Pre-Calculus For Dummies, 2nd Edition (2012)
Part IV. The Part of Tens
In this part . . .
This part presents two opposite sides of the spectrum in terms of preparation for calculus: good math habits to take into calculus and bad habits to break before calculus. Both ends of this spectrum are critical for success because the problems get longer and the sympathy from teachers for algebra errors gets shorter.
Chapter 16. Ten Habits to Develop as You Prepare for Calculus
In This Chapter
Preparing to solve a problem
Working through a problem
Verifying your accuracy after solving a problem
Going the extra mile to ensure pre-calc success
As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle calculus. In this chapter, we outline ten habits that should be a part of your daily math arsenal. Perhaps your teachers have been singing the praises of certain tasks since elementary school — such as showing all your work — but other tricks may be new to you. Either way, we’re confident that if you remember these ten pieces of advice, you’ll be ready for whatever calculus throws your way.
Figure Out What the Problem Is Asking
Often, math teachers test students’ reading comprehension (we know, unfair!) and ability to work with multiple parts that comprise a whole, which is the essence of the concepts behind mathematics. When faced with a math problem, start by reading the whole problem or all the directions to the problem. Look for the question inside the question. Keep your eyes peeled for words like solve, simplify, find, and prove, all of which are common buzz words in any math book. Don’t begin working on a problem until you’re certain of what it wants you to do.
For example, take a look at this problem:
The width of a rectangular garden is 24 inches longer than the garden’s length. If you add 3 inches to the length, the width is 8 inches more than twice the length. How long is the new, bigger garden?
If you miss any of the important information, you may start to solve the problem to figure out how wide the garden is. Or you may find the length but miss the fact that you’re supposed to find out how long it is with 3 inchesadded to it. Look before you leap!
Underlining key words and information in the question is often helpful. We can’t stress this tip enough. Highlighting important words and pieces of information solidifies them in your brain so that as you work, you can redirect your focus if it veers off-track. When presented with a word problem, for example, first turn the words into an algebraic equation. If you’re lucky and are given the algebraic equation from the get-go, you can move on to the next step, which is to create a visual image of the situation at hand.
Draw Pictures (And Plenty of ’Em)
Your brain is like a movie screen in your skull, and you’ll have an easier time working problems if you project what you see onto the paper. When you visualize math problems, you’re more apt to comprehend them. Draw pictures that correspond to a problem and label all the parts so you have a visual image to follow that allows you to attach mathematical symbols to physical structures. This process works the conceptual part of your brain and helps you remember important concepts. As such, you’ll be less likely to miss steps or get disorganized.
If the question is talking about a triangle, for instance, draw a triangle; if it mentions a rectangular garden filled with daffodils for 30 percent of its space, draw that. In fact, every time a problem changes and new information is presented, your picture should change, too. (Among the many examples in this book, Chapters 6 and 10 illustrate how drawing a picture in a problem can greatly improve your odds of solving it!)
If you were asked to solve the rectangular garden problem from the previous section, you’d start by drawing two rectangles: one for the old, smaller garden and another for the bigger one. These pictures get labels in the next section, where we begin to plan how to get the solution (see Figure 16-1).
Plan Your Attack
When you know and can picture what you must find, you can plan your attack from there, interpreting the problem mathematically and coming up with the equations that you’ll be working with to find the answer. If you follow the path we’ve shown below, you’ll be solving word problems in a jiffy!
Try making a “let x =” statement to start. In the garden problem from the last two sections, you’re looking for the length and width of a garden after it has been made bigger. Start by defining some variables:
Let x = the garden’s length now.
Let y = the garden’s width now.
Now add those variables to your picture of the old garden (see Figure 16-1a).
You know that the new garden has had 3 inches added to its length, so you can modify your equations:
Let x + 3 = the garden’s new length.
Let y = the garden’s width (which doesn’t change at all).
Now add these labels to the picture of the new garden (see Figure 16-1b).
Figure 16-1:Picturing the old garden and the new garden helps you plan your attack.
Be sure to write down the given information somewhere near the question. You can write the following information for our example problem:
The width is 24 inches more than the length.
This fact becomes the algebraic equation y = x + 24.
When the length has 3 more inches added on, the width is 8 inches more than twice the length.
This info becomes the algebraic equation y = 2(x + 3) + 8.
By planning your attack, you’ve identified the equation that you need to solve.
Write Down Any Formulas
If you start your attack by writing the formula needed to solve the problem, all you have to do from there is plug in what you know and then solve for the unknown. A problem always makes more sense if the formula is the first thing you write when solving. Before you can do that, though, you need to figure out which formula to use. You can usually find out by taking a close look at the words of the problem.
In the case of the garden problem from the previous section, the two equations you write — y = x + 24 and y = 2(x + 3) + 8 — become the formulas that you need to work with. At times, you won’t have to do so much thinking to come up with the formula you need, but you should still write down the formula.
For example, if you need to solve a right triangle, you may start by writing down the Pythagorean theorem (see Chapter 6) if you know two sides and are looking for the third. For another right triangle, perhaps you’re given an angle and the hypotenuse and need to find the opposite side; in this situation, you’d start off by writing down the sine ratio (also in Chapter 6).
