Teaching multi-step equations the first time to 150 8th graders was my most epic fail as a teacher. Teaching it to them the second time was my 2nd most epic fail as a teacher. It was embarrassing because I was co-teaching with a new teacher and I was supposed to be the expert. Yikes!
After the two huge fails we had a choice either to move on or to try something new. That’s when I got into my mad scientist mode and tried to figure out a new way to teach it. We were teaching it the way we had learned it, but math came easy to both of us. And it came easy to a few of our students, too. But for the rest… well, they still had that deer in the headlights look after two weeks.
I started analyzing how we’d taught solving multi-step equations, looking for some clue as to what went wrong. I looked through the process we’d taught of subtracting numbers from both sides and then dividing both sides to isolate the variable. It didn’t seem that hard. So, what was the problem?
I realized that it’s just a lot for students to remember. And they didn’t really get why they were doing the same thing to both sides. Also, they made a lot of mistakes keeping the two sides separate. I knew that I needed to find a way to help students organize their thinking and keep track of what they were doing. That’s when a picture of a box came into my head.
What if the students could set-up their own graphic organizer and keep the everything organized? Instead of performing the inverse operations to both sides they could simply move the positive or negative integer to the other side and change to the inverse operation.
We had two more days to try this new method and my co-teacher was such a good sport. We didn’t have much to lose, so we went for it. More students got it after two days with our new boxes method than had gotten it over two weeks with the traditional method. We both knew that the following year we would start the year with these method and we would have so much more success.
Ever since then we’ve both taught hundreds of kids the boxes method and every math teacher at our school uses it as well. Our kids become bosses as far as solving multi-step equations with variables on both sides is concerned. Throughout this post I’ll show you examples we worked through as a class to solve multi-step equations using what this boxes method.
Teaching with I Can Statements
To chunk out the learning for this topic, I first introduce students to the series of I Can Statements that breaks down our learning objections for this unit of study.
Last year I felt that I wasn’t using the interactive notebooks in my classroom to their full potential. Sometimes their notebooks were just a place to put some notes and graphic organizers and I wanted more. So, we started writing the objectives to the unit we were studying on the the first page of the unit in the form of I Can Statements. It was such a hit, that now I can’t imagine teaching a topic without having this to outline and reflect on our learning. To download your own free copy of I Can Statements for 20+ topics, click here:
I Can Statements for Solving Multi-Step Equations:
- I can use the distributive property to expand an expression.
- I can combine like terms in an equation.
- I can solve for x in a multi-step equation.
- I can solve for x in multi-step equations with fractions.
These are the four objectives that I want students to be able to do by the end of the unit. The first two are prerequisite skills that students should be familiar with. The last two are new to most of my students.
#1 I can use the distributive property to expand expressions
The first thing we review when getting started with solving multi-step equations is the distributive property. The distributive property challenges students because they have to remember it over time. They seem to get it in isolation, but when it comes to solving equations it feels like zombies have eaten their brains and all the practice disappeared. What a great opportunity to show kids that some things we learned before show-up in other places. We do a lot of practice with the distributive property and even some number talks about them.
Of course, when negative numbers and variables are thrown into the mix some students struggle. We have references for the rules and patterns with distributive property in our interactive notebook. Students use these references until they don’t need them anymore.
We practice with distributive property throughout the year. We use a lot of whiteboards, Quizizz games, and mazes as practice. I like to make sure that it is fun. When students forget parts of the process, we do quick mini reviews. Once we’ve reviewed expanding expressions with the distributive property, we’re ready for the next step.
#2 I can combine like terms in an equation.
Combining like terms and distributive property go hand in hand. These are the actions students have to take before they they start solving. If they can’t combine like terms then it becomes very difficult to solve multi-step equations. Being able to combine like terms is a key part of solving multi-step equations. In fact, in the boxes method that we use students will visually collect like terms on one side of the equation into a box. Then, they’ll collect integers on the other side in a separate box.
The biggest pitfall for combining like terms is when there are negative numbers. It students are solid with adding and subtracting negative numbers this can throw them for a loop. Make sure that you have scaffolding in place for them. I refer them to their notebook where they have strategies for adding and subtracting integers.
Just like with distributive property, we practice combining like terms throughout the year. We use Quizizz, whiteboards, and mazes to get as much practice as we need. I figure they can never be too good at something. With a solid foundation in place, students are ready to start tackling these equations.
#3 I can solve for x in a multi-step equation.
Now, we get to the fun part. What use to be a dreaded topic for me is now one of my favorite topics because I know that students are going to get it. From day one we teach them using the boxes. Many of them remember it from 7th grade because the teachers in 7th grade also use it.
Below is an example of how the boxes works:
First, student draw the two boxes underneath the two sides of the equation. These boxes always have 3 boxes within them- two smaller boxes in the top row, and a larger box in the bottom row.
Next, students line up the variables, or the x’s, on the left. Anything without a variable goes to the right. This helps students keep everything organized as they’re working through multiple steps.
If a number or variable crosses the equal sign, students need to change the sign (from positive to negative, or negative to positive).
Then, students combine the x’s and put that number in the larger box on the bottom. They do the same thing to simplify the numbers on the other side.
Finally, to get the variable by itself, divide both sides by the coefficient. You can see in the picture above how students show this process.
In the end, students can simply write the solution as they traditionally would.
I can solve for x in a multi-step equation with fractions.
Let’s be honest, fractions freak kids out. I’m not exactly sure why, but I’m sure a little bit is a transfer from unconfident teachers teaching fractions and sharing their fear or disdain of fractions with the kids. I try not to perpetuate that hate for fractions. Actually, I kinda love fractions. I’ve never struggled to understand them and they don’t scare me. I try to demonstrate that to my students.
That being said multi-step equation have a lot of complexity to begin with. And then we add fractions. I like to introduce the fractions after students are very comfortable solving for x when there are variables on both sides. Then I talk about how we are going to now solve equations with fractions and how it’s not that much harder. I emphasize that they just tackled multi-step equations and this is just one step more. Remember to frame things in a positive light.
Below is an example of how we work through a multi-step equation with fractions:
To solve the equation (2/5)x + 2= x – 4, students begin by multiplying everything by the denominator, or 5.
Then, students simplify the equation to 2x + 10 = 5x – 20. Now they’re ready to draw their magic boxes underneath the two sides of the equation.
Next, students follow the same process as before. They line up the variables, or the x’s, on the left and move anything without a variable to the right. They change the signs as numbers and variables travel across the equal sign. Students combine like terms to simplify both sides of the equation. Finally, students divide both sides by the coefficient.
The biggest emphasis during this section of the unit is the fact that when you multiple by a fraction you multiple by the numerator and divide by the denominator. I try to hit hard on the idea that fractions are related to division. I find that you have to work on students’ schema so that they start to make those connections themselves.
Take this to your classroom
I hope that there’s something new in this post that can help you and the students in your classroom. There’s a lot of trial and error in teaching. It’s really the essence of learning. Not everything here will work for every student every time. If you have something that has worked well for you please, I’d love to hear about it in the comments. A few of the resources I use to get students practice with these skills are shown below. Check them out!
Remember that even the most difficult things for us to teach are teachable. This simple thought helps me keep perspective, even when I have epic failures like I did when teaching multi-step equations the first time around. But I do believe that all students can solve multi-step equations like bosses, just like mine have. Thanks for reading! Until next time.