Solving Equations – Progressions

For the past few weeks my students have been exploring equations. The current unit of study introduces equations by showing different visual models end eventually ending with an inverse operations strategy. Students initially see equations through solving for ? or x by using trial-and-error. Up to this point in time that’s how they’ve solved equations. There hasn’t really been a formal procedure until this particular unit. As the unit progresses the class uses bar models, pan-balances, hanger models and inverse operations. This post is designed to review the different models that are introduced.

Bar-Model

Using a bar model is fairly new for most of the students that I teach. Students separate a box with a line. The left side of the equation goes on the top and the right on the bottom . Students use logical and spatial reasoning to solve for x. This was a jump in challenge for students. The spatial piece of being able to visualize how much space the variable will take has the potential to be confusing. My class ended up spending about two sessions reviewing this strategy.

Hanger Model

Students have already been introduced to Solve Me mobiles so this wasn’t as much of a stretch as a bar model strategy. This was the first time that students started to “balance” terms with a hanger. Another two lessons were spent here. Students enjoyed working on this although it was quite challenging when students reached the mastery level on the solve me mobiles site

Pan-balances

The next strategy involved pan-balances. This model involves more operations and steps. Students tended to thrive with this and it was great to use in breakout rooms. Students took items away from both sides of the equations and strategy played a role. As students discussed their strategy they found not all methods to solve them were efficient.

Inverse operations

Near the end the unit students were introduced to the inverse operations strategy. This is generally what students come to class knowing, but they’re unsure of why it works. Up to this point students have relied on visual models and are continuing to make sense of equations. They also reviewed how to combine like terms and integers during this process.

The progressions of how students see equations starts to really shine through between the pan-balances and inverse operations strategy. After reviewing all of the different strategies I surveyed my students and most are now more favorable to using the inverse operations strategy. I even had a few students comment that the strategy actually depends on the equation. Bingo!

I’m looking forward to reviewing the solving equations unit after spring break.

Here are a number of Desmos activities that I used throughout and to review the solving equation strategies:

Reviewing all the strategies

Combining like terms

Solving one-step equations

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