Solving equations is an essential Algebra 1 skill that comes back in so many ways throughout the year, from rearranging equations to slope-intercept form before graphing a line, to solving systems of equations and even solving quadratic equations.
Solving multi-step equations is at the heart of what we ask Algebra 1 students to do – it is certainly a Top 5 unit if we were ranking them all by importance.
In this unit, we dive into the nuts and bolts of how to solve multi-step equations of all sorts of different forms, as well as work on some application problems to extend and further their learning…all the while, making sure to also emphasize how to properly show and check their work, which are each separate skills entirely.
Click here to jump to your list of tips on what to include in your unit of notes (and why), or keep on scrolling below.
The Wisdom of “The Line”
Looking back, 10 years ago I would’ve rolled my eyes at this tip, but I suggest that you encourage students to draw a line down the equal sign of their equation when they are first learning how to solve equations. This isn’t something they need to do forever, but is great for this unit since it is so early in the year.
When I first started teaching Algebra 1, I remember feeling internally annoyed (ANNOYED!) by students for insisting, “Don’t forget the line!” who clearly learned that as a solving step from their middle school teachers before me. I shudder thinking about that now.
After teaching Algebra 1 Support for a few years and then actually teaching some lower middle school courses as well, I finally saw the wisdom of encouraging students to draw The Line down the equal sign when they first start learning how to solve multi-step equations and admit I was totally wrong to be annoyed before.
I’d wager that most students are quick to take to heart the golden rule of solving equations – that whatever they do to one side of the equation must be done to the other side. Although drawing the line through the equal sign makes it easier for students to not forget this rule, that’s not what got me to finally buy into it.
Over the years and across grade levels (even in my Algebra 2 classes), I noticed a common pattern of students accidentally adding/subtracting the same thing twice on one side of the equation and then not at all on the other. They were actually trying to follow the golden rule of equation solving but were somehow going amiss.
I finally connected that this mistake was overwhelmingly happening when students were dealing with visually lopsided equations, where one side of the equation had a longer expression and the other side had a very short expression, usually just a constant.
Here’s an example of a visually lopsided equation:
14w – (6w – 9) + 1 = 90
Notice the expression on the left is significantly longer than the expression on the right and the equal sign falls toward the far right of the entire equation.
For many students, they just assume that the equal sign goes in the middle of the two sides of the equation whether or not that’s where it actually falls (in the example above, that would be around where the 9 is). This can result in students doing the same operation twice to the same side of an equation if they are going off of where they think the equal sign should be.
After making this connection and going all in on The Line (it’s Step 0 of our solving process now), I noticed a marked improvement in the accuracy of my students’ solving skills.
This is totally a step that they will outgrow, but is an excellent practice while they are getting their sea legs, so to speak. Usually after the first or second quiz, I give them permission to stop doing it if they feel they no longer need to do it.
Tips for Planning Your Solving Equations Unit
If you’re new to teaching Algebra 1, figuring out what is essential to include in your unit notes can be overwhelming. In the section below, I’ll walk through every topic that you should include in this unit, and I’ll also provide direction about what to emphasize while covering that topic to ensure your students are being exposed to common mistakes & misconceptions and are able to develop a deep sense of conceptual understanding.
Use this list as a prescriptive guide for your planning.
At the bottom, I have included a YouTube video that shows every page of notes I give my students during this unit to give you a better sense of how I apply all of these planning tips to create a cohesive unit of guided notes for my students.
What to Include (and WHY)
In my Algebra 1 class, solving multi-step equations is our second unit of the year. Here are the topics I include, and what I recommend emphasizing while teaching this unit and creating your guided notes.
Unit 2 – Solving Multi-Step Equations
Topic 2.1 – Solving Multi-Step Equations (Variables on ONE Side)
Cover problems with variables only on ONE side of the equation – include examples where students must distribute and/or combine like terms before continuing with solving. Emphasize how all equations turn back into 2-step equations after they distribute and combine like terms.
Establish how to show all of their solving steps, as well as how to check their answers. Showing work is so incredibly important because it makes it easier for students to find their mistakes if they need to, and showing their work is a crucial part of justifying their answers. It also prevents (or at least minimizes) careless solving mistakes from going too quickly.
