Polya Model Problem Solving

* First. You have to understand the * First. You have to understand the problem. * What is the unknown? What problem. * What is the unknown? What are the data? What is the condition? * Is are the data? What is the condition? * Is

it possible to satisfy the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the condition sufficient to determine the unknown? Or is it insufficient? Or the unknown? Or is it insufficient? Or redundant? Or contradictory? * Draw a redundant? Or contradictory? * Draw a figure. Introduce suitable notation. * figure. Introduce suitable notation. *

Separate the various parts of the Separate the various parts of the condition. Can you write them down?condition. Can you write them down?

• * Second. Find the connection between * Second. Find the connection between the data and the unknown. You may be the data and the unknown. You may be obliged to consider auxiliary problems obliged to consider auxiliary problems if an immediate connection cannot be if an immediate connection cannot be found. You should obtain eventually a found. You should obtain eventually a plan of the solution. * Have you seen it plan of the solution. * Have you seen it

before? Or have you seen the same before? Or have you seen the same problem in a slightly different form? * problem in a slightly different form? * Do you know a related problem? Do Do you know a related problem? Do you know a theorem that could be you know a theorem that could be

useful? * Look at the unknown! And useful? * Look at the unknown! And try to think of a familiar problem try to think of a familiar problem

having the same or a similar unknown. having the same or a similar unknown. * *

Here is a problem related to yours and Here is a problem related to yours and solved before. Could you use it? Could you solved before. Could you use it? Could you use its result? Could you use its method? use its result? Could you use its method?

Should you introduce some auxiliary Should you introduce some auxiliary element in order to make its use possible? * element in order to make its use possible? * Could you restate the problem? Could you Could you restate the problem? Could you

restate it still differently? Go back to restate it still differently? Go back to definitions. * If you cannot solve the definitions. * If you cannot solve the

proposed problem try to solve first some proposed problem try to solve first some related problem. Could you imagine a more related problem. Could you imagine a more accessible related problem? A more general accessible related problem? A more general

problem? A more special problem? An problem? A more special problem? An analogous problem? Could you solve a part analogous problem? Could you solve a part

of the problem? Keep only a part of the of the problem? Keep only a part of the condition, drop the other part; how far is condition, drop the other part; how far is the unknown then determined, how can it the unknown then determined, how can it

vary?vary?

Could you derive something useful Could you derive something useful from the data? Could you think of from the data? Could you think of

other data appropriate to determine other data appropriate to determine the unknown? Could you change the the unknown? Could you change the

unknown or data, or both if unknown or data, or both if necessary, so that the new unknown necessary, so that the new unknown and the new data are nearer to each and the new data are nearer to each

other? * Did you use all the data? other? * Did you use all the data? Did you use the whole condition? Did you use the whole condition? Have you taken into account all Have you taken into account all essential notions involved in the essential notions involved in the

problem? problem?

3. CARRYING OUT THE 3. CARRYING OUT THE PLAN PLAN * *

Third. Carry out your Third. Carry out your plan. * Carrying out plan. * Carrying out

your plan of the your plan of the solution, check each solution, check each

step. Can you see step. Can you see clearly that the step is clearly that the step is correct? Can you prove correct? Can you prove

that it is correct? that it is correct?

4. Looking Back *4. Looking Back *

* Fourth. Examine the solution * Fourth. Examine the solution obtained. * Can you check the obtained. * Can you check the

result? Can you check the result? Can you check the argument? * Can you derive the argument? * Can you derive the solution differently? Can you see solution differently? Can you see it at a glance? * Can you use the it at a glance? * Can you use the result, or the method, for some result, or the method, for some

other problem?other problem?

ThanksThanks

Prepared byPrepared by

MUHAMMAD BIN BASARMUHAMMAD BIN BASAR

Maths DepartmentMaths Department

IPGM KTJBIPGM KTJB