Show Each Step of Your Work
Yes, you’ve been hearing it forever, but your third-grade teacher was right: Showing each step of your work is vital in math. Writing each step on paper minimizes silly mistakes that you can make when you calculate in your head. It’s also a great way to keep a problem organized and clear. It takes precious time to write every single step down, but it’s well worth your investment.
Neglecting to show your steps can keep you from getting partial credit. If you show all your work, your teacher may reward you for the knowledge you show, even if your final answer is wrong. But if you get a wrong answer and show no work, you’ll receive no credit.
Know When to Quit
What’s worse than a pop quiz or snoring after you’ve fallen asleep in class? Glaring at a multiple-choice question that doesn’t list your answer as a possibility. It will leave you screaming. No, the test isn’t wrong, though you really want to believe it is. Sometimes a problem has no solution. If you’ve tried all the tricks in your bag and you haven’t found a way, consider that it may have no solution at all. You may want to ask your teacher to check the problem for typos or errors. Teachers do occasionally make mistakes.
Some common problems that may not have a solution include the following:
Equations with the variable under a square-root sign
Quadratic equations (which may have solutions that are complex numbers; see Chapter 11)
On the other hand, you may get a solution for some problems that just doesn’t make sense. Watch out for the following situations:
If you’re solving an equation for a measurement (like length or an angle in degrees) and you get a negative answer, either you made a mistake or no solution exists. Measurement problems include distance, and distance can’t be negative.
If you’re solving an equation to find the number of things (like how many books are on a bookshelf) and you get a decimal answer, that just doesn’t make any sense. How could you have 13.4 books on a shelf?
Check Your Answers
Even the best mathematicians make mistakes. When you hurry through calculations or work in a stressful situation, you tend to make mistakes more frequently. So check your work. Usually, this process is very easy: You take your answer and plug it back into the equation to see if it really works. Making the check takes very little time, and it guarantees you got the question right, so why not do it?
As an example, we’ll go back and solve the garden problem from earlier in this chapter by looking at its system of equations:
y = x + 24
y = 2(x + 3) + 8
Solving this system (using the techniques we describe in Chapter 13), you get the ordered pair (10, 34). Plug x = 10 and y = 34 into both of the original equations, just to be sure; when you do, you’ll see that they both work:
34 = 10 + 24
34 = 2(10 + 3) + 8
Practice Plenty of Problems
You’re not born with the knowledge of how to ride a bike, play baseball, or even speak. The way you get better at difficult tasks is to practice, practice, practice. And the best way to practice math is to work the problems. You can seek harder or more complicated examples of questions that will stretch your brain and make you better at a concept the next time you see it.
Along with working along with us on the example problems in this book, you can take advantage of the For Dummies workbooks, which include loads of practice exercises. Check out Trigonometry Workbook For Dummies, by Mary Jane Sterling, Algebra Workbook and Algebra II Workbook For Dummies, both also by Mary Jane Sterling, and Geometry Workbook For Dummies, by Mark Ryan (all published by Wiley), to name a few.
Even your textbook from class is great for practice. Why not try some (gulp!) problems that your teacher didn’t assign, or maybe go back to an old section to review and make sure you’ve still got it? Typically, textbooks show the answers to the odd problems, so if you stick with those you can always double-check your answers. And if you get a craving for some extra practice, just search the Internet for “practice math problems” to see what you can find! For example, to see more problems like the garden problem from previous sections, we searched the Internet for “practice systems of equations problems” and found more than 3 million hits. That’s a lot of practice!
Make Sure You Understand the Concepts
Most math classes build on previous material so students can understand new concepts. So typically, if you miss an idea in Chapter 1 of a book, this problem will affect you for the rest of the course. For this reason, you should always remember not to move on in math until you’ve mastered each concept.
Your teacher is your best resource, because he or she knows what’s important and what’s coming up in the rest of the course. Find your teacher after class and get the help that you need until the concept makes sense to you. Because these tutorial types of meetings generally are one on one, your teacher can take personal time to explain the concept to you in ways you understand.
Pepper Your Teacher with Questions
There’s no such thing as a silly math question. If you don’t understand a concept a teacher has presented, or if you simply can’t figure out a problem on your own, seek guidance. Ask 20 questions about one problem until you completely understand it. Your teacher won’t care that you’re asking a lot of questions if you’re earnestly trying to understand! (If you’re asking questions just to goof off in class, though, your teacher will know that and will indeed care.)
Here are just a few examples of really good math questions you may want to ask your teacher:
Why does this work the way that it does?
Why am I taking this step here?
What are the steps, in general, to solving this type of equation?
Can you give me another example to try?
When will I use this information in the future of this class? In the real world? In my life?
Even if you still struggle to understand, simply asking questions can help your grade. Given two students — one who isn’t doing well but is trying to seek out assistance and one who isn’t doing well but doesn’t ask questions after class — who do you think the teacher will have more time, patience, and sympathy for? Showing your face equals showing you care.