Learning to Solve Problems
· To understand how to solve problems by asking and answering a series of questions.
Imagine today is the first day of your new job. The problem is that you don’t know how to get there. However, as luck would have it, a friend does know the way and offers to drive you. What should you do while you sit in the passenger seat? If your goal is simply to get to work today, you might not pay attention to how to get there. However, you will need to get there on your own tomorrow, so you should pay attention to distances, signs, and turns. The difference between these two approaches is the difference between taking a passive role (going along for the ride) and an active role (learning how to do it yourself). In this section, we will emphasize that you should take an active role in reading the text, especially the solutions to the practice problems.
One of the great rewards of studying chemistry is that you become a good problem solver. Being able to solve complex problems is a talent that will serve you well in all walks of life. It is our purpose in this text to help you learn to solve problems in a flexible, creative way based on understanding the fundamental ideas of chemistry. We call this approach conceptual problem solving . The ultimate goal is to be able to solve new problems (that is, problems you have not seen before) on your own. In this text, we will provide problems, but instead of giving solutions for you to memorize, we will explain how to think about the solutions to the problems. Although the answers to these problems are important, it is even more important that you understand the process—the thinking necessary to get to the answer. At first we will be solving the problem for you (we will be “driving”). However, it is important that you do not take a passive role. While studying the solution, it is crucial that you interact—think through the problem with us, that is, take an active role so that eventually you can “drive” by yourself. Do not skip the discussion and jump to the answer. Usually, the solution involves asking a series of questions. Make sure that you understand each step in the process.
Although actively studying our solutions to the problems is helpful, at some point you will need to know how to think about these problems on your own. If we help you too much as you solve the problems, you won’t really learn effectively. If we always “drive,” you won’t interact as meaningfully with the material. Eventually you need to learn to drive by yourself. Because of this, we will provide more help on the earlier problems and less as we proceed in later chapters. The goal is for you to learn how to solve a problem because you understand the main concepts and ideas in the problem.
Consider, for example, that you now know how to get from home to work. Does this mean that you can drive from work to home? Not necessarily, as you probably know from experience. If you have only memorized the directions from home to work and do not understand fundamental principles such as “I traveled north to get to the workplace, so my house is south of the workplace,” you may find yourself stranded. Part of conceptual problem solving is understanding these fundamental principles.
Of course, there are many more places to go than from home to work and back. In a more complicated example, suppose you know how to get from your house to work (and back) and from your house to the library (and back). Can you get from work to the library without having to go back home? Probably not, if you have only memorized directions and you do not have a “big picture” of where your house, your workplace, and the library are relative to one another. Getting this big picture—a real understanding of the situation—is the other part of conceptual problem solving.
In conceptual problem solving, we let the problem guide us as we solve it. We ask a series of questions as we proceed and use our knowledge of fundamental principles to answer these questions. Learning this approach requires some patience, but the reward is that you become an effective solver of any new problem that confronts you in daily life or in your work in any field.
To help us as we proceed to solve a problem, the following organizing principles will be useful to us.
1. First, we need to read the problem and decide on the final goal. Then we sort through the facts given, focusing on keywords and often drawing a diagram of the problem. In this part of the analysis, we need to state the problem as simply and as visually as possible. We can summarize this process as “Where Are We Going?”
2. We need to work backward from the final goal in order to decide where to start. For example, in a stoichiometry problem we always start with the chemical reaction. Then as we proceed, we ask a series of questions, such as “What are the reactants and products?,” “What is the balanced equation?,” and “What are the amounts of the reactants?” Our understanding of the fundamental principles of chemistry will enable us to answer each of these simple questions and eventually will lead us to the final solution. We can summarize this process as “How Do We Get There?”
3. Once we get the solution of the problem, then we ask ourselves: “Does it make sense?” That is, does our answer seem reasonable? We call this the Reality Check. It always pays to check your answer.
Using a conceptual approach to problem solving will enable you to develop real confidence as a problem solver. You will no longer panic when you see a problem that is different in some ways from those you have solved in the past. Although you might be frustrated at times as you learn this method, we guarantee that it will pay dividends later and should make your experience with chemistry a positive one that will prepare you for any career you choose.
To summarize, a creative problem solver has an understanding of fundamental principles and a big picture of the situation. One of our major goals in this text is to help you become a creative problem solver. We will do this first by giving you lots of guidance on how to solve problems. We will “drive,” but we hope you will be paying attention instead of just “going along for the ride.” As we move forward, we will gradually shift more of the responsibility to you. As you gain confidence in letting the problem guide you, you will be amazed at how effective you can be at solving some really complex problems, just like the ones you will confront in real life.
An Example of Conceptual Problem Solving
Let’s look at how conceptual problem solving works in practice. Because we used a driving analogy before, let’s consider a problem about driving.
Estimate the amount of money you would spend on gasoline to drive from New York, New York, to Los Angeles, California.
Where Are We Going?
The first thing we need to do is state the problem in words or as a diagram so that we understand the problem.
In this case, we are trying to estimate how much money we will spend on gasoline. How are we going to do this? We need to understand what factors cause us to spend more or less money. This requires us to ask, “What Information Do We Need?,” and “What Do We Know?”
Consider two people traveling in separate cars. Why might one person spend more money on gasoline than does the other person? In other words, if you were told that the two people spent different amounts of money on gasoline for a trip, what are some reasons you could give? Consider this, and write down some ideas before you continue reading.
Three factors that are important in this case are
· The price of a gallon of gasoline
· The distance of the trip between New York and Los Angeles
· The average gas mileage of the car we are driving
What do we know, or what are we given in the problem? In this problem, we are not given any of these values but are asked to estimate the cost of gasoline. So we need to estimate the required information. For example, the distance between New York and Los Angeles is about miles. The cost of gasoline varies over time and location, but a reasonable estimate is a gallon. Gas mileage also varies, but we will assume it is about miles per gallon.
Now that we have the necessary information, we will solve the problem.
How Do We Get There?
To set up the solution, we need to understand how the information affects our answer. Let’s consider the relationship between the three factors we identified and our final answer.
· Price of gasoline: directly related. The more a gallon of gasoline costs, the more we will spend in total.
· Distance: directly related. The farther we travel, the more we will spend on gasoline.
· Gas mileage: inversely related. The better our gas mileage (the higher the number), the less we will spend on gasoline.
It should make sense, then, that we multiply the distance and price (because they are directly related) and then divide by the gas mileage (because it is inversely related). We will use dimensional analysis as discussed in Chapter 2. First let’s determine how much gasoline we will need for our trip.
Notice how the distance is in the numerator and the gas mileage is in the denominator, just as we determined they each should be. So, we will need about gallons of gasoline. How much will this much gasoline cost?
Notice that the price of a gallon of gas is in the numerator, just as predicted. So, given our information, we estimate the total cost of gasoline to be . The final step is to consider if this answer is reasonable.
Reality Check Does our answer make sense? This is always a good question to consider, and our answer will depend on our familiarity with the situation. Sometimes we may not have a good feel for what the answer should be, especially when we are learning a new concept. Other times we may have only a rough idea and may be able to claim that the answer seems reasonable, although we cannot say it is exactly right. This will usually be the case, and it is the case here if you are familiar with how much you spend on gasoline. For example, the price to fill up the tank for an average car (at per gallon) is around to . So, if our answer is under , we should be suspicious. An answer in the thousands of dollars is way too high. So, an answer in the hundreds of dollars seems reasonable.