Learning how to solve problems is a vital part of math. When a problem stumps you at first, check this list of ideas. It will help you break down the problem-solving process by suggesting things you can ask yourself each step of the way. [Revised from How To Solve It, by George Polya, 1945.]
Step 1: Understanding the Problem
First, you have to clearly understand the problem.
What are you being asked to find? It often helps to ‘define variables’: “Let x =” this quantity.
What information are you given? It might help to organize the information, maybe in a table.
Draw a picture. Try to show the relationships; don't worry about good artwork.
Step 2: Planning Your Path
Second, you want to come up with a plan for the solution steps. You’ll need to find connections between the information given and what you're being asked to find.
Can you restate the problem in your own words? (If not, discuss with someone else.)
Have you seen a problem like this one before?
Can you solve part of the problem? (If so, do it and state a new problem from what's left to solve.)
If you can't solve this, can you solve an easier problem with the same structure? (If so, make one up now to solve. That will give you insight that may help you with this one.)
Can you add information to your picture that will make it easier to solve your problem?
Can you figure out anything interesting from the given information?
Can you say: "If I knew _____, then I could solve this" ? (If so, state a new problem in terms of trying to find _____.)
Are you sure you've used all the given information?
Is there information that may be implied but not stated outright? Perhaps it’s buried in a definition?
Step 3: Carrying Out Your Plan
Third, carry out your plan.
While carrying out your plan, check each step.
Can you see clearly that the step is correct?
Can you prove that it's correct?
Step 4: Looking Back
Fourth, examine your solution.
Did you answer the question that was asked?
Can you check your answer?
Can you check the steps and the reasoning?
Is there another way to figure out the answer?
Can you see it at a glance now?
Can you use your answer, or these methods, for some other problem?