I spent much of this week focused on solving equations and inequalities. I started out the week reviewing how to solve one-step equations, moved on to solving one-step inequalities (linking one-step equations and graphing inequalities) and then spent the remainder of the week solving two-step equations. I really liked the fact that we kept coming back in all these scenarios to the same big idea of isolating the variable.
The first thing that struck me was how good my students were at solving the equations intuitively but how haphazard their work was. I know that this will not serve them well as the problems become more complex, so I embedded a fair amount of focus on getting them to use a more structured approach to the task.
Day One and Day Two were quite frankly on the boring, dry side. Students took notes and then practiced the process. I had them work in partners using Kagan’s Showdown Cooperative Learning Structure so that they would have some discourse, even if it wasn’t on a rich task. Students had to analyze each others’ work and discuss/resolve discrepancies. I spent my time circulating and working with kids individually to improve the “organization” of their work. Naturally, I met with a certain amount of resistance. From their perspective, they could get the right answer so that should be good enough, right? Alas, to their dismay, it was not. As they saw from the daily feedback on their entry cards, I was holding pretty firmly to my expectation on how they must show their work.
Day Three and Day Four were all about two-step equations. I had students start out by taking Cornell notes on two-step equations. I used color-coding to make the process more visual since a large percentage of my students are visual learners.
Next, I demonstrated how to use algebra tiles to model a two-step equation. Then, they had to correctly solve two two-step equations before moving on to work in equation stations.
I structured the stations so that there was work developing some conceptual understanding in the early stations and then more abstract work as they progressed. I also structured them so that there was a gradual release. In early work with a concept, students worked with a partner, then worked individually with a teacher check, and then worked individually with a self-check mechanism.
Using the stations allowed me to incorporate some movement into the class period, which is always good. It’s hard to sit still for 6 1/2 hours when you are eleven or twelve. The stations also allowed me to differentiate instruction. Each student was working at his or her own pace, at his or her own level of understanding.
Station 1: Algebra Tiles
Students had three tw0-step equations to solve using algebra tiles. The use of the algebra tiles gave them a concrete representation for an abstract concept and helped to build some conceptual understanding of the process.
Station 2: Two-step equation puzzles
Students worked with a partner to build a set of puzzles. Each puzzle had a two-step equation problem. Students had to put the pieces in the correct order to complete the puzzle. Each color was a different puzzle. When students thought they had all the puzzles correctly put together, I would check and tell them “OK, you’ve got it go on to the next station” or “No, you have a mistake in the pink puzzle”. I was intentional about not telling them what the mistake was. This forced students to compare and contrast (a Marzano high-yield strategy) and analyze their work to find the error.
A pdf file of four of the puzzles is below. I print each puzzle in a different color piece of card stock. Then, cut the steps apart. Mix up the steps and paperclip them together. Four puzzles go into a single ziploc bag as a task.
Station 3: Problems with a teacher-check
Students had to correctly, independently solve three problems. I would check their work as they progressed through the problems. If their were errors, I worked with them 1:1. For each problem where there was an error, they had to do an additional problem. They did not exit this station until they had correctly and independently solved three problems.
Station 4: Two-step equation fortune tellers
Students worked independently at this station. They had to solve all of the problems on the fortune teller. The use of the fortune teller allowed them to self-check their work. While the check mechanism in solving two-step equations should ensure that they know when they make an error, I have found that a lot of kids who make a mistake in their solution seem to also make a mistake in their check mechanism. So, the fortune teller provides a secondary check mechanism.
A pdf of the two step equation fortune teller is below.
On Day Four, I continued to focus on two-step equations. I wanted to increase the difficulty level of the problems (incorporating negative numbers and fractions) and bring in some more problem-solving. I began by playing the Employee Lounge puzzle in MIT’s Education Arcade Lure of the Labyrinth game with the class. Students had to decide what actions to take in the game. Since they have been playing the game since the beginning of the year, there were many thoughts on the best moves. As we played the game, we discussed the equations necessary to solve the problem, de-privatizing the thinking and linking the math we were doing in class to the math they were doing in the game.
After some game play, I had students solve two two-step equation incorporating rational numbers. After successfully doing so, they moved on to a second day of stations. Students returned to whichever station at which they had been working the previous day (I keep track of this using an index card system). I also added several new stations for those students who were progressing more rapidly through the stations.
Station 5: Combining Like Terms Pyramid
Students had to combine terms from adjacent bricks to fill the brick above. When they reached the peak, they had to compare their final expression with their partners as a checking mechanism. If there was a discrepancy, they had to go back and find the errors and fix them before moving to the next station.
A friend gave me this activity and I would like to credit the source. However, I’m not sure where she got it. I recreated it and the files are attached below. I used the “Easy” version for this station. Later, I will bring out the challenge version for those students who are ready for additional challenge
Station 6 – Combining Like Terms Uno
This would have been the final station of the day, but no one got quite that far. I didn’t really expect that they would, but had it ready as a “just in case” station. We will use it later in the spring when we come back to these ideas again.
As I look back on the week, I see some changes I want to make for next year.
- I want to pull the Lure of the Labyrinth game play forward to the first day of the sequence. The initial problems in the game play are one-step equations and would lead nicely into the lesson. It also turns the sequence back to a place where the students are figuring out how to do things in the first part of the lesson, which is what I would much rather have. I think a lesson in which they work with an idea first and then I help them bring their ideas together and formalize them gives them a lot more ‘sense-making” and a much deeper understanding of the material.
- I want to find a way to make the solving/graphing inequalities lesson a little more interesting. Students grasped the concept well, so it was “successful”. It was just boring and math should not be boring. I also think I need to bring in more problem-solving here.
- I would like to develop/find some foldables relating to these topics to use as part of a summary at the end of the lesson. I think they are a nice way to bring things together and are a good study tool as kids prepare for quizzes and tests.