“Experience is what you get when you didn’t get what you wanted.” Randy Pausch
That certainly held true for me the first time I taught students the properties of inequalities. I was teaching three different courses that year, each of them for the first time. I didn’t like it but, I knew that I was going to have to make some conscious choices about where and how I spent my time preparing lessons.
- Course number 1 was a regular 8th grade math course. There were students with significant gaps, students with language barriers, and students with learning disabilities, all of whom needed to be adequately prepared for algebra the next year.
- Course number 2 was a gifted math 6 class that fell into my lap at the 20 day count because the teacher decided to leave a week after school started. They spent the first month of the academic year with a substitute teacher and then finally landed in my lap because I had the necessary endorsement for the course. They were great kids with tremendous potential but the loss of the first month of school was definitely going to take work to fix.
- Course number 3 was a high school Algebra class for 8th The students were advanced and would be OK so long as I taught the material.
Given the options, I decided to teach the Algebra course pretty much straight out of the book so that I would have more time (not enough, but more) to address the needs of the students who had the greater need.
Even with that choice, it was all I could do to keep my head above water that year, which brings me to the properties of inequalities. That first year, I gave my algebra students very clear notes on the properties of inequalities. I modeled problems for them and gave them practice and called it good. I called it good, but it really wasn’t. Some of them “got it”, but some of them didn’t. Not what I wanted, so I guess I should call it experience.
Constructing the properties of inequalities
This year, I used a set of stations to allow my students to construct an understanding of the properties of inequalities. Students worked collaboratively at each station to construct inequalities using a pair of dice. The task card at the station then directed them to perform certain actions on each side of the inequality and decide if the inequality should be preserved (stay the same) or be reversed.
- Station 1 – Students start with an inequality in which one number is positive and one number is negative. They perform five different problems. In each problem, they add or subtract a positive or negative value to/from each side of the inequality. They discover that the inequality is always preserved with addition or subtraction.
- Station 2 – Students start with in inequality in which one number is positive and one number is negative. They multiply both sides of the inequality by -1. They create a new problem by rolling the dice (one is positive and one is negative). Again, they multiply both sides by -1. After repeating this process several times, students discover that multiplying an inequality by -1 results in the need to reverse the inequality.
- Station 3 – Students start with an inequality in which both numbers are negative. They perform five different problems. In each problem, they multiply or divide by a positive number. They discover that the inequality is always preserved.
- Station 4 – Students start with an inequality in which one number is positive and one number is negative. They perform five different problems. In each problem, they multiply or divide both sides of the inequality by a negative number. In so doing, they discover that the inequality must be reversed whenever one multiplies or divides by a negative number.
Student groups were randomly assigned to an initial station. Groups rotated through the stations in 6 minute intervals.
I found these stations in EngageNY. I modified them slightly. In the original EngageNY version, the inequalities showed -3< 2, (-1)(-3)< (-1)(2), 3>-2. The middle step in the sequence is not true, so I changed the inequality symbol in the middle step to be a “?”. I wanted to students to be asking the question of whether the initial state of the inequality was true. I also don’t want them to establish a habit in which they write things that are not mathematically true. The “?” mark replacement seemed like a reasonable substitution. I will provide a link to download the station task cards soon. (WordPress is not allowing me to attach a PDF right now. I will have to investigate)
After students completed work at all four stations, the class came together for a discussion of their findings. This was a chance to de-privatize student thinking and make sure that everyone had access to the main ideas in the lesson. I think it can be powerful to bring these ideas out through student voice.
The last few years, I have made a real effort to conclude the lesson with a summary. Sometimes, this is something that I give verbally and students put into their own words in their notebooks. For this lesson, I used a foldable to summarize the ideas. The foldable compares and contrasts the properties for each of the operations, employing a Marzano high-yield strategy. (I will provide a link to download it soon.)
Applying the properties of inequalities
The following day, I had students apply the properties of inequalities as they solved word problems. This gave them the opportunity to use the properties in a real context. It also revisited the work they had done writing and solving equations to solve word problems prior to the work with inequalities. I spent two days in class working on word problems with inequalities. I felt my students were pretty good at “solving” the problems but not very good at writing an equation to do so. This was a chance to revisit that while also working with the properties of inequalities.
On the second day, I required students to also graph the inequality. They had learned to graph inequalities in a previous unit (without the context of a word problem) and had been quite successful with it. I expected this to be just a straightforward extension of the word problems, giving them some additional practice with a previously learned skill. It wasn’t. Some students correctly wrote the inequality for the problem, solved the inequality and then graphed something completely random. They randomly pulled numbers from the problem and graphed an inequality using those numbers. Because I had students work together with Kagan’s Numbered Heads Together Cooperative Learning Structure, the issues they had with the inequality were resolved through their discussions within their table groups. However, I am still grappling with the “why” behind the randomness of their graphs. The “model” should add meaning to the solution. Clearly this handful of students did not recognize that. I am left with the question of whether I have not done well enough at conveying the “why” behind models or whether I did not write the “question” well enough. Another bit of “experience” to ponder and improve.