Can I think of Algebra like this?
My question is: Is there any disadvantages to thinking about Algebra like this? Is there anything later in my math education that will require me to know that I am subtracting or adding 2x to get rid of it on this side?
As a college algebra instructor, I'm involved with remediation efforts for hundreds of students each year who have graduated high school but can't get started with college math, mostly due to incorrect concepts picked up in their prior schooling. So I would say "yes". There are some shortcuts that teachers can take to get students to pass some specific tests or programs that they are involved in; but the incorrect concepts definitely make things more difficult for students, sometimes overwhelmingly so, later on. (A majority of students that land in college remediation programs never get college degrees.)
The first thing that I would point out is that the "apply inverse operations to both sides" idea is generalizable to any mathematical operation; this allows you to cancel additions, subtractions, multiplications, divisions, exponents, radicals... even exponential, logarithmic, and trigonometric functions. (With appropriate fine print: no division by zero, square roots to both sides creates two plus-or-minus solutions, trigonometric inverses creates infinite cyclic solutions, etc.)
In contrast, the "move over and change the sign" method is not generalizable, as it only works for addend terms. This sets students on a course that requires memorizing many apparently different rules, one for each operation, which is much harder. When solving $2x = 10$, how is the multiplier of 2 canceled out? Must we remember to move it and turn it into the reciprocal 1/2? Will the students mistakenly change the sign and multiply by -1/2? Or add or multiply by -2 (I see this a lot)? How do we remove the division in $\frac{x}{2} = 5$ (probably some other rule)? How will we remember the seemingly totally different rule to solve $x^2 = 25$?
By way of analogy, I have college students who never memorized the times tables; they did manage to get through high school by repeatedly adding on their fingers, and can get through perhaps the first part of an algebra course that way. But then we start factoring and reducing radicals: "What times what gives you 54?" I might ask; "I have no idea!" will be the answer (this happened this past week; and here's a student who has effectively no chance of passing the rest of the course).
In summary: There are shortcuts or "tricks" that can get a student through a particular exam or test, which prove to be detrimental later on, as the "trick" fails in a broader context (like in this case, with any operations other than addition or subtraction). This then sets a student on a road to memorizing hundreds of little abstract rules, instead of a few simple big ideas, and at some point that complicated ad-hoc structure comes crashing down. Be polite and don't fight with your teacher to change things; but make sure to pick up a broader perspective for yourself, and share it with other students if they're willing, because you will need it later on. Take the opportunity to think about how you could improve on teaching the material, and then you may be on the path to being a master teacher yourself someday, and helping lots of people who need it.
It's important to understand both points of view. In particular, if you ever need to ask yourself why it's okay to "move things around", you could always think about it and remember that all you're really doing is applying the same change to two sides of an equation.
What we have here is a common occurrence in mathematics: two different points of view of the same problem are each useful for different reasons. In the end, it is by synthesizing these points of view that we gain a deeper understanding of the problem at hand and grow as mathematicians.
As for those who don't realize that you're "really" doing the same thing to two different sides: I wouldn't worry too much about it. In all likelihood, it will come up in a math problem one day and suddenly the light in their head will switch on and they'll realize "oh, those two different things that I've learned are really the same". I know that the same thing happens to me on a regular basis.
And for those who never have that epiphany: well, some people insist on remembering things the hard way, and go on to complain that math is a bunch of ad hoc and arbitrary rules. If you find a way to change that, do let me know.
One of the unfortunate things about math in school is that teachers are often tasked with getting students to understand one particular thing. So they'll do whatever they can to teach that one thing so the students can pass the test, even if it means that the students may not truly understand the underlying concept, making it harder for them to learn more advanced topics. (By that time, they'll have moved on to a different teacher...)
In this case, your teacher is tasked with getting you to understand addition and subtraction in algebra. This can be tough to understand, so he invents this cute story about things becoming negative when they move through the equals sign. Ok. That will work for solving stuff like $x+5 = 10$.
But what about equations like $6x = 18$? At this point, you have to multiply or divide, where the cute story about moving stuff through the equals sign falls flat on its face and your teacher will have to make up a new cute story. Eventually, all of these cute stories will accumulate to create profound confusion, and after years of studying algebra, the students will still not understand the underlying concepts.
