This post is also available as the following pdf Being Strategic in Solving Equations
In a previous post under Solving Equations, I shared with you the Do/Undo Method for basic equations. In this post, I’d like to explore more advanced equation that have a variety of starting points for solving. It’s important to not require students to be so routine and procedural when solving multistep equations. I do say that with a little hesitation, because we’ve all had those students, usually not the strongest mathematics students, who thrive on procedures, steps, and routines. It is perfectly fine for students to utilize a procedure or follow steps, so long as they understand the purpose behind them. This involves students having flexibility in number sense and solving basic equations.
Here is an interesting opening problem/demonstration that you may want to use prior to starting a unit on solving equations with the variable on both sides of the equals.
Start with a basic two step equation, say 3x + 5 = 8. Inform students to simply watch and evaluate your mathematical reasoning. Don’t allow for students to call out any observations until you give the signal. Begin by subtracting 3x from both sides to end up with 5 = 8 – 3x. Then subtract 5 from both sides to get 0 = 3 – 3x. Then subtract 3 from both sides to get -3 = -3x. Finally, divide both sides by -3 to get x = 1. You could keep this going for some time if you’d like. However, the purpose of this demonstration is to show students that so long as they are mathematically correct in their solving process and in their integer operations, they are free to take as many steps as they’d like to determine the value of their variable. I’ve often called it “Taking the Scenic Route” to my students, which is a polite way of saying, “You didn’t have to do that many steps to solve the problem”. It’s okay to take the scenic route once and a while. In fact, there are things that can be learned by doing so; however, it’s not the most efficient way to solve a problem.
Let’s look at a multistep equation with the variable on both sides.
3x – 7 = 8 – 2x
At quick glance, there’s four ways to begin this problem without the use of the commutative property. I personally like to get one representation of the variable as my first step, and then proceed with solving the two step equation. I teach this method first so that students see the connection to the previously learned material. Before I combine my variable terms, I need to decide which term to move and what I desire to be the value of the combined terms. If I remove the 3x , I must subtract 3x from both sides of the equation, producing a -5x on the right-hand side.
3x – 7 = 8 – 2x
– 3x – 3x
– 7 = 8 – 5x
If I remove the -2x, negative 2x through the definition of subtraction, I must add 2x to both side of the equation, producing a 5x on the left-hand side.
3x – 7 = 8 – 2x
+ 2x + 2x
5x – 7 = 8
Notice the differences between these two resulting equations. Each term in the second equation is simply the opposite of each of the terms in the first equation. This is a characteristic that you can share with your students as you work out problems. As these equations become more and more complex, and the number of methods of solution increase, you need to give students the tools to assess their own understanding. If not, you will find yourself doing the same problem over and over again, wasting instruction time.
Now let’s compare these two equations on another level. As you’re illustrating to students how to solve an equation like this, it’s important to engage students in this type of conversation and planning before jumping in and doing steps procedurally. Generally speaking, students are more likely to have success in solving equations when negative values are minimal. The more students have to keep up with negative signs as they solve an equation, the more likely they will make a mistake. This is not researched based, although I believe the research is there to be found, but rather this is my 10 years of classroom experience talking. Moreover, if students understand they have a choice in what term they move, and the ramifications of that move, then students are more likely to choose the step that results in the positive value. With that said, in many cases it is impossible to combine variable terms and avoid negative results, simply by the nature of integer addition and subtraction rules.
When solving these multistep equations during instruction or guided practice with students, good questions to pose are the following:
- “What are your options?”
- “Do you want to combine your variable terms first or your constant terms?”
- “What will happen if you subtract/add ____ from both sides of the equals?”
- “If you do that first step, do you end up with a positive or a negative term?”
Repeatedly asking students these questions throughout instruction will actually cause students to start to internalize them. Soon they will be asking themselves these questions as they work independently. You are simply providing them the research tools they need to investigate future problems for themselves. This is a good thing.
I want to caution you on one thing. I would stay away from creating the habit of ending up with the variable term always on the left side, or the right side. Many misconceptions can be developed from doing so, and can have a negative impact on student performance in later topics and in later courses.
I’m going to stop here for now and let you digest these ideas. I will come back another time and talk about different scenarios that provide us a bit more strategic opportunity. Stay tuned!