Problem Solving

1. Problem Solving Goal-oriented, state-to-state search Most notes from Dr. Michael Wick

2. Applied Strategy: Draw a Diagram

3. Curly used a shovel to dig his own swimming pool. He figured he needed a pool because digging it was hard work and he could use it to cool off after working on it all day. He also planned to build a rectangular concrete deck around the pool that would be 6 feet wide at all points. The pool is rectangular and measures 14 feet by 40 feet. What is the area of the deck? The Pool Deck

4. Try ideas. 52 feet + 26 feet + 52 feet + 26 feet = 156 feet 156 feet x 6 feet = 936 square feet Counts each corner twice!

5. Keep drawing diagrams. Two lengths: 40 ft x 6 ft x 2 = 480 sq ft Two widths: 14 ft x 6 ft x 2 = 168 sq ft Four corners: 6 ft x 6 ft x 4 = 144 sq ft Total 792 sq ft

6. And another . . . 52 ft x 6 ft = 312 sq ft 312 sq ft x 2 = 624 sq ft for extended lengths 14 ft x 6 ft = 84 sq ft 84 sq ft x 2 = 168 sq ft for widths Total = 624 sq ft + 168 sq ft = 792 sq ft

7. And another . . . Area of entire figure = 52 ft x 26 ft = 1352 sq ft Area of pool alone = 40 ft x 14 ft = 560 sq ft Area of deck = 1352 – 560 = 792 sq ft

8. Applied Strategy: Make a Systematic List

9. The object of the game Frisbin is to throw three Frisbees at three different-sized bins that are set up on the ground about 20 feet away from the player. If a Frisbee lands in the largest bin, the player scores 1 point. If a Frisbee lands in the medium-sized bin, the player scores 5 points. If a Frisbee lands in the smallest bin, the player scores 10 points. Kirk McCoy is playing the game. If all three of his Frisbees land in bins, how many different total scores can he make? The Game of Frisbin

10. Alternative Systematic Lists

11. Use Matrix Logic

12. Tom, John, Fred, and Bill each brought his favorite food to a dinner. From the clues below, determine each man’s occupation and favorite food. Tom is neither the nurse nor the teacher. Fred and the pilot play in a jazz band together. The burger lover and the teacher are not musically inclined. Tom brought hot dogs. Bill sat next to the burger fan and across from the steak lover. The secretary does not play an instrument or sing.

13. Make a chart (matrix).

14. Mark traits(ignoring musical cues).

15. Combine with musical clues (perhaps using adjunct list).

16. Eliminate possibilities.

17. Act on eliminated possibilities and eliminate more.

18. Substitute Fred’s food marks at bottom of nurse column.

19. Act on eliminated possibilities and eliminate more.

20. Cross-correlate “Secretary” and “John.”

21. Eliminate possibilities and act on eliminated possibilities.

22. Read off solution. Tom Pilot Hdog John Scty Burg Fred Nurse Steak Bill Tchr Chkn

23. Identify Subproblems

24. A mixture is 25% red paint, 30% yellow paint, and 45% water. If 4 quarts of red paint are added to 20 quarts of the mixture, what is the percentage of red paint in the new mixture? Subproblems

25. What do you need to know? How many quarts of red paint are in the new mixture? How many quarts of paint are in the new mixture? What percentage of the new mixture is red paint?

26. What do you need to know to answer these questions? How many quarts of red paint are in the new mixture? How many quarts of red paint are in the original mixture? And how many quarts of paint are in the original mixture?

27. Rearrange the list in an order that the questions can be answered. How many quarts of paint are in the original mixture? How many quarts of red paint are in the original mixture? How many quarts of red paint are in the new mixture? How many quarts of paint are in the new mixture? What percentage of the new mixture is red paint?

28. Solve the subproblems in order. How many quarts of paint are in the original mixture? How many quarts of red paint are in the original mixture? How many quarts of red paint are in the new mixture? 20 (given) 25% of 20 = 5 quarts 5 + 4 = 9 quarts

29. Keep solving. How many quarts of paint are in the new mixture? What percentage of the new mixture is red paint? 20 + 4 = 24 quarts 9 / 24 = 0.375 = 37.5%

30. Applied Tactic: Solve an Easier Problem First

31. Common Ways to Create an Easier Problem Use a number instead of a variable Use smaller or easier numbers Do a set of specific easier examples and look for a pattern Do a specific easier example and figure out an easier process Change, fix, or get rid of some conditions Eliminate unnecessary information

32. Averages The average of a group of quiz scores is 31.8. There are k quiz scores in the group. The average of 10 of these quiz scores is 24.3. Find the average of the remaining quiz scores in terms of k.

33. The average of a group of quiz scores is 31.8 30. There are k quiz scores in the group. The average of 10 of these quiz scores is 24.3 25. Find the average of the remaining quiz scores in terms of k. Use a smaller or easier number.

34. The average of a group of quiz scores is 30. There are k50 quiz scores in the group. The average of 10 of these quiz scores is 25. Find the average of the remaining quiz scores in terms of k 50. Use a number instead of a variable.

35. Sum of all scores is 30 x 50 = 1500. Sum of 10 scores is 25 x 10 = 250. Sum of other scores is 1500 – 250 = 1250. Average of those 40 scores is Solve the simpler problem.

36. Apply the process to the original problem.

37. In this election, there are 29 issues and candidates. In the last election, there were 28,311 voters, representing 18,954 households, and they voted at 14 polling places. This time there will be 34,892 voters. How many polling places will be needed? SimpleTown Elections

38. Use smaller or easier numbers. • Polling places (last election): 15 • Voters (last election): 30,000 • Households: 20,000 • Issues: 30 • Voters (this election): 35,000 • Polling places (this election): ?

39. Polling places (last election): 15 Voters (last election): 30,000 Households: 20,000 Issues: 30 Voters (this election): 35,000 Polling places (this election): ? Eliminate unnecessary information.

40. Solve the simpler problem. voters/polling place polling places (this election)

41. Apply the process to the original problem. 28,311 14 ≈ 2,022.2 voters per polling place

42. The divisors of 360 add up to 1170. What is the sum of the reciprocals of the divisors of 360? Divisors & Reciprocals

43. Use divisors of 24, which are 1, 2, 3, 4, 6, 8, 12, and 24. Their sum is 60. The sum of reciprocals is Do an easier problem to develop a process.

44. Try another easier problem. The divisors of 10 are 1, 2, 5, and 10. Their sum is 18. The sum of reciprocals is 0

45. Apply the process to the original problem.

46. How Many Squares How many squares are there on a checkerboard?

47. Do a set of easier examples and find a pattern.

48. Generalize the pattern to solve the original problem. 1 8x8 square 4 7x7 squares 9 6x6 squares 16 5x5 squares 25 4x4 squares 36 3x3 squares 49 2x2 squares 64 1x1 squares total: 204 squares

49. Next Train East A train leaves Roseville heading east at 6:00 a.m. at 40 miles per hour. Another eastbound train leaves on a parallel track at 7:00 a.m. at 50 miles per hour. What time will it be when the two trains are the same distance away from Roseville?

50. Change, fix, or get rid of some conditions. One possibility: Change the condition so that the first train travels for an hour and then stops, and the second travels at 10 miles per hour.