Chapter 2 Chapter 2: Problem Solving 1

Chapter 2

Problem Solving

Chapter 2 Problem Solving 2

Problem Solving

A great discovery solves a great problem but there is a grain ofdiscovery in the solution of any problem. Your problem may bemodest; but if it challenges your curiosity and brings into playyour inventive faculties, and if you solve it by your own means,you may experience the tension and enjoy the triumph ofdiscovery. Such experiences at a susceptible age may create ataste for mental work and leave their imprint on mind andcharacter for a lifetime.

– George Pólya

How To Solve It

Chapter 2 Problem Solving Process 3

Problem Solving Process

Phase 1: Understanding the problem

Phase 2: Devising a plan

Phase 3: Carrying out the plan

Phase 4: Looking back

Chapter 2 Phase 1: Understanding the problem

4

1: Understanding the problem

What is the unknown? What are the data?

What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown?

Draw a figure. Introduce suitable notation.

Chapter 2 Phase 2: Devising a plan 5

2: Devising a plan

How you seen the problem before? Do you know a related problem?

Look at the unknown. Think of a problem having the same or similar unknown.

Split the problem into smaller sub-problems.

If you can’t solve it, solve a more general version, or a special case, or part of it.

Chapter 2 Phase 3: Carrying out the plan 6

3: Carrying out the plan

Carry out your plan of the solution. Check each step.

Can you see clearly that the step is correct?

Can you prove that it is correct?

Chapter 2 Phase 4: Looking back 7

4: Looking back

Can you check the result?

Can you derive the result differently?

Can you use the result, or the method, for some other problem?

Chapter 2 Area of circle 8

Area of circle

What is the data? Side of square = 2a

What is the unknown? Area of circle, C.

What is the condition? If radius r is known, C can be calculated.

How to obtain r?

2a

Chapter 2 Area of circle 9

Area of circle

Pythagoras’ theorem

r2 = 2 * a2

Area of circle

C = * r2 = * 2 * a2

a

a r

Chapter 2 Pascal's triangle 10

Pascal’s triangle

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

nCk = n! / (k! * (n-k)!)

Chapter 2 NE-paths 11

NE-paths

To find number of NE-paths between any two points.

C

A

Chapter 2 Creative thinking 12

Creative thinking

Most new discoveries are suddenly-seen things that were always there. A new idea is a light that illuminates presences while simply had no form for us before the light fell on them.

Susan Langer

Chapter 2 Creative thinking 13

Creative thinking

Myths

Creativity requires genius

You have to be odd

Creative thinking isn’t rigorous, but uses simple processes

Chapter 2 Homework 14

Homework

Try exercises behind chapter 2.