Problem Solving

1. Problem Solving Mr. Wesley Choi Mathematics KLA

2. How do you study mathematics? - Memorize the formula sheet - Learn a series of tricks from textbook and teachers Trick A for Type A problem; Trick B for Type B problemand so on - Do Chapter & Revision Exercises / Past papers - Follow the above routine

3. Learning Outcome You are - NOT engaging in the real process of solving a problem - NOT able to tackle unfamiliar situations - NOT able to apply the subject in other areas - NOT enjoying learning

4. Your role in learning You are -Observer - Routine follower - Passive learner

5. George Polya (1887 – 1985) • Hungarian-Jewish Mathematician • Professor of Mathematics in Stanford University 1940 - 1953 • Maintain that the skills of problem solving were not inborn qualities but something that could be taught and learnt.

6. “How to solve it?” – G Polya (1945) • Translated into more than 17 languages • For math educators • Describe how to systematically solve problem • Identified 4 basic principlesof problem solving

7. 4 Basic Principles of Problem Solving • Understand the problem • Devise a plan • Carry out the plan • Look back

8. Self-asking questions • Understand the problem • Do I understand all the words used in stating the problem? • What is the question asking me to find? • Can I restate the problem in my own words? • Can I use a picture or diagram that might help to understand the problem? • Is the information provided sufficient to find the solution?

9. Self-asking questions • Devise a plan • Have I seen this question before? • Have I seen similar problem in a slightly different form? • Do I know a related problem? • If yes, could I apply it adequately? • Even if I cannot solve this problem, can I think of a more accessible related problem? For example, more specific one. • Or can I solve only a part of it first?

10. Self-asking questions • Carry out the plan • Can I see clearly the step is correct? • Are these steps presented logically? • Can you prove that it is correct?

11. Self-asking questions • Look back • Can I check the result? • Can all my arguments pass? • Can I derive the result differently? • Can I still solve it if some conditions change? • Can I use the result, or the method, for some other problems?

12. List of Strategieson devising a plan • Look for a pattern • Draw a picture • Solve simpler problem • Use a model • Work backwards • Use a formula • Be ingenious • … • Make an orderly list • Guess and Check • Eliminate possibilities • Use symmetry • Consider special cases • Use direct reasoning • Solve and equation

13. Problem 7 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people?

14. First Principle UNDERSTAND THE PROBLEM

15. Self-asking question Do I understand all the words used in stating the problem?

16. Understand the problem 7 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? No one shakes with oneself

17. Understand the problem 7 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? No one shakes with oneself Each one shakes with everyone

18. Understand the problem 7 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be doneso that everyone should have a chance of handshaking all people? No one shakes with oneself Each one shakes with everyone No repeated handshake by any two persons

19. Self-asking questions What is the question asking me to find? Can I restate the problem in my own words?

20. Define notations for each person A B C D E F G Handshake by A and D can be represented by AD

21. Define notations for each person A B C D E F G Handshake by A and D can be represented by DA

22. Define notations for each person A B C D E F G Handshake by C and F can be represented by CF

23. Define notations for each person A B C D E F G Handshake by C and F can be represented by FC

24. Self-asking question Can I use a picture or diagram that might help to understand the problem?

25. Draw a diagram and introduce notations

26. Draw a diagram and introduce notations Handshake by A and D

27. Draw a diagram and introduce notations Handshake by C and F

28. Second Principle DEVISE A PLAN

29. Count the number of 2-letter combinations among the letters A B C D E F G Plan A Handshake by A and B can be represented by DA

30. Count the total number of Line segments in the diagram Plan B

31. List of Strategies on devising a plan • Look for a pattern • Draw a picture • Solve simpler problem • Use a model • Work backwards • Use a formula • Be ingenious • … • Make an orderly list • Guess and Check • Eliminate possibilities • Use symmetry • Consider special cases • Use direct reasoning • Solve and equation

32. Self-asking question Even if I cannot solve this problem, can I think of a more accessible related problem? For example, more specific one.

33. Make it a smaller value 3 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? B C A Counting by “listing out” A B B C C A No. of handshakes = 3

34. A bigger value 4 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? B C D A Counting by “listing out” A B C A B C B D C D D A No. of handshakes = 6

35. List of Strategies on devising a plan • Look for a pattern • Draw a picture • Solve simpler problem • Use a model • Work backwards • Use a formula • Be ingenious • … • Make an orderly list • Guess and Check • Eliminate possibilities • Use symmetry • Consider special cases • Use direct reasoning • Solve and equation

36. Immediate Reflection Can we count in a more systematic way?

37. Make it a specific one 4 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? B C D A Counting by “listing out systematically” A B B C C D A C B D A D No. of handshakes = 6

38. Make it a specific one 4 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? B C D A Counting by “listing out systematically” A B B C C D A C B D A D No. of handshakes = 3 + 2 + 1 = 6

39. Third Principle CARRY OUT THE PLAN

40. Carry out Plan A 7 people goes to a party and start shaking hands with each other. At least how many times of handshakes should be done so that everyone should have a chance of handshaking all people? G B C D E F A Counting by “listing out systematically” … A B B C C D F G … … … C G B G A G No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21

41. Carry out Plan B

42. Carry out Plan B

43. Carry out Plan B

44. Carry out Plan B

45. Carry out Plan B

46. Carry out Plan B

47. Carry out Plan B

48. Carry out Plan B No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21

49. Devise Plan C

50. Carry out Plan C