Solving Word Problems George Polya’s Four-step Problem-Solving

Process

As part of his work on problem solving, Polya

developed a four-step problem-solving process

similar to the following:

A. Understanding the Problem

1. Can you state the problem in your own

words?

2. What are you trying to find or do?

3. What are the unknowns?

4. What information do you obtain from the

problem?

5. What information, if any, is missing or not

needed?

B. Devising a Plan

The following list of strategies, although not

exhaustive, is very useful:

1. Look for a pattern.

2. Examine related problems and determine if

the same technique can be applied.

3. Examine a simpler or special case of the

problem to gain insight into the solution of

the original problem.

4. Make a table.

5. Make a diagram.

6. Write an equation.

7. Use a guess and check.

8. Work backward.

9. Identify a sub goal.

C. Carrying out the Plan

1. Implement the strategy in Step 2 and

perform any necessary actions or

computations.

2. Check each step of the plan as you proceed.

This may be intuitive checking or a formal

proof of each step.

3. Keep an accurate record of your work.

D. Looking Back

1. Check the results in the original problem. In

some cases, this will require a proof.

2. Interpret the solution in terms of the original

problem. Does your answer make sense? Is

it reasonable?

3. Determine whether there is another method

of finding the solution.

4. If possible, determine other related or more

general problems for which the techniques

will work.

(Source:

http://www.drkhamsi.com/classe/polya.html)

Solution Strategies & Tips for Particular Word

Problems

A. Representations for Number Problems

1. If the sum of two numbers is s and x is one

number, then the other number is s x .

2. If the difference between two numbers is d

and x is the smaller number, then the larger

number is d x .

3. If the first number is x and another is k times

the first, then the other number is kx .

4. If the first number is x and another number

is k more than the first, then the other

number is x k .

5. If the first number is x and another

number is k less than the first, then the

other number is x k .

6. Consecutive integers: If x is the first

integer, then 1x is the 2nd

, 2x is the

3rd

, etc.

7. Consecutive even/odd integers: If x is the

first even/odd integer, then 2x is the

2nd

even/odd integer, 4x is the 2nd

even/odd integer, etc.

8. Digits. A two digit number can be written

in the from 10T U , where T is the ten’s

digit and U is the unit’s digit. A three-

digit number can be written in the form

100 10H T U , where H is the

hundred’s digit.

N.B. If the digit of a two-digit number

10T U is REVERSED, the new number

becomes 10U T . Reversing a three-digit

number means reading the number backwards;

i.e., 100 10H T U becomes 100 10U T H .

B. Age Problems

If a is the present age of person A, then a y

is the age of A y years ago, while a y is the

age of A y years from now (or hence).

Age table for age problems:

Persons Age now

Age some

years ago or

from now

Mathematics Reviewer

WORD PROBLEMS IN ALGEBRA

C. Mixture/Collection Problems

For mixtures, the general equation is

amount of % concen- amount of

× = solution tration substance

while for those involving prices or money,

price or cost totol cost

quantity × = per unit or price

Types of

quantities

No. of

units

Amount/price/percent

per unit

Total

amount

TOTAL = sum of all entries at last column

D. Motion Problems

General Formula: distance rate time

d rt

Situations:

1. Overtaking = equal distances covered

2. Bodies moving in opposite directions = the

distance apart is the sum of the distances

traveled by each body

3. Bodies moving toward each other = the sum of

the distance between the origin of 1st body and

2nd

body to the meeting point is equal to the sum

of the distances between the origin of each point

4. For bodies traveling in air/current:

rate against the rate in still rate of wind

wind/current wind/water or current

rate with the rate in still rate of wind

wind/current wind/water or current

E. Work Problems

If person A can finish a job alone in a time units

and B can finish the same job alone in b time

units, then

1. A can finish 1

a of the job in 1 time unit,

while B can finish 1

b of the job in 1 time

unit. So after k time units, A and B can

finish 1

ka

and 1

kb

of the job,

respectively.

2. Together, A and B can finish 1 1 1

a b x of

the job in 1 time unit, where x is the no. of

time units that A and B can finish the job

together.

3. Suppose that A started working the job

along in the first p time units then B joined

for x more time units until they finished the

job, then 1 1 1

1p xa a b

where 1

represents the whole job.

4. If A and B were doing the job together for

the first q time units until B left, letting A

finish the job alone in x more time units,

then 1 1 1

1q xa b a

.

Note: If B undoes what A does, the “+” becomes a

“” between 1/a and 1/b in #s 2, 3, and 4.

The situations described above can be extended for

3 or more persons doing a job.

F. Some Formulas in Geometry

1. Perimeter Formulas

a. square: 4 , length of one sideP s s

b. rectangle: 2 2P l w (l = length, w

= width

c. triangle: P a b c (a, b, and c are

the sides of the triangle)

d. circle (circumference): 2C r (r =

radius)

2. Area formulas

a. square: 2A s

b. rectangle: A lw

c. triangle: 1

2A bh

d. circle: 2A r

3. Volume Formulas

a. cube: 3V s

b. rectangular solid/prism: V lwh

c. sphere: 34

3V r

d. right circular cylinder: 2V r h

e. right circular cone: 21

3V r h

Anything you can solve in five minutes should not be considered a problem.

-- George Polya

© gjnabueg 2 August 2003

Revised 19 July 2009

All rights reserved

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