Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 4.1 Read the Instructions

An important aspect of mathematics is the attention to detail and precision required to do it well. However, for many students (and even many teachers) the importance of that precision is often overlooked in favor of repetitive execution. The most prevalent example of imprecision in language for mathematics at this level is the word "solve." Unfortunately, this word is used as a placeholder for a wide range of mathematical procedures.

Here are some examples:

• Solve \(5 + 8\text{.}\)

• Solve \(5x + 8x\text{.}\)

• Solve \(5x + 8 = 23\text{.}\)

• Solve \(x^2 + 6x + 9\text{.}\)

Only one of these reflects the proper usage of the word "solve." Do you know which one it is? Can you come up with a description of what that word means?

When one word comes to represent so many different activities, it becomes extremely confusing to know what's being asked of you. But by taking the time to be precise with our language, we can create mental categories to help us keep information organized. Here are the same calculations, but given with proper instructions.

• Calculate \(5 + 8\text{.}\)

• Simplify \(5x + 8x\text{.}\)

• Solve \(5x + 8 = 23\text{.}\)

• Factor \(x^2 + 6x + 9\text{.}\)

There are even more words that can be used as instructions. Evaluate, compute, estimate, round, rewrite, verify, prove, plot, sketch, and graph are just some of them. Some of these are similar to each other, and some are very different.

The purpose of this section is to help you to make sense of some of these words to create some of the mental structures that will help you think through the ideas more effectively. As usual, we will start with a couple definitions.

Whether or not something is meaningful depends on your level of knowledge. And the number of symbols that are meaningful should grow over time in the same way that your vocabulary grows over time. At this point, you should recognize that "\(5 + \strut\)" is not meaningful, but you might not be able to determine whether "\(2 \sin \pi - 1\)" is if you've never seen trigonometry (and maybe even if you have).

Numbers (such as 5 and and monomials (such as and are expressions. Expressions may also consist of calculations (such as \(5 + 8\)) or polynomials (such as \(x^2 - 8x + 12\)). Between these last two examples, there is an important distinction. There is a shorter way to write \(5 + 8\) with fewer symbols (namely, as the number 13), but we cannot do the same with \(x^2 - 8x + 12\text{.}\)

This is a broad category for a lot of mathematical concepts. For example, the process of combining like terms is one way to simplify an expression, since we can reasonably see that \(5x + 8x\) is not as simple as \(13x\text{.}\) We can also use this word for arithmetic calculations such as \(8 + 6 \cdot 7 - 12\text{.}\) You will sometimes see words like calculate or compute for these types of problems.

We didn't bring any attention to it, but in the last section we demonstrated the presentation format for problems where the goal is simplification.

You can think of this as a long line of small manipulations. It's helpful to go over to the right before going down so that you can more easily visualize the idea that the top expression on the left side is the thing that is equal to all of the expressions on the right. But it's not strictly necessary to do that. The more important thing is to recognize that each new line is a continuation from the previous one, and not an entirely new equation.

The reason the distinction in presentation is important is because the types of things you're allowed to do when simplifying an expression are different from the things you can do when solving an equation.

The equal sign can be read as "represents the same quantity as," so that \(A = B\) can be read as "\(A\) represents the same quantity as \(B\text{.}\)"

If we think of expressions as phrases, then equations are sentences. The important distinction here is that equations make a declaration that is either true or false, where as expressions are just ideas. For example, "the dog" is just an expression. If we say "the dog" without any context, the response is, "What about it?" But we can turn it into a complete thought, such as "the dog is brown," which is going to be either true or false. We won't know whether it's true or false until we pick a particular dog to look at.

In the same way, is a mathematical expression. It just represents a number, like 5. It's not until we put it into a mathematical equation (such as \(2x = 10\)) that we can start to analyze the truth value. Depending on the specific value of the equation might be true or it might not be true. If \(x = 4\) then the equation is not true, but if \(x = 5\) then the equation is true.

We're not going to spend a lot of time discussing truth values and just rely on your intuition. The quation \(2 + 3 = 5\) is true and the equation \(7 - 4 = 2\) is false. We can apply the same ideas to equations with variables. It's true that if \(x = 2\) then \(3x - 1 = 5\text{,}\) and it's false that when \(x = 2\text{,}\) \(3x - 1 = 7\text{.}\) This means that \(x = 2\) is a solution of the equation \(3x - 1 = 5\text{,}\) but is not a solution of the equation \(3x - 1 = 7\text{.}\)

When working with equations, the key idea is that we have to maintain the equality. If we perform an arithmetic operation on one side of the equation, then we must do it to the other side. As you get further in mathematics, the types of manipulations become more complex, and there are other types of properties that you have to keep in mind when working with equations.

The presentation format when working with equations looks like the following:

When reading the meaning of this, what we're saying is that if the original equation is true, then the first manipulated equation is true because of the reason in explanation 1. Then the next equation is true because of the next manipulation, and so forth.