If there’s one thing I’ve learned about mathematics during my many years at school, it’s that having a solid foundational understanding of the main components can go a long way towards learning new subjects within mathematics. Unfortunately, this is what is often lacking for students, and it can have the knock-on effect of making later concepts more difficult to grasp.
As a tutor, the concept I see students struggling the most in secondary is a basic understanding of algebra and how equations work. This becomes evident to me because the concepts I’m supposed to help students with aren’t necessarily knowing how to manipulate equations, yet this is exactly the aspect they struggle with. I feel like most of the students I help don’t have a solid grasp of what it means to have an equation (or solve a system of equations), which is why I wanted to go over the common mistakes I see.
Mistake One: Flipping Signs
Here’s an equation:
If you want to solve this, the step-by-step method involves bringing the 7 over to the other side, followed by bring the 5 over as well, giving us an answer for $x$. However, the mistake I often see here is one of the following:
Bringing the 7 to the other side like this:
Or bring the 5 over like this:
In both of these cases, the mistake is one of signs. I remember when I was first learning this, I would try to simply remember that multiplication and division doesn’t include flipping signs when “crossing” over to the other side of an equation, and that I would have to flip signs if it was addition or subtraction. This is how I learned it at the beginning, but it definitely wasn’t a good method.
The reason is that mathematics is logical. If you can’t figure out why something ends up being the way it is in an answer, it’s because you don’t fully understand it. In this case, I was content to just remember my rule for flipping the signs, but that made me disregard real reason why my rule worked. Now, I have a better understanding of how these equations work, and so I don’t have to remember my rule.
I think the trouble stems from the way we talk about these equations. When we’re solving for $x$ as in the above equation, we talk about “bringing” numbers to the right-hand side in order to isolate our variable. However, the word “bringing” is pretty vague, and ends up having a different meaning depending on if we are talking about the 7 or the 5 in our example. What we really mean when we say, “bring 7 to the other side” is to subtract 7 from both sides of the equation, like so:
From here, it’s much more clear what’s going to happen. The 7s on the left-hand side will cancel, and the one on the right will be subtracted from 2. There’s no need to think about flipping signs. Instead, it’s all about what it will take to get the seven away from one side of the equation.
It’s also important to note that this works because you’re doing something to both sides of the equation. I always find it’s useful to think of an equation as a balance. Therefore, if you add something to one side, you have to add the same amount to the other if you want the balance to remain in equilibrium.
Likewise, once we have isolated $5x$ and want to find out what $x$ is, we simply need to divide both sides of the equation by 5. When thinking about equations this way, it makes perfect sense that you won’t be flipping the sign, since you want to isolate $x$ through dividing both sides by 5.
Mistake Two: Not Following The Order of Operations
This one’s more common when students are first learning, but it’s still something I see quite often. Let’s return to our example:
Sometimes, I see a student do something along these lines:
To the student, this makes sense, since I they want to “bring” the 5 to the other side of the equation. Unfortunately, this disregards the 7, which is a problem. To help them see this, I ask, “What did you do to bring the 5 to the other side?” Usually, the response will be something along the lines of, “I brought the 5 to the other side, so it’s now a division.”
Once again, this shows the danger of using language like “bringing over”, because it masks what the student is actually doing. Once I tell them that in order to “bring” something to the opposite side of an equation, they have to do the same operation on both sides of the equation, I show them how that includes dividing the 7 as well. It’s not necessarily intuitive to them at first, but once I get them on board with the idea of operating on both sides of the equation, they begin to see how they cannot just transport a number to the other side and do the inverse operation. They have to do it on both sides. This also shows them how isolating for a variable and then dividing or multiplying each side usually avoids this common mistake.
These are just a few mistakes I’ve seen through tutoring students and getting them used to manipulating equations. It’s not always easy for me to see their difficulties, since years of practice makes manipulating equations for me an easy process, but I try to find concrete arguments and reasons to show them where their mistakes are and why they are mistakes. It can be horribly confusing to be told that your answer is wrong, while not being given any explanation. As such, I try to be mindful that the best way to make students understand their mistakes is to show them situations in which their mistake obviously makes no sense. From there, I can lead them to becoming more comfortable with algebra and equations.