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Problem SolvingMr. Wesley Choi
Mathematics KLA
How do you study mathematics?
- Memorize the formula sheet- Learn a series of tricks from textbook and
teachersTrick 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
ADHandshake by A and D can be represented by
Define notations for each person
AB CDE F G
DAHandshake by A and D can be represented by
Define notations for each person
AB CDE F G
CFHandshake by C and F can be represented by
Define notations for each person
AB CDE F G
FCHandshake by C and F can be represented by
Self-asking question
Can I use a picture or diagram that might help to understand the problem?
Draw a diagram and introduce notationsA
A
B
B
C
C
D
D
E
E
F
F
G
G
Draw a diagram and introduce notationsA
A
B
B
C
C
D
D
E
E
F
F
G
G
Handshake by A and D
Draw a diagram and introduce notationsA
A
B
B
C
C
D
D
E
E
F
F
G
G
Handshake by C and F
Second Principle
DEVISE A PLAN
Count the number of 2-letter combinations among the letters
AB CDE F G
DAHandshake by A and B can be represented by
Plan A
Count the total number of Line segments in the diagramA
A
B
B
C
C
D
D
E
E
F
F
G
G
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
A BB CC A
No. of handshakes = 3
Counting by “listing out”
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
A BB CC D
No. of handshakes = 6
Counting by “listing out”D
C AB DD A
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• …
Can we count in a more systematic way?
Immediate Reflection
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
A BA CA D
No. of handshakes = 6
Counting by “listing out systematically”D
B CB D
C D
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
A BA CA D
No. of handshakes = 3 + 2 + 1 = 6
Counting by “listing out systematically”D
B CB D
C D
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
A B
…
A GNo. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21
D
B C
B G
C D
E F G
… F G……C G
Counting by “listing out systematically”
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
Carry out Plan B
A
BC
D
E
FG
No. of handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21
Devise Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0 1
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0 1 3
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0 1 3 6
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0 1 3 6
+ 1 + 2 + 3
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0 1 3 6 10
Carry out Plan C
No. of persons 1 2 3 4 5 6 7
No. of handshakes 0 1 3 6 10 15
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 handshakes = 1247 + 1246 + … + 2 + 1 = ?
No. of persons 1 2 3 4 5 6 7 … 1248
No. of handsha
kes0 1 3 6 10 15 21 … ?
A NEW Analysis
No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?
No. of persons 1 2 3 4 5 6 7 … 1248
No. of handsha
kes0 1 3 6 10 15 21 … ?
Times 2 0 2 6 12 20 30 42
A NEW Analysis
No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?
No. of persons 1 2 3 4 5 6 7 … 1248
No. of handsha
kes0 1 3 6 10 15 21 … ?
Product of
integers
A NEW Analysis
No. of handshakes = 1247 + 1246 + … + 2 + 1 = ?
No. of persons 1 2 3 4 5 6 7 … 1248
No. of handsha
kes0 1 3 6 10 15 21 … ?
Formula
BINGO!!
No. of handshakes = 1247 + 1246 + … + 2 + 1
No. of persons 1 2 3 4 5 6 7 … 1248
No. of handsha
kes0 1 3 6 10 15 21 … ?
Formula …
=
BINGO!!
No. of handshakes = 1247 + 1246 + … + 2 + 1
No. of persons 1 2 3 4 5 6 7 … 1248
No. of handsha
kes0 1 3 6 10 15 21 … ?
Formula … 778128
= = 778128
Further investigation
A B C D E F G
A
B
C
D
E
F
G
Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ?
A B C D E F G
A
B
C
D
E
F
G
A B C D E F G
A
B
C
D
E
F
G
Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ?
A B C D E F G
A
B
C
D
E
F
G
Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ?
A B C D E F G
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 ?
Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ?
Why 1 + 2 + 3 + 4 + 5 + 6 = 6 7 2 ?
6
7
1 + 2 + 3 + … + n =
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
Message of the Day
Thank you !
Problem solving were not inborn qualities but something that could be taught and learnt.
