Polya Model (strategic Problem Solving)

STRATEGIC PROBLEM SOLVING NON-ROUTINE

GEORGE POLYA

BASIC MATHEMATICS

HOW TO SOLVE IT?

UNDERSTANDING THE PROBLEM  

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First. You have to understand the problem. What is the unknown? What are the data? What is the condition? Detect the variables involved in the problem. Know the relationship between the variables which have been ascertained. Understand which variable needs to be thoroughly searched or answered. Draw a figure. Introduce suitable notation.

1) DEVISING A PLAN Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution. Consider the following strategies:

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1. select suitable operations 2. 3. 4. 5. 6.

use suitable diagram use analogy use the unitary approach guess and check construct table

7. working backward 8. simplify the problem 9. using experiment 10. identify sub goal 11. simulation 12. identify of math pattern

1) CARRYING OUT THE PLAN  

Third. Carry out your plan. Carrying 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?

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Looking Back @ Checking Answers

Fourth. Examine the solution obtained. Use another way to solve the same problems. Adopting the inverse method. E.g.: division multiplication Can you use the result, or the method, for some other problem?

Problem 1 Jacinski’s Hardware has a number of bikes and tricycles for sale. There are 27 seats and 60 wheels all together. Determine how many bikes there are and how many tricycles there are.

answer Step 1 : Understand the problem • each bike has 2 wheels • each tricycle has 3 wheels • 1 bike 1 seat, 1 tricycle 1 seat • there are 27 seats = no. of bike + no. of tricycle

Step 2 : Devise a plan • Strategy 1 : Construct a table • Strategy 2 : Draw a diagram

Step 3 : Carry out the plan • Strategy 1: Construct a table No. of bikes

No. of tricycle

No. of wheels

15

12

(15x2)+(12x3)=66

16

11

(16x2)+(11x3)=65

17

10

(17x2)+(10x3)=64

21

6

(21x2)+(6x3)=60

+4 21 bikes and 6 tricycles

PATTER N -4

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Strategy 2 : Draw diagram

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First, draw all the 27 seats. Then add a wheel to each seat and when it reaches to the 27th seat, repeat back until there are 60 wheels. Note : Each circle is a seat and each leg is a wheel.

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The grey circle The green circle tricycle.

is the bike. is the

There are 21 diagram of bikes and 6 diagram of tricycles.

Step 4 : Check the Answer •

Use the inverse method:

multiplication    

(21x2)+(6x3)=60 21x2=42 and 6x3=18 42÷2=21 and 18÷3=6 21+6=27 seats

Proven true!!

division

Problem 2 How many rectangles are there in each of these figures?

answer Step 1 : Understand the problem • each figures is a rectangle

Step 2 : Devise a plan • Strategy 1 : Draw a diagram • Strategy 2 : Look a pattern • Strategy 3 : “gauss’ trick”

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Step 3 : Carry out the plan

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Strategy 1 : Draw diagram

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First, draw all the rectangles. The first figure has only 1 rectangle. Then add with the number of rectangle below it. It continues till the end of the figure. 1 rectangles 3 rectangles 6 rectangles 10 rectangles 15 rectangles

Strategy 2 : look for a pattern    

1 1 1 1

+ + + +

2 2 2 2

= + + +

Pattern -3 3 Pattern -4 3=6 3 + 4 = 10 Pattern -5 3 + 4 + 5 = 15

There are 15 rectangles

Strategy 3 : “Gauss’ trick”     

1 1 1 1 1

x x x x x

5 4 3 2 1

1 + 2 + 3 + 4 + 5 = 15 rectangle

For this last one : 1 + 2 + 3 +

+ 14 + 15 + 16

17 x 8 = 136

Step 4 : check the answers 

Add all the rectangles in the figures.

+ + + +

1 2 3 4 5 15

 It is proven that there are 15 rectangles in the figures.

Problem 3 In three bowling games, Lulu scored 139, 143, and 144. What score will she need in a fourth game in order to have an average score of 145 for all four games?

answer Step 1 : Understand the problem • three bowling games, lulu score 139, 143, and 144 • average score is 145 for all four games.

Step 2 : Devise a plan • Strategy 1 : Algebra • Strategy 2 : Logic • Strategy 3 : Make a chart

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Step 3 : Carry out the plan

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Strategy 1 : Algebra

X = unknown score 139 + 143 + 144 + x 4 X = 154

= 145

Strategy 2 : Logic 

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If average needs to be 145, and there are 4 scores. The sum is 4 x 145 =580 From 580, subtract 139, 143, 144. X = 580-139-143-144 = 154

The missing score is 154.

Strategy 3 : make a chart GAME

SCORE

AWAY FROM AVERAGE

TOTAL AWAY

1

139

-6

-6

2

143

-2

-8

3

144

-1

-9

On the 4th games, it needs to be +9 over average.

Step 4 : check the answers 

Multiply the average score with 4 games

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145 x 4 =(average) 580

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139 + 143 + 144 + x = 580

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X = 580 – 139 – 143 – 144 X = 154#

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