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There are four basic types of problem in mathematics:
- Those that test your memory
- Those that test your skills
- Those that ask you to apply your skills to different situations
- Those that extend your skills and theory in unfamiliar situations
As you study mathematics you will meet more complex problems: in early work, problems often require just one step to find a solution; as you progress you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece - divide and conquer!
In most cases you cannot solve a problem just by looking at it! If you cannot see how to start then you need to use a pencil and paper and try various things.
Apply Pólya's four-step process:
- The first and most important step in solving a problem is to understand the question, that is, identify exactly which quantity the problem is asking you to find or solve for (make sure you read the whole question).
- Next you need to devise a plan, that is, identify which skills and techniques you have learned can be applied to solve the problem at hand.
- Carry out the plan.
- Look back and reflect: Does the answer you found seem reasonable? Also review the problem and method of solution so that you will be able to more easily recognise and solve a similar problem.
Some problem-solving strategies: use one or more variables, complete a table, consider a special case, look for a pattern, trial and improvement, draw a picture or diagram, make a list, solve a simpler related problem, use reasoning, work backward, solve an equation, look for a formula, use coordinates.
The practice you get doing set work and reviewing will make exam problems easier to tackle.
When you tackle problems, write out complete solutions, as if you were taking an exam. Don't just scratch out a few lines and check the answer in the back of the book.
If your answer is not right, rework the problem; don't just do some mental gymnastics to convince yourself that you could get the correct answer. If you can't get the answer, get help.
'Word' problems are really 'applied' problems. The term "word problem" has only negative connotations. It's better to think of them as "applied problems". These problems should be the most interesting ones to solve. Sometimes the "applied" problems don't appear very realistic, but that's usually because the corresponding real applied problems are too hard or complicated to solve at your current level. But at least you get an idea of how the maths you are learning can help solve actual real-world problems.
Solving a word problem:
- First convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem.
- If possible, start by drawing a picture. Label it with all the quantities mentioned in the problem.
- If a quantity in the problem is not a fixed number, name it by a variable.
- Identify the goal of the problem.
- Complete the conversion of the problem into math, i.e., find equations which describe relationships among the variables, and describe the goal of the problem mathematically.
- Solve the maths problem you have generated, using whatever skills and techniques you need (refer to the four-step process above).
- As a final step, you should convert the answer of your maths problem back into words, so that you have now solved the original applied problem.