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Mr. Wesley Choi Mathematics KLA

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-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 -Follow the above routine

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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

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You are -Observer -Routine follower -Passive learner

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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.

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Translated into more than 17 languages For math educators Describe how to systematically solve problem Identified 4 basic principles of problem solving

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Understand the problem Devise a plan Carry out the plan Look back

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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?

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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?

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Carry out the plan – Can I see clearly the step is correct? – Are these steps presented logically? – Can you prove that it is correct?

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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?

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

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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?

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UNDERSTAND THE PROBLEM

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Do I understand all the words used in stating the problem?

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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

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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

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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 No repeated handshake by any two persons

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What is the question asking me to find? Can I restate the problem in my own words?

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ABCDEFGABCDEFG AD Handshake by A and D can be represented by

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ABCDEFGABCDEFG DA Handshake by A and D can be represented by

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ABCDEFGABCDEFG CF Handshake by C and F can be represented by

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ABCDEFGABCDEFG FC Handshake by C and F can be represented by

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Can I use a picture or diagram that might help to understand the problem?

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Handshake by A and D

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Handshake by C and F

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DEVISE A PLAN

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ABCDEFGABCDEFG DA Handshake by A and B can be represented by Plan A

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Plan B

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

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

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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? A BC A B B C C A No. of handshakes = 3 Counting by “listing out”

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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? A BC A B B C C D No. of handshakes = 6 Counting by “listing out” D C A B D D A

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

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Can we count in a more systematic way?

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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? A BC A B A C A D No. of handshakes = 6 Counting by “listing out systematically” D B C B D C D

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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? A BC A B A C A D No. of handshakes = 3 + 2 + 1 = 6 Counting by “listing out systematically” D B C B D C D

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CARRY OUT THE PLAN

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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? A BC A B … A G No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21 D B C B G C D EF G … F G … … C G Counting by “listing out systematically”

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No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21

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No. of persons 1234567 No. of handshakes

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No. of persons 1234567 No. of handshakes 0

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No. of persons 1234567 No. of handshakes 01

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No. of persons 1234567 No. of handshakes 013

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No. of persons 1234567 No. of handshakes 0136

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No. of persons 1234567 No. of handshakes 0136 + 1 + 2 + 3

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No. of persons 1234567 No. of handshakes 013610

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No. of persons 1234567 No. of handshakes 01361015

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No. of persons 1234567 No. of handshakes 0136101521

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LOOK BACK

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NOT simply a check of the correctness of the solution An extension of mental process of reexamining the result and the path that led to it Is a process that may consolidate your knowledge and develop the real ability of problem solving

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Can I still solve it if some conditions change?

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There are 1248 students in the hall and they 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. of handshakes = 1247 + 1246 + … + 2 + 1 = ?

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No. of persons 1234567… 1248 No. of handsha kes 0136101521…?

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No. of handshakes = 1247 + 1246 + … + 2 + 1 = ? No. of persons 1234567… 1248 No. of handsha kes 0136101521…? Times 2 02612203042

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No. of handshakes = 1247 + 1246 + … + 2 + 1 = ? No. of persons 1234567… 1248 No. of handsha kes 0136101521…? Product of integers

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No. of handshakes = 1247 + 1246 + … + 2 + 1 = ? No. of persons 1234567… 1248 No. of handsha kes 0136101521…? Formula

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No. of handshakes = 1247 + 1246 + … + 2 + 1 No. of persons 1234567… 1248 No. of handsha kes 0136101521…? Formula …

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No. of handshakes = 1247 + 1246 + … + 2 + 1 No. of persons 1234567… 1248 No. of handsha kes 0136101521…? Formula … 778128 = 778128

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ABCDEFG A B C D E F G

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ABCDEFG A B C D E F G

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ABCDEFG A B C D E F G

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ABCDEFG A B C D E F G

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ABCDEFG A B C D E F G

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6 7

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Can I use the result, or the method, for some other problems?

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-“Hug-Hug” problem -Combination problem of selecting 2 objects from n different objects -Line intersection problem – find maximum number of intersections made by n straight lines -Series Sum problem – find the sum of 1 + 3 + 5 + … + 2013 = ?

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Will try to occasionally incorporate problem solving tasks in the lesson Will encourage and facilitate you to think more on approaching problems Provide some recreational math problems

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Willing to take the first step Develop good mental habit Experience yourself in different strategies Accumulate the experiences of independent work You are not solely solving a problem, but developing an ability to solve future problems

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Thank you ! Problem solving were not inborn qualities but something that could be taught and learnt.

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