“Simple” equations? Not always so simple.
Have you ever wondered how students can make mistakes with equations that appear extremely simple to solve, equations as basic as: 9 – x = 11, or: – 9 + x = 11?
To me this was rather baffling until I started to see these equations through students’ eyes, thanks to some tutorees who told me why they were making some mistakes here.
Through this experience, I’ve come to realize that I should be careful before I label any equations “simple equations.”
Here’s a case in point, which I’ll call Example #1:
9 – x = 11
If you ask students how the 9 in this problem is connected to the left side of this equation, a distressingly LARGE NUMBER will say that the 9 is connected by SUBTRACTION. If you just think about it, it’s a very understandable mistake. There’s a big glaring negative sign to the the right of the 9. When students look at that negative sign, many think it tells them that the 9 is connected by SUBTRACTION. Using this analysis, and armed with the knowledge that you do “the opposite operation,” these students will blithely ADD 9 to both sides of the equation, but they’ve just gotten off on the wrong track, getting: 18 – x = 20.
Similarly, given an equation as “simple” as this, Example #2:
– 9 + x = 11
many students will tell you that the – 9 is connected by addition. Why? Because of the big glaring + sign to the right of the – 9. . Reasoning thus, these students will gleefully do the opposite operation and subtract 9 from both sides of the equation, landing in an equally sticky puddle of wrong thinking, getting: – 18 + x = 2
How can we help students avoid these common algebraic pitfalls? Here’s what I’ve found helps.
Tell students that if a number is connected to a variable by an ADDITION or SUBTRACTION sign, you have to look to the LEFT of the number — not to the right — to determine how it is connected to that side of the equation.
To see how this works, let’s look back at Example #1: 9 – x = 11
A student who understands the correct process knows she must look to the LEFT of the 9 to see how it is connected to this side of the equation. Since there’s NO VISIBLE SIGN to the LEFT of the 9, that means that there’s an invisible + sign, so the problem can be re-written, for sake of clarity as: + 9 – x = 11
Now the student is getting somewhere. Using the idea of looking to the LEFT, she looks to the left of the 9 and sees this + sign. This tells her that the 9 is connected by ADDITION, not by subtraction. Following the rule of inverse operations, she now knows to SUBTRACT 9 from both sides of the equation, to get: – x = 2
From there it’s just a matter of multiplying both sides by (– 1) to get the variable alone in its positive form, obtaining the answer: x = – 2
Similarly, Example 2 can be solved using the Look to the LEFT rule of thumb.
This problem is: – 9 + x = 11
The student must ignore the + sign to the right of the 9, and instead look to the LEFT of the 9, finding a – sign. That – sign tells the student that the 9 is connected to the left side of the equation by SUBTRACTION. Using the rule of inverse operations, he knows to add 9 to both sides of the equation, getting: x = 20 And that is the final answer.
As I think about this area of confusion, I think it occurs because we teachers sometimes tell kids that they must “get rid of the number that is connected to the variable.” I know I’ve been guilty of talking this way (but I am trying to reform myself).
When a number is multiplying or dividing a variable, it is perfectly appropriate to tell kids to “get rid of the number next to the variable.” For example, in: 3x = 21, students need to get rid of the 3 which is connected to x through multiplication. Likewise, in x/5 = 6, students must “get rid” of the 5 that is connected through division.
But my recent algebra epiphany is that when numbers are connected through addition or subtraction, as in these problems:
10 – x = 23
x + 7 = – 5
– 8 + x = 13
– 2 – x = 8
we need to be more subtle in our instructions. We must tell students to Look to the LEFT of the number on the same side of the variable. And then tell students to let the sign they see to the LEFT guide them as to how to get rid of the number term that is in the way.
Just a little self-confession that I thought I’d pass along.
Good news: I’ve already started using this “Look to the Left” approach, and it is clearing up lots of confusion for students.