Why do we write $x$ instead of $1x$?

Question

I am currently student teaching for an Integrated Math 1 class (which is similar to Algebra 1) that consists of 9th graders. I have been teaching my students how to solve linear systems using Elimination.

Here is one example we did in class:

$$\begin{cases} x-y=10 \\ 2x+y=5 \end{cases}$$

The students recognized that they should add the second equation from the first. However, a common mistake I noticed was that some students did not combine like-terms correctly (which they have previously learned). For example, one student (who has an IEP) added the equations and wrote $$2x+y=15$$

Clearly, this student did not recall how to add $x+2x$ and $-y+y$, but he did correctly add $10+5=15$ (presumably because the $10$ and the $15$ were "visible"). After discussing with my cooperating educator (CE), she suggested that I rewrite the original system as $$\begin{cases} \color{red}{1}x-\color{red}{1}y=10 \\ 2x+\color{red}{1}y=5 \end{cases}$$ to make the coefficients stand out.

But then it occurred to me: why is that in math, we do not normally write, for instance, $x$ as $1x$?

Now the obvious reason may be that any number times $1$ is itself, so it is more efficient to write $2$ instead of $2$ times $1$, and consequently, it is more efficient to write $x$ instead of $1x$. But why explain this to a 9th grader when they can just simply get in the habit of always writing $1x$? If anything, this will help the students see the coefficient and make less mistakes when adding or subtracting like-terms. Otherwise, I find myself explaining to the students "Don't forget that there's an invisible $1$ next to $x$", which doesn't really stick with them. This issue would be avoided if the students were accustomed to writing $1x$ instead of just $x$.

Recognizing that my students have a variety of skill levels (and the fact that I am young and naive), it might be overkill for some of them to always write $1x$ instead of $x$ if they already know what they are doing, so I understand that my proposal will depend the students' mathematical abilities. Hence, my questions are: For the purpose of teaching, should we normalize writing $1x$ instead of $x$? Or is this approach better suited for students with special learning needs instead of forcing all students to write this way?

Answer 1

I think the basic answer is that there are all sorts of things that we could write, but don't. We usually leave off things that are redundant, but we can add them back in when convenient.

For instance, instead of $1x$ we could also write $1\times 1\times 1\times x$. Or we could write $x + 0$ or even $0x^3 + 0x^2 + x + 0$. We could also write $0yz + 0y^2 + 0x^2 + 1\times 1\times x + 23 \times 0$.

Sometimes it is even helpful to add these things in. For instance, sometimes it is helpful to write $x$ as $2\times\frac{1}{2}\times x$. The point is that there are a lot of equivalent ways to write things, and we choose the ones that are more helpful for us or help us think about the question more clearly.

However, in absence of other considerations, we like to write things in the shortest way possible, which is why we leave off the 1.

Answer 2

I would like to comment on the question: For the purpose of teaching, should we normalize writing 1x instead of x? not for attempting an answer but to ask why one should consider it a (not so) reasonable idea.

I think that behind such question there is the idea that if I write $1x$ rather than $x$ then the algorithm for sum would be clearer. I think what's wrong in this position is the idea of solving teaching problems always via algorithms: if I give my students the universal working algorithm they'll just need applying it. Which is however the opposite of understanding.

It's a teaching direction that does not work, in my opinion, for two reasons:

1. it is practically impossible to write decent math without such notational abuses (then shouldn't we more completely write $1\cdot x$)? Attempts to eliminate them all are a path to failure.
2. learning how to deal with such abuses is a good way to improve your understanding. And one should not free the path of a student from learning opportunities.

It is however important to develop awareness: a teacher should be aware that for someone such ambiguities are source of misunderstandings so should be ready to unravel and explain. Students should grow in awareness that at a number of points notational shortcuts can be dangerous ans should reflect on them.

Answer 3

Part of our job is to teach students to read and write mathematics properly. We shouldn't be turning out students who say "derive" when they mean "differentiate," say "cancel" when they mean "eliminate," write $\sin(x)$ instead of $\sin x$, or who don't know that $\exp$ is a notation for the exponential function. Therefore it would be a disservice to any student to encourage them to write weird stuff like $1x$ as a routine thing instead of $x$. By the same token, you should not be teaching the student to write $1x^1+0$ instead of $x$.

I also don't understand why anyone would hypothesize that misunderstanding something like this would lead to the particular answer that the student wrote. Now if a student wrote $3x+x=3x$, and explained that they reasoned there "was nothing in front of the $x$," then you would have reason to explain to them that $3x+x$ is the same as $3x+1x$, but only as an explanation, not as something they should always make a habit of writing.

Answer 4

Clearly, this student did not recall how to add $x+2x$ and $-y+y$, but he did correctly add $10+5=15$ (presumably because the $10$ and the $15$ were "visible"). After discussing with my cooperating educator, she suggested that I rewrite the original system as $$\begin{cases} \color{red}{1}x-\color{red}{1}y=10 \\ 2x+\color{red}{1}y=5 \end{cases}$$ to make the coefficients stand out.

I don't think it makes sense to make everyone in the class write $1x$ and $1y$ instead of $x$ and $y$, the main reason being that generally speaking, people and textbooks don't use $1x$ and $1y$.

Did you get to the bottom of how the student got to: $2x+y = 15$ ? Then you can simply explain to the student why their specific working out was incorrect and help them do it the right way. You can also explain to him/her that you can write $x$ as $1x$, but in mathematical textbooks and literature, it is written as $x$ for brevity, so it is better to stick to the most common way to write it, for communication's sake.

Answer 5

It is important to become comfortable with disambiguation. For example, in listening we must use context to distinguish between the homophones “reed” and “read” as in “I will read a book tonight.” In reading, we must use context to know when to pronounce “read” with a long e sound as in “reed” or with a short e sound as in “red.” The brain is an amazing disambiguation machine: it is much better than a search engine such as Google.

We do students a disservice by trying to insulate them from exercising disambiguation skills. Maybe instead of trying to change the common practice of writing $x$ for $1x$, you could approach this issue by taking $2x$ and writing it as $x+x$. If one is having difficulty seeing that $2x+x$ is $3x$, then the teacher could write $(x+x)+x$ instead of $2x+1x$.

When students are learning rules of exponents and getting confused with $x^nx^m$ and $(x^n)^m$ it can help to trot out the examples $a^2a^3$=$(aa)(aaa)$ and $(a^2)^3=(aa)(aa)(aa)$. Eventually the brain becomes quick at sorting out $a^{n+m}$ and $a^{nm}$.

We cannot avoid disambiguation. In fact we need to embrace it, because it underlies important skills.

Answer 6

I wouldn't introduce the notation 1x, which they won't see on any standardized test or in any higher level class, unless it is necessary for student with an IEP. Instead I suggest the following solution (to be proposed to your CE).

You say in the comments that you hadn't taught combining like terms to the class, but they had learned it before. It would seem to me that they have forgotten what they learned and need a review of the material and problems to help drill the concepts.

Tell the class that many have forgotten how to combine like terms which is essential for all the work you will be doing so you are going to review. When reviewing working with like terms: 6x - x, I would say that since x is the same as one x, then 6x - x = 5x. You've explained it without writing. If they need a concrete example - have a 6 pack of soda and take away one can of soda, leaving 5 cans.

Once you have reviewed it, give them some problems for drill. Walk around and check that they are doing it correctly. If needed give a few drill problems at the start of each class, until the class has gotten it. I would think this would be enough for mastery.