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Problem Solving. Mr. Wesley Choi Mathematics KLA. 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 problem and so on - Do Chapter & Revision Exercises / Past papers

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## Problem Solving

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**Problem Solving**Mr. Wesley Choi Mathematics KLA**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**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**Your role in learning**You are -Observer - Routine follower - Passive learner**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.**“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**4 Basic Principles of Problem Solving**• Understand the problem • Devise a plan • Carry out the plan • Look back**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?**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?**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?**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?**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**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?**First Principle**UNDERSTAND THE PROBLEM**Self-asking question**Do I understand all the words used in stating the problem?**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**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**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**Self-asking questions**What is the question asking me to find? Can I restate the problem in my own words?**Define notations for each person**A B C D E F G Handshake by A and D can be represented by AD**Define notations for each person**A B C D E F G Handshake by A and D can be represented by DA**Define notations for each person**A B C D E F G Handshake by C and F can be represented by CF**Define notations for each person**A B C D E F G Handshake by C and F can be represented by FC**Self-asking question**Can I use a picture or diagram that might help to understand the problem?**Draw a diagram and introduce notations**Handshake by A and D**Draw a diagram and introduce notations**Handshake by C and F**Second Principle**DEVISE A PLAN**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**Count the total number of Line segments in the diagram**Plan B**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**Self-asking question**Even if I cannot solve this problem, can I think of a more accessible related problem? For example, more specific one.**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**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**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**Immediate Reflection**Can we count in a more systematic way?**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**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**Third Principle**CARRY OUT THE PLAN**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**Carry out Plan B**No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21

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