GEORGE POLYA

STRATEGIC PROBLEM SOLVING

NON-ROUTINE

BASIC MATHEMATICS

HOW TOHOW TO

SOLVE IT? SOLVE IT?

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)1)UNDERSTANDING THE PROBLEMUNDERSTANDING THE PROBLEM

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: 1. select suitable operations 7. working backward

2. use suitable diagram 8. simplify the problem 3. use analogy 9. using experiment 4. use the unitary approach 10. identify sub goal 5. guess and check 11. simulation 6. construct table 12. identify of math pattern

2)2) DEVISING A PLANDEVISING A PLAN

3)3) CARRYING OUT THE CARRYING OUT THE PLANPLAN

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?

4) Looking Back @ Looking Back @ Checking AnswersChecking 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?

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.

Problem 1Problem 1

answeranswer

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

+4PATTER

N-4

21 bikes and 6 tricycles

• Strategy 2 : Draw diagram

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.

The greygrey circle is the bike. The greengreen circle is the

tricycle.

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

Step 4 : Check the Answer

• Use the inverse method:

multiplication division

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

Proven true!!

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

answeranswerStep 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”

• Step 3 : Carry out the plan• Strategy 1 : Draw diagramStrategy 1 : Draw diagram

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 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10 1 + 2 + 3 + 4 + 5 = 15

Pattern -3

Pattern -4

Pattern -5

There are 15 rectangles

Strategy 3 : “Gauss’ trick”

1 x 5 1 x 4 1 x 3 1 x 2 1 x 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 3Problem 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?

answeranswerStep 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

• Step 3 : Carry out the plan

• Strategy 1 : Algebra

X = unknown score

139 + 143 + 144 + x = 145

4

X = 154

Strategy 2 : Logic

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.The missing score is 154.

Strategy 3 : make a chartStrategy 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

145 x 4 =(average) 580 139 + 143 + 144 + x = 580

X = 580 – 139 – 143 – 144 X = 154#

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