Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

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.

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.

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.

PDF Version of these Worksheets

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Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

We spent a lot of time in this section discussing how mathematics is presented. One of the main challenges of this is that there are no formal rules that tell you how much or how little writing is required. How much work you show depends on the audience that will be reading the work.

Reading the instructions of a problem should give you a sense of what the problem is asking you to do, and from that information you can also make decisions about what steps are important and which ones are less so.

In this section, we talked about the difference between equations and expressions. But there’s another type of mathematical statement that is like an equation, but instead of declaring that two quantities represent the same value, we want one to be greater than or less than the other. These statements are known as inequalities.

We often call the first two symbols strict inequalities, emphasizing that we are interested in the more stringent condition. Interestingly, we don’t have a formal name for the other two. You will sometimes see them called non-strict, inclusive, or weak inequalities, but there’s no consensus term among mathematicians.

Manipulating inequalities is almost identical manipulating equations, though there is one very important distinction. When multiplying or dividing, the axioms only allow for doing this with positive values. You should compare this definition with Definition 1.1.

You might remember from your previous experiences that when you multiply or divide by a negative number, that you’re supposed to flip the inequality. But that’s not listed here! What’s happening is that this property is not an axiom. We can actually show that this property is a logical consequence of the listed properties. We will show how the negative sign appears in a simple calculation and let you try to prove the general property on your own.

To prove the general case, note that if \(c \lt 0\text{,}\) then \(-c \gt 0\) so that you can multiply both sides by \(-c\) in the first step using the axioms of inequality. From there, you’ll need to make a similar manipulation to the one above.

We can expand our definition of what it means to solve an equation so that it applies to inequalities. You should compare this definition with Definition 4.4.

Notice that the change happened right at the step where we divided by -2. It doesn’t happen before that step, and it doesn’t happen on its own line after that step. It also does not happen because the 9 is negative. It happens because the algebraic manipulation called for dividing by a negative number. Furthermore, if you use the improper presentation from the first section of the book, inequalities lead to some rather unusual and nonsensical mathematical statements. We put such a heavy emphasis on presentation in order to avoid many of the errors that can arise from these things.

A key distinction is that solutions to inequalities are usually not a single value, but (usually) an infinite collection of values. For example, the inequality \(x > 0\) is true when \(x = 1, 2, 3, \ldots\text{.}\) But we have to remember that we’re thinking about all possible values, so we have to include decimals such as \(0.1, 0.01, 0.001, 0.0001, \ldots\text{.}\)

The second collection of values hints at a very unusual feature about numbers. There is no smallest solution to the inequality \(x > 0\text{.}\) Another way of saying this is that there is no smallest positive number. For any positive number you can think of, there’s always another that’s less than your number but still positive. If you’ve never really thought about why this happens, it’s worth taking some time to think about it.

We won’t go into further detail about any of these topics here, as it goes way beyond the scope of this course. But it is interesting to reflect on how some very basic questions can lead to very profound mathematical ideas.