Solving equations is foundational for middle and high school math. Students actually have been doing this since first grade! However, students can make careless errors or struggle when it comes to following the many procedural steps required to solve an equation.
The Standard
In 6th grade CCSS, students are asked to solve one-step equations with positive whole numbers (6.EE.7). Seventh graders are asked to solve two-step equations (including distributing) with rational numbers (7.EE.7). Eighth graders are expected to solve equations with rational number coefficients, distributing, and collecting like terms.
By the time students are in Algebra 1, they will use all this previous knowledge to:
- Solve linear equations and inequalities in one variable (A.REI.3)
- Create equations and inequalities in one variable and use them to solve problems. (A.CED.1)
- Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (A.CED.4)
Why the struggle?
Solving equations is very revealing. If your students struggle with integer operations, then it will show up again when solving equations. If your students struggle with rational number operations, then it will show up again when solving equations.
Here are some other misconceptions:
- Combining like terms incorrectly
- Dividing/multiplying a negative coefficient. Students trying to add instead.
- Not distributing when they should. Example: 5 – 2(x + 2) = 10 simplifying to 3(x +2) = 10
- Students completing inverse operations on the same side of the equal sign instead of combining like terms (see below for picture)
Does this sound familiar?
Tips 1: Use Algebra Tiles (No, really, use them!)
I was resistant! Manipulatives can cause undue stress (tiny items + 30 children); it is a lot! However, I think that if I had used algebra tiles in my classroom, students would have been way more engaged! I occasionally drew a model to demonstrate solving equations, but then Noelle showed me all the ways algebra tiles could be used to combine like terms, distribute, and solve two-step equations (even quadratics). I was on board!
Before using algebra tiles to introduce students to solving equations, use algebra tiles to demonstrate combining like terms. Combining like terms can be challenging for students, but when you ask students to combine all of the long green tiles (x), the long red tiles (-x), the small yellow tiles (+1), and the small red tiles (-1), it provides excellent concrete practice. It will also assist with zero pair understanding, which will be essential to solving equations with algebra tiles.
What would you rather combine as a student? The terms or the algebra tiles?
Let’s move on to solving equations. Look at the following example:
Before exposing your students to two-step equations with variables on both sides, let them practice with one-step equations first. Scaffolding is key. I chose this example to show you how versatile algebra tiles are (variables on both sides and negative values)
To isolate the variable, students would need to recognize that they needed to get rid of positive 2. To do this, they would add two negative tiles to make a zero pair. Adding negative 2 to one side would mean they would add to the other side to keep the equation balanced.
After students have removed the two zero pairs on the left side of the equation, they might be a little stuck. They are exploring, after all. Remind students that we want all variables on one side, since we want to find out what one x is equal to. Students would then add a negative x to both sides to create another zero pair.
Lastly, to get the solution for one x, students would divide the remaining red tiles among the two x tiles: X = -4.
Students are actually solving for x by undoing the problem, by using inverse operations (adding -2 to both sides and dividing by 2), and by keeping the equation balanced (they are adding tiles to both sides). All the procedural steps that might mean nothing to students in a traditional problem have meaning when students have been exposed to practicing with algebra tiles.
Remember that algebra tiles (like most manipulatives) exist to make the math visual. They provide conceptual understanding. Eventually, students will move into the algorithm. When students are exploring, make sure all of the solutions are integers (you can’t break the tiles into pieces).
To learn more about students making the jump from concrete to abstract, please check out our posts about the CRA Framework – Part 1 and Part 2.
Another tip I recommend is to have students write out what is happening as they are using algebra tiles to solve an equation. If they are combining 3 green tiles with two red tiles, then they would need to write 3x-2x with evidence of only 1x remaining.
If you don’t have access to algebra tiles, students can use this website.
Be sure to grab our Getting Started with Algebra Tiles freebie to learn more about using algebra tiles to tackle simplifying expressions, the distributive property, solving linear equations, adding and subtracting polynomials, multiplying and dividing polynomials, and factoring polynomials.
If you don’t have tiles or access to personal devices, then have students make their own algebra tiles. You only need access to green, yellow, and red paper to get started. Students could easily make their own set that lives in a plastic baggy in their folder.
Idea 2: Use Amplify (formerly Desmos) Activities
Balancing a Hanger: In this activity, students will determine the value of an unknown by “balancing the hanger.” The problems are one step only, but this is a great intro activity for students since the hanger creates a visual. With 24 problems, and 3 open-ended “create your own” problems, this activity would be perfect for student pairs or groups to increase the mathematical discourse. This activity is ideal for 6th grade students.
Working Backwards: In this activity, students will use inverse operations to solve multi-step equations using a “number machine.” The activity has built-in discussion questions and asks students to use substitution to check their work. This activity would be great for 7th and 8th graders.
Helpful Tips
These teacher tips might help your students:
- Draw a line to separate the two sides of the equation.
- Do Undo Line – this is another strategy that can help students.
- Color-coding to help with combining like terms
- Making sure to actually say (and make students say), “2 times x equals 5” as opposed to “2x = 5.”
For more resources, check out these units and bundles.
WHAT IS MANEUVERING THE MIDDLE?
If you find this information helpful, consider checking out more of our resources! At Maneuvering the Middle, we design and develop standards-based math resources for grades 5 – Algebra 1. Our curriculum provides high quality, engaging resources for students and provides teachers with planning resources and plenty of training.
- All Access: standards-based, on-level curricula available for grades 5 – Algebra 1
- Maneuvering Math: a skill-based intervention program for middle school math students
What additional tips do you have? Do you use algebra tiles in your classroom?
Maneuvering the Middle has been sharing content for over 10 years. This post was originally published in 2019. It has been updated for accuracy and relevancy.
