At the start of the unit on percents, I use this opportunity to revisit solving equations, focusing on more complex situations. Normally this topic is limited to accelerated Math 7 classes, but I have included it in regular classes.
When I first started teaching, I did not include it, but then I decided to do so. At first I did not mention that it was a topic of accelerated math in case that made them think that they wouldn’t be able to do it. But I noticed that sometimes students in the regular class would be curious about what the kids in “the advanced math class” were doing. I changed my tactic and would preface the topic for this particular lesson by saying that this was what the accelerated class was doing. Some students who felt like they were relegated to the “slower class” felt challenged that they could do whatever the accelerated class was doing. For the others whose confidence was lacking, I would say “I think you can do this.”
Both classes would have difficulty at first, and part of it was sometimes a matter of organization and neatness. Not everyone in the regular Math 7 class caught on, but at least they gave it a try. I didn’t put a large amount of weight on the topic in terms of tests, but I wanted them to at least have a chance at it.
Some of the difficulty that students have with equations that have variables and numbers on both sides, is organizing their steps in a way that they can read them and know where they are in the problem. In this lesson I provide a way to do the steps in the problem in an orderly way that doesn’t take up a lot of space and allows students to retrace their steps as necessary.
Warm-Ups. I focus the warm-ups on two step equations that I use in the lesson itself as a segue to solving equations with variables on both sides.
1. Solve. 3x +2 = 8 Answer: 3x = 6; x = 2
2. Solve. (2/3)m = 10 Answer: m = 10 × 3/2 = 152
3. Solve. (2/3)t – 5 = 15. Answer: (2/3)t = 20; t = 20 × 3/2 = 30
4. Write an equation and solve. Four less than three times a number is 8. Answer: 3x -4 = 8; 3x = 12; x = 4
5. Solve. 4x = 3x + 9 Answer: x = 9
All of the problems are two-step equations. Problem 5 is one in which there are variables on both sides. There will be questions on this, and a typical hint might be “How can I isolate the “9” on the right hand side? How would I eliminate the 3x from that side?” If they are still stuck, I tell them to look at problem 1 and I ask “How did you get rid of the “2” on the left hand side?” Upon hearing that they subtracted it from both sides, I’ll say “Can you do the same thing with 3x?” Sometimes I will hear “Can you do that?” to which I will respond “Why not?”
Overall Approach. Problem 5 of the warm-ups serves as the conduit to the day’s lesson. In going over the problem, I explain that they had difficulty with this only because it was something they hadn’t seen before.
“The name of the game with problems like these is to get the variables on one side of the equation, and numbers on the other.” I add that these problems are similar to the two-step equations they have been doing, with the exception that now they either add or subtract variables in addition to numbers.
Taking off from the warm-up problem, I write a problem on the board and ask them what I should do:
“We want to get the variables on one side. What should I do?”
This is a similar enough problem to the warm-up that students should see that they can subtract 3x from both sides, resulting in:
The fact that 3x subtracted from 3x equals zero will confuse students at first. Why? Because in the equations they have been doing, such as 3x + 5 = 14, when subtracting 5 from both sides, the 5 on the left side simply disappears. What really happens is that we have 3x + 0 = 9. Since there’s no need to write it that way, they don’t.
One way to make this more obvious is an approach which also eliminates messiness and confusion in writing down steps. This is the typical way students write down steps:
Instead, students can write it like this:
The 3x – 3x on the right hand side, although it still seems strange to many students, makes it more obvious that the result is zero.
The next step is then to add 24 to both sides. They can use this same method of writing down the step:
The resulting equation 6x = 24 is then easily solved.
I use a few more examples of this type:
“What should we do first?”
Ideally, someone will suggest adding y to each side:
But maybe someone suggests subtracting 5y from each side first. And if no one does, I suggest it: “What do you think would happen?” I’ll write down the equation using the notation just discussed: 12 –y -5y = 12 - 5y. “Let’s combine terms,” I suggest, and we end up with:
“What do I do now?”
Students may be confused. Is there an order, a rule we should follow? Should we subtract 12 from each side, or add 6y to each side. I assure them that we can do either. I’ll have one student subtract 12 from each side, and another add 6y to both sides. We end up with two equations:
For the first equation we divide each side by -6. For the second, we divide each side by positive 6. “What do we end up with?” After students see they are the same, I’ll ask “Which method did you find easier? The first where we added y to each side, or the second where we subtracted 5y from each side?”
They will likely say they prefer the first because there are fewer steps. I will tell them that it’s a matter of practice before they start to see what the easiest way is, but in any event, they will end up with the same answer. Similarly when we have something like -6y = -12, it may seem strange, but the answer will be the same. “As a rule,” I say, “I like to end up with a positive variable term, but after doing these for a while, you will get comfortable with either one.”
To cement the process in I give them a problem where the simplest method is obvious:
Adding t to both sides results in 30t = 120, whereas subtracting 29t from each side adds an extra step. And yes, there are students who will subtract 29t from each side.
Variables and Numbers on Both Sides. After they get comfortable doing maybe two more examples, I move on to equations where there are both variables and numbers on both sides. We start with a problem on the board:
Students’ general confusion is that they see a lot of numbers and variables and resemble either deer in the headlights or the poster from the 1960 movie “Village of the Damned”.
One area of confusion that I address immediately is that it doesn’t matter whether we work with numbers or variables first.
“We can work with the numbers first, or the variables first. The name of the game is to get numbers on one side and variables on the other. I like to look at the variables first because I want to know how I can get a positive variable term on one side rather than a negative one.”
From there we look at the equation and ask what choices I have for moving the variable terms. To make it less distracting, I have sometimes placed a post-it note over
one of the numbers—let’s say it is -20 on the right hand side, so that students now see this:
“Now what would you do? Subtract 4y from each side or 10y from each side?”
The general consensus is usually to subtract 4y, since that results in a positive variable term. “Let’s look at the whole equation now,” I’ll say, removing the Post-It note. They can now see the whole equation with the subtraction step written in the style talked about earlier:
“Someone simplify this for me,” I will say.
The result is 16 = 6y -20.
“We have 6y on the left hand side; what now?”
Some students are still confused, and I have had students suggest subtracting 6y from both sides. If this happens, I will ask if others agree. Some will say no and there might be a hub-bub of conversations. To prevent a Mathematical Tower of Babel situation, I will intercede and say that we worked hard to get a single positive variable term on one side and to now let’s work with the numbers.
There is only one choice, which is to add 20 to both sides, though I have had students suggest subtracting 16 from both sides. Again, if that happens I might say “But we’re so close to getting it where we want it—variable term on one side and number on the other—why would we subtract 16 from both sides?”
Finally we end up with what we want:
I remind students that it could be stated as 6y=36—it doesn’t matter. More examples follow with my help diminishing after about the second, so they are working independently. They should do their work in their notebooks. I walk around the room to see what they’re doing, offering help and guidance when needed and answering questions.
Continued in next post…