Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

In the previous sections, we have discussed the idea of both numbers and arithmetic being represented by movement. We’re going to formalize this idea a bit more in this section.

We will begin by thinking about numbers as arrows that indicate movement. When we were looking at integer chips, we talked about taking a certain number of steps to the left or to the right. It is actually more useful to think about this as taking a certain number of steps forward or backward. Imagine that you are standing on the number line facing to the right:

Every number represents movement. Positive numbers represent forward steps, negative numbers represent backwards steps, and zero represents taking no steps. It is important to focus on the idea that this is an instruction about movement, not location. This means that 3 means 3 steps forward no matter where you are on the number line.

Similarly, means steps backwards no matter where you are on the number line.

It is very important that the orientation remain consistent. The default direction is to face to the right, so that the default for "forward" is to the right. From this idea, we can actually reduce the entire picture to just arrows (though we’ll stick with including the figure for emphasis).

With this representation in place, we can very quickly understand and visualize a particular relationship between addition and subtraction. Notice that facing right and walking backward results in the same type of movement as facing left and walking forward. Similarly, facing right and walking forward is the same as facing left and walking backward.

These diagrams show us the following relationships:

These ideas give us a clear representation of "subtraction is addition of the opposite." Walking forward (no matter which way you’re facing) gives you the same result as turning around and walking backward. At a very deep level, this is just another way to see that addition and subtraction are fundamentally related concepts.

PDF Version of these Worksheets

¹

external/worksheets/29-Worksheets.pdf

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 have reached the end of the exploration of addition and subtraction. We have covered a variety of ways of thinking about those mathematical operations. Remember that the point is not that you need to use all of these ideas all of the time, but rather that having that diverse toolbox of ideas and approaches will give you more flexibility to solve problem you come across in the future.

Many students think of math as a set of rules that you need to follow, rather than a diverse set of concepts that are related to each other. That conceptual diversity in many ways reflects the diversity of students. Different students come to mathematics with different experiences and intuitions. Perhaps one student finds the movement diagrams easier to understand while another student thinks that blocks are easiest. It doesn’t really matter that much. The goal is for students to come to their own experience of understanding of mathematical thinking and the diversity of approaches in the last several sections helps to reflect the diversity of perspectives that exist within mathematical reasoning.

We’ll close this section with a very important note. If you find one way of looking at something to be difficult to understand, that’s usually a signal to probe more deeply into it. It’s not a signal that you should simply do something else. It is far better to take an intellectual posture that seeks to understand rather than seeks to not understand. Having a broader understanding of ideas only stands to benefit you. Maybe you will eventually understand it, or maybe not. But if you don’t try, you definitely won’t.

At this point, we have a wide range of tools for representing numbers and arithmetic. We’ve used movement on the number line, base-10 blocks, and integer chips to help build out our understanding of addition and subtraction. But it turns out that this is just the beginning.

What we have been discussing is a way to represent numbers. In particular, we’ve been representing numbers in the service of thinking through arithmetic. There’s a completely different way to represent numbers where we can begin to understand numbers on their own terms and make discoveries about the numbers themselves.

We’re going to strip everything back to one of the most primitive representations of numbers that we have. We can think of numbers as the quantity represented by a collection of objects. We will use dots as those objects. A key idea here is that the organization of those dots is not what defines the number, but simply the quantity of dots. This means that we can have multiple representations of the same number. Here are some examples:

As we create more representations of numbers, we can find that some numbers share features with each other, and this creates a pattern that we can explore. Consider the following patterns:

This pattern is highlighting a basic feature of numbers, which is that some of them are even and some of them are odd. This is something inherent to the numbers themselves. In some sense, we do not make the even numbers even and the odd numbers odd. It’s just the case that some numbers can be broken into two equal-sized groups, and that others can’t. And the ones that can’t will always have one extra object by itself. This seems to be an intrinsic property of the numbers.

This isn’t the only place that these types of number patterns show up. Here are a few more examples.

As we construct these geometric patterns, it’s helpful to come up with algebraic expressions that can represent them. The square numbers are somewhat obvious. The square number is just which we can write as \(S_n = n^2\text{.}\) The almost-square numbers are a little bit trickier, but we can look at the pictures and see that there’s always one more row than the number of columns, and so the almost-square number is We can write this as \(A_n = n(n+1)\text{.}\)

But the triangular numbers are trickier. One way we can express those numbers is by writing them as a sum of the dots in each column. For example:

This gives us a computational method for calculating triangular numbers. One drawback to this is that if we wanted to find a very large triangular number, such as the one millionth one, we would end up having to add up a million numbers. Mathematicians see these situations as a challenge. Is there a better way to calculate triangular numbers?

It turns out that there is, but it’s a little bit tricky. And this is where a lot of creative mathematical thinking starts to come into play. Look at the following diagram:

If you try drawing a few other pictures, you’ll see that this pattern persists. This means that we have the relationship \(2T_n = A_n\text{,}\) which can be written as \(T_n = \frac{A_n}{2}\text{.}\) But we established earlier that \(A_n = n(n+1)\text{,}\) and so we must have \(T_n = \frac{n(n+1)}{2}\text{.}\) And this gives us a different formula for the triangular numbers, but now we’re able to easily compute the millionth triangular number.

This line of thinking can lead us to explore other possibilities. Instead of adding every number, what happens if we only add the odd numbers together? For example, we can see that \(1 + 3 + 5 = 9\) and \(1 + 3 + 5 + 7 = 16\text{.}\) Do those numbers look familiar? This hints at a mathematical relationship between the odd numbers and perfect squares. We’re not going to elaborate any further on that relationship, but here is a diagram to contemplate as you think about why this relationship seems to exist.

There is an endless supply of questions like these. The exploration of these number patterns is at the heart of an area of mathematics known as number theory, which is one of the oldest topics of mathematical study. We’ll have to leave the topic here, but hopefully this brief introduction will encourage you to explore some new ways of thinking about mathematics.