Introduction to Problem Solving

This web site contains a collection of problems, with solutions and strategies, laid out in a systematical manner to assist students / teachers with problem solving. These problems are designed with grade 9 - 12 students in mind; however, they may be suitable for an even wider range of people.

A mathematical problem, like any problem in life, is defined as a problem because it causes us much difficulty in attaining a solution. If the solution, or even the procedure for solving it, is obvious to you then it is no longer a problem but just an exercise. Much of our classroom mathematics is composed of repetitive exercises. ( This teaching method does have a useful purpose but it should not be all that mathematics is about. )

A question is a problem if the procedure or method of solution is not immediately known to you but requires you to apply creativity and previous knowledge in new and unfamiliar situations. In a problem, you are not aware of any algorithm that will guarantee a solution.

"To have a problem means to search consciously for some action appropriate to attain some clearly conceived but not immediately attainable aim. To solve a problem means to find such an action." (George Polya)

Algorithms are special methods specifically designed for solving a certain type of question. We have all learnt algorithms for specific situations, such as, the FOIL method for multiplication of binomials. This method has been developed for that specific situation.

Heuristics are general suggestions that may be applicable to all types of questions. They contain a series of tasks, each containing a series of decisions, that are loosely combined to form a model which can assist in problem solving.
The set of heuristics below is based on the work of Krulik & Rudnick (1989).
 

The five components above do not guarantee a solution. They are simply meant as a guide, not an algorithm, since problem solving is a process which has no set method. However, the use of a set of heuristics and familiarity with various strategies can combine with previous knowledge, creativity and perseverance to form a strong arsenal for attacking problems.

• Starting at Problem 1, you are encouraged to first attempt to solve the problem.
• Do not be discouraged, they are meant to be problems and do require time!
• If unsuccessful after trying for some time, read the hint provided. (You must try first)
• After a good effort you should view the strategy and solution page.
• The strategy shown on the solution page represents just one of the various ways in which the problem could be solved. Most problems can be solved in various different ways.
• Study the process so that you may use the strategy in future problems.
• After being introduced to three problems and strategies, try the review problems.
• These involve the strategies which you have previously been introduced to.
• Hopefully you will find the review problems easier since you are now equipped with strategies for solving them. However, remember they are problems and therefore they should take time so be persistent and do not get discouraged. Rise to the challenge!

The following is a list of common strategies that can be used in solving the problems within this web site. This is not meant to be an exhaustive list of strategies.

• draw a diagram
• look for a pattern
• Simplify: solve a simplified problem
• solve for critical or extreme cases
• make an organized list or table
• Estimation: guess and check
• work backwards
• use logical reasoning
• write equations or ratios

Teacher Resource Page: Check out my list of helpful books on problem solving.