Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Multiplication can be represented in several different ways. For this section, we’re going to focus on the idea of multiplication as the area of a rectangle. The area of a rectangle with length and width is given by \(A = \ell w\text{.}\) This can also be seen by drawing out the grid and counting the squares, though it’s a waste of time to do it in practice. So we will often represent this symbolically by simply using a box to represent the idea of the calculation.

One of the advantages of reducing the idea to just a representation is that it creates a space for us to work with abstract ideas. Once we stop focusing on specific numbers, we can start to use symbols to represent the idea of creating a grid. Consider the following examples:

We may not know the particular values of the variables and but we see that if we knew what those values were, then we would be able to draw out the grid to count out the number of squares, and that the number of squares would correspond with the expression inside the box.

We can push this even one step further and think about products that don’t even have a proper physical representation:

We will now start thinking about how to extend this idea to products of polynomials. Just as before, we will start with concrete values before looking at abstract ideas. Let’s say that we wanted to calculate \(18 \cdot 6\text{.}\) We can draw out the picture, but there are a whole lot of squares to count.

So we might start to think about how we can organize this in a more sensible manner. With a little bit of thinking, you might realize that breaking up the into \(10 + 8\) might be helpful. Notice that this doesn’t change the number of boxes. However, it does rearrange the information in a way that’s more useful to us because we can work with more familiar multiplication calculations.

There are times when it’s helpful to make variables a particular length so that they’re different from numbers. For example, we could physically represent the product \((x + 2)(x + 1)\) using the following diagram.

We will be using this representation to help us to think about factoring in the next section.

You are probably familiar with FOIL (First-Outer-Inner-Last) as a way to do this product. One of the downfalls to FOIL is that students end up confused when there are more than two terms in the parentheses. Many are so trapped by FOIL that they try to force it to happen even in problems that do not call for it. The basic problem is that students do not have an organized sense of what FOIL is supposed to accomplish, and so they treat it like a rule to be blindly followed.

However, this grid method extends very naturally regardless of the number of terms in the parentheses. And it’s completely built around the basic idea of thinking of multiplication as an area. Having the right organizational scheme frees you from having to memorize more and more rules.

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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.

In this section, we saw the last of the algebraic properties of arithmetic.

If this looks familiar, it’s because we saw something very similar to it a few sections ago. It turns out that addition also has these two properties (Definition 3.5). And when we combine them together with the distributive property (Definition 3.4), we get a pretty thorough description of how arithmetic and algebra works.

It’s important to notice that we’ve never proven any of these properties. We’ve relied on our experiences going all the way back to when we were first learning basic arithmetic as our foundation. And from those experiences, we’ve built out our understanding that this is how numbers work. The world would be a very different place if these things weren’t true.

So at a very basic level, we believe these statements are true because our experiences with reality tell us they should be true. These are not rules that mathematicians came up with and told everyone else they had to follow. Mathematicians simply created a language to describe these things.

When it comes to college level mathematics, it can be helpful to think about the work you’re doing with that framework in mind. As much as you can, try to ground all the work that you do in terms of practical reality. You won’t always be able to do it. Sometimes, the work that you’re doing is an abstraction or generalization of a concept. If that’s the case, try to bring yourself back to the last example that made sense, and see if you can use that knowledge to help you build the next piece.

This isn’t always going to be easy. Some ideas will take more time to sink in than others. That’s part of the nature of learning. But if you invest the time and the energy to slow down and learn to think clearly about what you’re doing, you’ll have a much better chance at being successful.

For many students, one of their struggles with mathematics is that all the concepts feel extremely abstract to them. This can make math particularly difficult for students that are more visual or kinesthetic in their learning. A number of educational manipulatives have been created to help students connect to mathematical thinking, but not all teachers are familiar with them or have encouraged their students to embrace them. We will be discussing a few of these as we work our way through the content to help bring deeper insights into the material.

Algebra tiles were introduced in this section to help visualize products, but they can be used more generally to help represent a range of algebraic concepts. We will start by describing the tiles themselves.

There are three basic shapes of tiles, and each one comes in two colors. The three shapes are a small square, a rectangle, and a large square. The dimensions of the different shapes correspond to each other, so that the narrow side of the rectangle matches with the small square and the long side of the rectangle matches with the large square. The actual relationship between the sides of the small square and large square are irrelevant. They just need visibly different in size. The small square is a unit tile, the rectangle is an \(x\) tile, and the large square is an \(x^2\) tile. The two colors represent either a positive or negative version of the various tiles.