# Mr. Wesley Choi Mathematics KLA. -Memorize the formula sheet -Learn a series of tricks from textbook and teachers Trick A for Type A problem; Trick B.

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Mr. Wesley Choi Mathematics KLA

-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

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

You are -Observer -Routine follower -Passive learner

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.

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

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

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?

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?

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

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?

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 …

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?

UNDERSTAND THE PROBLEM

Do I understand all the words used in stating 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

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

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

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

ABCDEFGABCDEFG AD Handshake by A and D can be represented by

ABCDEFGABCDEFG DA Handshake by A and D can be represented by

ABCDEFGABCDEFG CF Handshake by C and F can be represented by

ABCDEFGABCDEFG FC Handshake by C and F can be represented by

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

Handshake by A and D

Handshake by C and F

DEVISE A PLAN

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

Plan B

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 …

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

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”

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

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 …

Can we count in a more systematic way?

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

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

CARRY OUT THE PLAN

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”

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

No. of persons 1234567 No. of handshakes

No. of persons 1234567 No. of handshakes 0

No. of persons 1234567 No. of handshakes 01

No. of persons 1234567 No. of handshakes 013

No. of persons 1234567 No. of handshakes 0136

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

No. of persons 1234567 No. of handshakes 013610

No. of persons 1234567 No. of handshakes 01361015

No. of persons 1234567 No. of handshakes 0136101521

LOOK BACK

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

Can I still solve it if some conditions change?

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 = ?

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

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

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

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

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

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

ABCDEFG A B C D E F G

ABCDEFG A B C D E F G

ABCDEFG A  B  C  D  E  F  G 

ABCDEFG A  B  C  D  E  F  G 

ABCDEFG A  B  C  D  E  F  G 

6 7

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

-“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 = ?

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

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

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

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