Include examples where students must distribute a -1.
- Example: 4x-(3x+2)=5
Topic 2.2 – Solving Multi-Step Equations with Variables on BOTH Sides
Level up your solving by introducing variables on both sides of the equation.
Ease overwhelmed students by showing them this is just one baby step extra than the topic before, and that all equations will still turn back into a 2-step equation.
Include examples where students may be tempted to add/subtract before distributing and combining like terms.
- Example: 7-5(x+1)=12. Students will be tempted to do 7-5 first, instead of distributing the -5.
Topic 2.3 – Special Case Solutions
Connect back to the definition of a solution – a value that makes an equation true.
Topic 2.4 – Applications with Consecutive Integers & Perimeter
These two areas are classic yet structured ways to apply students’ newly gained solving equations skills.
These applications also emphasize key vocabulary and help to further develop their substitution skills.
Topic 2.5 – Writing & Solving Word Problems
Continue to apply your students’ skills by having them write an equation to represent a scenario and then solve it for the desired piece of missing information.
Encourage students to work on defining their variables in a way that is relevant to each unique scenario by writing “let” statements.
- Example: “Let x = # points scored by Team A”
Topic 2.6 – Algebraic Proofs
Doing 2-column algebraic proofs is an incredible way to solidify students’ understanding of how and why we solve equations the way that we do. It also sets them up to be more successful down the road when they take geometry so a 2-column proof isn’t so scary.
If you are running short on time, this is a topic that can be cut but I would encourage you to rearrange your unit to make time for this one. It is most helpful for students who still need that aha moment for solving equations to click.
Topic 2.7 – Ratios & Proportions
On its own, this may not be the most exciting topic but it is a very helpful solving skill for percent change applications as well as down the road when solving rational equations. It’s even useful for scale factor problems in geometry!
Refresh students’ memories on what a ratio is and what it means for two ratios to be proportional.
Show students how to solve equations using the cross-product property and that, once again, each equation turns into 2-step equations at some point.
Topic 2.8 – Percent of Change
Continue solving proportions by introducing the percent of change ratio and use it to solve interesting real-world problems.
- Examples: Housing market changes, college tuition increases, and more!
Topic 2.9 – Absolute Value Equations
2.9a – Intro to Solving Absolute Value Equations
Help students understand why there are 2 solutions to most absolute value equations and investigate the special case where there are no solutions.
Build a conceptual understanding of how to solve an absolute value equation.
Introduce “extraneous solutions” into their vocabulary.
2.9b – Solving Absolute Value Equations
Give students plenty of additional practice in solving absolute value equations – especially where students must isolate the absolute value before they continue solving.
Topic 2.10 – Writing Absolute Value Equations
If you are short on time, this can be cut. If you are able to spend the time, focusing on writing equations will be worth the day spent because it helps your student understand the solving process on a deeper conceptual level.
Cover writing absolute value equations given the solutions and from scenarios.
Topic 2.11 – Solving Literal Equations
Start by showing parallels between solving a 1-variable equation vs multi-variable equations (also known as literal equations) that are similar in structure.
- Example: 2x+ 5 = 9 vs 2x + 5y = 9
This is a great opportunity to incorporate formulas from science, finance, and geometry.
Emphasize solving linear equations for a given variable. This will be invaluable when students have to graph linear equations in slope-intercept form as well as when they solve systems of equations by substitution.
See Every Page of Notes
If you want to take a look at every page of notes I include in Unit 2 of my Algebra 1 interactive notebook, check out this video. You can speed it up or pause at any point to view a specific page.
Want it D-O-N-E for you?
Let me lighten your planning load!
If you’d like to grab the daily guided notes and recap warm-ups I have created for this unit, you can find the interactive notebook version here and the binder note version here.
All of the amazing planning tips from above have been incorporated to make an incredibly cohesive and thorough unit that will help your students reach grade-level standards.
This unit includes all of the notes you would need to teach your daily lessons, along with color-coded answer keys for every set of notes. There are also daily recap warm-ups included along with a pacing guide so you know just what to cover each day!
You can grab my entire year of interactive notes & warm-ups for Algebra 1 here. You can also get the exact same set of notes in a print-and-go binder format here.
Math by the Mountain 