphasizing understanding, planning, execution, reflection, generalization, extension, and tackling similar problems for improvement and mastery.