17099989 Polya Model Strategic Problem Solving

8/2/2019 17099989 Polya Model Strategic Problem Solving

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GEORGE POLYA

STRATEGIC

PROBLEMSOLVING

NON-ROUTINE

BASIC MATHEMATICS


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HOW TO

SOLVE IT?


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First. You have to understandthe 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 variableswhich have been ascertained.

Understand which variable needs to be

thoroughly searched or answered. Draw a figure. Introduce suitable notation.

1)UNDERSTANDING THE PROBLEM


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Second. Find the connection between thedata and the unknown. You may be obliged toconsider auxiliary problems if an immediateconnection cannot be found. You should obtaineventually aplan 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 goal5. guess and check 11. simulation

6. construct table 12. identify of math pattern

2) DEVISING A PLAN


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3) CARRYING OUT THE PLAN

Third.Carry outyour plan.

Carrying out your plan of thesolution, check each step. Can yousee clearly that the step is correct?

Can you prove that it is correct?


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4) Looking Back @ CheckingAnswers

Fourth.Examine the solutionobtained.

Use another way to solve the sameproblems.

Adopting the inverse method.

E.g.: division multiplication Can you use the result, or the

method, for some other problem?


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Jacinskis Hardware has anumber of bikes and tricycles

for sale. There are 27 seatsand 60 wheels all together.Determine how many bikes

there are and how manytricycles there are.

Problem 1


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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


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Step 3 : Carry out the plan

Strategy 1: Construct a table

No. ofbikes

No. oftricycle

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


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Strategy 2 : Draw diagram

First, draw all the 27 seats. Then add a wheel to eachseat and when it reaches to the 27th seat, repeat backuntil there are 60 wheels.

Note : Each circle is a seat and each leg is a wheel.


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The grey circle is the bike.

The green circle is thetricycle.

There are 21 diagram of bikesand 6 diagram of tricycles.


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Step 4 : Check the Answer

Use the inverse method:

multiplication division

(21x2)+(6x3)=60

21x2=42 and 6x3=18 422=21 and 183=6

21+6=27 seats

Proven true!!


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Problem 2How many rectangles are there in each of thesefigures?


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answer


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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Strategy 1 : Draw diagram

First, draw all the rectangles. The first figure has only 1rectangle. Then add with the number of rectangle belowit. It continues till the end of the figure.

1 rectangles

3 rectangles

6 rectangles

10 rectangles

15 rectangles


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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


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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


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Step 4 : check the answers Add all the rectangles in the figures.

1

+ 2

+ 3

+ 4

+ 515

It is proven that there are 15 rectangles

in the figures.


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Problem 3

In three bowling games, Luluscored 139, 143, and 144. Whatscore will she need in a fourthgame in order to have anaverage score of 145 for all fourgames?


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answer


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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Strategy 1 : Algebra

X = unknown score

139 + 143 + 144 + x = 145

4

X = 154


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Strategy 2 : Logic

If average needs to be 145, and thereare 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.


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Strategy 3 : make a chart

GAME SCORE AWAYFROMAVERAGE

TOTALAWAY

1 139 -6 -6

2 143 -2 -8

3 144 -1 -9

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


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Step 4 : check the answers

Multiply the average score with 4games

145 x 4 =(average) 580

139 + 143 + 144 + x = 580

X = 580 139 143 144

X = 154#

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17099989 Polya Model Strategic Problem Solving

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