Teaching systems of equations is always a challenge for me. It’s a culmination of so many things that we’ve done during the year. Some students seem to memorize the steps, but I find that it doesn’t really stick unless they understand what they are doing. We use the interactive notebook throughout the unit. We add information, synthesize, practice, and reflect, all in the interactive notebook. Let’s break down what this whole systems of equations unit looks like in our interactive notebooks.
Today, I’d like to share some of the activities that we use. You don’t have to use all of these interactive notebook ideas in order to be successful. You can choose the ones that work for you and your students. I’ll break each component I use down, and you’ll see from the pictures an idea of how each piece works.
Systems of Equations I Can Statements
After we’ve completed our introductory activity, the first thing we do is add an I Can Statement to our INB (read more about I Can statements in this post). Then, we look back at them through out the unit. You can see that this outlines the main concepts that we’ll be working on with systems of equations. Students can look at this and reflect to themselves on whether or not they understand how to solve a system of equations with substitution and so forth.
When it comes times for the test, I have students write a reflection about how their learning has progressed. I like to to give them sentence stems to make their answers better. In this case I would have them start with a sentence stem like, “Learning about systems of equation was challenging because…” This can be scaffolded to give as much or as little support as your students need.
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Vocabulary and background building activity
After the I Can Statements, I like to add a vocabulary or building background activity. This is a type of warm-up for the whole unit. (again, you can read more about each of these components I use in my interactive notebooks in this post).
With this background building activity for working with systems of equations, I let students try to complete whatever they can on their own. We’ve usually completed a discovery lab at this point, so students have some background. If they need more help then I describe the different types of solving methods and let them write some things down. As another option, you could have the students interview you and ask the teacher questions to try and figure out the answers.
This activity gives you a chance to see who has a background with the topic. Also, students get a chance to talk about and write about systems of equations. It’s also a nice reference for students to use later.
The steps to solving systems of equations
Now students are ready to get into the “meat” of this concept. To take notes on how to solve systems of equations, I use a foldable graphic organizer for each approach. The goal of this note-taking is to help students learn each process. I also want them to have an easy-to-use reference to go back to when they’re practicing solving systems of equations.
Elimination
When solving a system of equations using elimination, we follow this process:
- Look for opposite value variables. If there are none, then make some.
- Add to eliminate one of the variables.
- Solve for the remaining variable.
- Solve for the other variable.
- State the solution in terms of a coordinate point.
The foldable that we use can be seen below. On the inside students work through a problem and have an example to refer to during practice.
Substitution
We use a different foldable and a different set of steps when it comes to solving a system of equations using substitution:
- Change both equations to slope intercept form.
- Put parentheses around all values of y.
- Substitute one value of y for another.
- Solve for x.
- Substitute and solve for y.
You have to be careful to not emphasize the words. Students really need to see the process. Also, as we’re walking through this process it’s important to remember that many students will come up with their own methods and strategies. This should be encouraged.
Graphing
Graphing to solve a system of equations brings together so many of the concepts and skills that we teach in 8th grade and Algebra I. There’s a catch, though- it only works in situations where students have graph paper. When we teach students all of the different methods, ultimately they have to learn when each one is appropriate. They have to choose whether to solve with substitution, elimination, or graphing. Here are the steps for solving a system of equations by graphing:
- Change both equations to slope intercept form.
- Plot the lines of the two equations on the graph.
- Find the point where they intersect. This is the solution.
The foldable that I use to teach about solving systems of equations through graphing walks students through one example. Later, you have to keep reminding students to go back to the notes and look at the worked example when they get stuck. They want you to tell them again, but they’ll learn better if they look at the notes themselves.
How many solutions?
Another part of this standard is for students to identify how many solutions a system of equations has. My colleague and I come up with a foldable that’s more or less like a flowchart. It presents a series of if-then statements that lead students to the answer.
Questions:
- Do both equations have the same slope and y-intercept?
- 2. Are the slopes the same and the y-intercepts different?
- 3. If you answered no to both questions then it is one solution.
On the inside of the foldable it has a place where students can draw an example and explain why. For some students who need more support I print this foldable with example already there.
Guided practice with systems of equations
Due to the difficulty of this topic, I try to introduce one of the methods for solving at a time. I have two pages that I work with to do guided practice. We only do the problems that go with the method that we’re learning that day. What I love about this is that we can do an I Do, We Do, You Do with each method. These problems serve as out worked problems throughout the instructional part of the unit, and students can go back to them when they forget what to do.
During this time of guided practice students get a chance to really see how you do something. I remember one particular student who developed her own logic and reasoning on how to solve with elimination. She had a chance to share it with the class and it helped other students learn. Those moments are always so cool.
The cheat sheet
All of the cheat sheets that I use are a little different from topic to topic. My design of them has morphed over time. Their main purpose is to be a reminder of things for kids (read more here). This cheat sheet in particular has two parts. One the top part you’ll see a cheat sheet for how many solutions. We teach this twice during the school year and this is a place where students can see it a second time. Also, we review how to graph a line. Both of these topics should be review, but they’re essential for students to keep straight when working with solving systems of equations. Even though the cheat sheet is at the end of the unit, for this topic it’s one of the first things we work on.
The other part of this cheat sheet is a mini foldable for slope intercept form. It’s another chance to practice and remind students of this core concept. I want them to see it so much that they get sick of seeing it, and think this is an easy way to review slope intercept form again.
You’ve got this
Now, it’s time to go and teach. Remember that these notes are the formal instruction, sandwiched between a discovery lab and lots of practice activities. I find this pattern really supports students.
I hope you’ve discovered some new ideas to help you teach solving systems of equations. Now let’s get out there and engage our students in rigorous instruction. Remember, you’ve got this!
All of the ideas and images here can be used as a jumping off point in your own notebook. Or, you can grab all the note pages I use for this topic in the Ultimate Interactive Notebook Bundle for Systems of Equations. Thanks so much for reading. Until next time!