I'll try to set the record straight. Algebra is (usually) about doing two things:
- Maintaining equality
- Solving for an unknown
You already know about the solving for an unknown part -- you're always trying to solve for $x$. That's the goal.
The first part is a little trickier to understand. We usually have two or more expressions (for example, $x+5$ and $10$) that are connected by an equals sign. This means that these two expressions are equal. The numbers they represent are the same. What we then do is manipulate these expressions, while maintaining their equality, to achieve our goal: solving for the unknown.
So when we have a statement like $5x = 10+2x$, we want to manipulate the expression on the left and the expression on the right, and keep them equal at all times. If our manipulation is skilful, then we'll eventually end up with just one $x$ on one side, and we'll have found out what it is equal to.
So let's take a statement like $a = b$. Fundamentally, maintaining equality means that whatever you do to $a$, you must do to $b$. It makes sense, right? If $a$ and $b$ are equal, then if you add $5$ to $a$ and want to keep $a$ equal to $b$, you must also add $5$ to $b$. If you multiply $a$ by $10$, you must also multiply $b$ by $10$. If you square $a$, you must square $b$. So on, so forth.
But there are caveats. What do we really mean when we talk about equality in this context? Take an equation like $x^2 = 25$. Either $x=5$ or $x=-5$ will be a valid solution: both of those values for $x$ will let the statement of $x^2=25$ be true. So when we solve $x^2=25$, we can't just take $\sqrt{25}=5$ and say we're done. We may have done the same thing to both expressions, but we did not maintain equality.
Or take a statement like $x=1$. We can square both sides and get $x^2 = 1^2 = 1$. Now, $x^2 = 1$ has solutions $x=1$ and $x=-1$. Again, we did the same thing to both sides, but we did not maintain equality. So what does it mean to maintain equality?
We start with a statement that is true, like $x^2 = 25$. The unknown in that statement is $x$. There is a set of solutions (that is, a bunch of numbers) for $x$ that will let $x^2=25$ be true. $5$ and $-5$ are both in that set. $6$ is not in that set because $6^2 = 36$, and $36 \neq 25$.
In the $x=1, x^2=1$ example, we did not maintain equality because we changed the solution set. There's only one value for $x$ that makes $x=1$ true, and that's $1$. But there are two values for $x$ that make $x^2=1$ true: those are $1$ and $-1$. Let's call those the potential solution set.
So we have to go back to our original equation, the one we know to be true, and check the values we got in the potential solution set. In this example, our equation was $x=1$. So we pick the first number from the potential solution set: $1$. We plug it in for $x$. $1=1$. Great. Now we try the other number: $-1$. Plug it in for $x$. $-1 = 1$. Nope.
So to summarize, we maintain equality between two expressions by doing two things:
- Doing the same things to both expressions.
- Watching out for operations that can change the solution set to an equation (these are most commonly squaring/square roots, i.e. exponentiation, or trigonometric operations, like $\sin$, etc.). When dealing with those kinds of operations, it's a good idea to take the potential solutions and to plug them into your original equation to see which ones work and which ones don't.
I would strongly encourage you to always solve these problems with the mindset of maintaining equality, rather than by using the stories your teacher may tell. Beyond what I've said above, there are two big reasons for this:
- By thinking about equality, you will think more about the underlying mathematics, rather than just about moving around symbols.
- The problems you will encounter in Algebra (and in math in general) will gradually become harder, so to be successful, you'll need to actually understand the underlying notions.
Anyway, let's solve the problem $3x + 5 = 2x + 10$ with the kind of mindset I've advocated. For an easy problem like this, it may be overkill, but it'll certainly be useful to think in these terms when you're facing harder problems. $$3x+5 = 2x+10$$ These statements are equal. Subtract $2x$ from each expression. $$3x-2x + 5 = 2x - 2x + 10$$ Simplify. $$x + 5 = 10$$ Subtract $5$ from each expression. $$x + 5 - 5 = 10 -5$$ Simplify. $$x = 5$$