PROBLEM SOLVING

The higher the math class more will be the problems of different types. In earlier classes, problems often required just one step to find a solution. Now 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!

Problem types
Problems testing memorization. Problems testing skills. Problems requiring application of skills to familiar situations. Problems requiring application of skills to unfamiliar situations. Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.

In earlier classes of school level you solved problems of types 1, 2 and 3. In College you are expected to do mostly problems of types 2 and 3 and sometimes of type 4. Later courses expect you to tackle more and more problems of types 3 and 4, and eventually of type 5. Each problem of types 4 or 5 usually requires you to use a multi-step approach, and may involve several different math skills and techniques.
When you work problems on homework, write out complete solutions, as if you were taking a test. 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. The practice you get doing homework and reviewing will make test problems easier to tackle.

Tips on Problem Solving
The first and most important step in solving a problem is to Understand the problem, that is, identify exactly which quantity the problem is asking you to find or solve for. 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. After that you Carry out the plan. Finally Look back: Does the answer you found seem reasonable? Also review the problem and method of solution so that you will be able to recognize more easily 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 Guess and test 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.

Word Problems are Really Applied Problems
The term word problem has only negative connotations. It is better to think of them as applied problems. These problems should be the most interesting ones to solve. Sometimes the applied problems do not appear very realistic, but that is 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 math you are learning can help solve actual real-world problems.
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. Then complete the conversions 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 math problem you have generated, using whatever skills and techniques you need. As a final step, you should convert the answer of your math problem back into words, so that you have now solved the original applied problem.

COMMENTS
Not too helpful comment: "I don't understand this section." The best you can expect in reply to such a remark is a brief review of the section, and this will likely overlook the particular thing(s) which you don't understand.

Good comment: "I don't understand why f(x + h) doesn't equal f(x) + f(h)." This is a very specific remark that will get a very specific response and hopefully clear up your difficulty.

Good question: "How can you tell the difference between the equation of a circle and the equation of a line?"

Okay question: "How do you do #17?"

Better question: "Can you show me how to set up #17?"Or "This is how I tried to do #17. What went wrong?" (The Instructor can let you try to finish the problem on your own).
Right after you get help with a problem, work another similar problem by yourself.