Problem Solving Mr Wesley Choi Mathematics KLA How

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 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 principles of 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 Strategies on devising a plan • • 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 …

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 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

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 AB CDE F G Handshake by A and D can be represented by AD

Define notations for each person AB CDE F G Handshake by A and D can be represented by DA

Define notations for each person AB CDE F G Handshake by C and F can be represented by CF

Define notations for each person AB CDE 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

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 Plan A AB CDE F G 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 • • 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 …

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? A B C Counting by “listing out” AB BC CA 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? A B C D Counting by “listing out” AB CA BC BD CD DA No. of handshakes = 6

List of Strategies on devising a plan • • 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 …

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? A B C D Counting by “listing out systematically” AB BC CD AC BD AD 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? A B C D Counting by “listing out systematically” AB AC AD BC BD CD 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? A B C D E F G Counting by “listing out systematically” … … AG … AB BC CD … FG C G BG No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21

Carry out Plan B

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

Devise Plan C No. of persons No. of handshakes 1 2 3 4 5 6 7

Carry out Plan C No. of persons 1 No. of handshakes 0 2 3 4 5 6 7

Carry out Plan C No. of persons 1 2 No. of handshakes 0 1 3 4 5 6 7

Carry out Plan C No. of persons 1 2 3 No. of handshakes 0 1 3 4 5 6 7

Carry out Plan C No. of persons 1 2 3 4 No. of handshakes 0 1 3 6 5 6 7

Carry out Plan C No. of persons 1 2 3 4 No. of handshakes 0 1 3 6 +1 +2 +3 5 6 7

Carry out Plan C No. of persons 1 2 3 4 5 No. of handshakes 0 1 3 6 10 6 7

Carry out Plan C No. of persons 1 2 3 4 5 6 No. of handshakes 0 1 3 6 10 15 7

Carry out Plan C No. of persons 1 2 3 4 5 6 7 No. of handshakes 0 1 3 6 10 15 21

Fourth Principle LOOK BACK

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

Self-asking question Can I still solve it if some conditions change?

Condition Changed 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 = ?

A NEW Analysis No. of persons 1 2 3 4 5 6 7 … 1248 No. of handsha kes 0 1 3 6 10 15 21 … ? No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?

A NEW Analysis No. of persons 1 2 3 4 5 6 7 … 1248 No. of handsha kes 0 1 3 6 10 15 21 … ? Times 2 0 2 6 12 20 30 42 No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?

A NEW Analysis No. of persons 1 2 3 4 5 6 7 … 1248 No. of handsha kes 0 1 3 6 10 15 21 … ? Product of integers No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?

A NEW Analysis No. of persons 1 2 3 4 5 6 7 … 1248 No. of handsha kes 0 1 3 6 10 15 21 … ? Formula No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?

BINGO!! No. of persons 1 2 3 4 5 6 7 … 1248 No. of handsha kes 0 1 3 6 10 15 21 … ? Formula … No. of handshakes = 1247 + 1246 + … + 2 + 1

BINGO!! No. of persons 1 2 3 4 5 6 7 … 1248 No. of handsha kes 0 1 3 6 10 15 21 … ? … 778128 Formula No. of handshakes = 1247 + 1246 + … + 2 + 1 = 778128

Further investigation A A B C D E F G

Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ? A A B C D E F G

Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ?

Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ? 7 6

Self-asking question Can I use the result, or the method, for some other problems?

Extend Induce NEW 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 = ?

Math teachers • 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

Your action 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 • •

How to create chocolate out of nothing?

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