Yesterday, three days into a segment on systems of linear inequalities, my plan to help more of my Algebra 1 class understand the topic, and its prerequisites: graphing lines and linear inequalities, flopped in a most surprising way.
Most students in the class still struggle with translating, visualizing, and conceptualizing inequalities while several cannot recall how to graph a line. My hypothesis is that students are so ingrained with the number line, the x-axis in a Cartesian coordinate system, that they typically believe anything to the right is considered “greater than” while anything to the left is “less than.” While that works for inequalities expressed as a function of y, or ax > by + c and ax < by + c, it does not for those expressed as a function of x, or by > ax +c and by < ax + c, where the latter is much more common on summative assessments such as quizzes, tests, etc.
Rather than have students continue to flounder in their discovery and ability to visualize systems of inequalities, I depicted the solutions to three problems given to them the earlier day which I used as a formative assessment for where the class was with respect to the learning segment. Since very few understood all problems, it was clear I needed to focus more intently on visuals for students. The figure to the right served as their warm up for the day. They simply needed to revise the earlier day’s work, which I returned ungraded with guiding comments. I hoped this would allow them to self-assess and think about what they did not understand or did understand.
Also, since I learned about a program called GeoGebra recently, I searched for a way to leverage it in class. Unfortunately, it does not support inequalities within the core program. However, I found a dynamic worksheet online called “Graphing Linear Inequalities” that looked like it would fit the bill; Daniel Kaufman created the applet for GeoGebra users. It runs in a simple web browser running Java. I used it to create the graphs shown in problems 2 and 3 of the preceding figure.
In addition to the static graphs in the warm up, I hoped to spark some discussion and interest in inequalities, using the dynamic worksheet capabilities, . I used it to create the graph, to the left, of a system of inequalities so student could see, step by step, how the system of inequalities is graphed.
Before I showed the above graph, I decided to show students graphical depictions of various inequalities, and systems of inequalities in a dynamic fashion; dynamic simply turned out to be “non-static” as opposed to “rousing” or “inspiring,” to my chagrin. Anyways, I started with the simplest inequality supported by the online tool, y < 2, as shown to the right; it did not support graphing inequalities that were a function of y though, so I could not depict vertical lines. This approach also helped me to re-teach graphing lines, for those that could not recall since I could highlight the slope and y-intercept each time.
I then showed them the following different cases for a linear inequality where the slope varied from 10 to 1 to 0 to -1 to -10, and all slopes in between, all for a y-intercept of 2, and in the form of y < f(x). While I varied these, I queried students, with little response or reaction. It seemed as if they were attempting to follow the rotating line with the shaded part of the graph changing in size and place as the line moved, however, there were no exclamations of understanding, confusion, or of any form.
Afterwards, I continued with my attempt to give them an alternate way to visualize how to build up to the solution graphically as shown below.
When finished, I asked if my demonstration helped make things clearer. One student said “Yes,” but a couple of others, more vocally, said “No, it did not help,” and that it was “too difficult to follow.” No one else spoke up, which is the subject of an upcoming post. The vocal opposers to my initial foray with GeoGebra also said they preferred it when I drew the solution for them at the board; the SMART Board, to be precise.
So, after hoping this use of technology might help much of the class see the subtlety in inequalities, especially systems, no such clarity existed. I quickly accepted their preference, and returned to writing manually at the board. I continued demonstrating an approach to interpreting inequalities and translating the symbolic expressions into graphs. At the end of class, I could tell that students still struggled, even some of the brightest in the class.
The next day, I decided to present the figure at the left as a visual aid and key for students to translate the symbolic text into correct graphical depictions. It seemed to resonate with students. While I presented the same graphic on the first day of instruction, they did not seem to be able to digest its ideas then, even though we covered graphing linear inequalities last marking period. However, after we wrestled with systems of inequalities for three days, the graphic made more sense for them.
I will know how effective the latter figure was, at least in some way, along with the three day’s worth of learning, when I grade their quizzes tomorrow. Until then, I can only hope that I helped some students grasp the concept of solutions for systems of linear inequalities. It is still extremely difficult to assess on the fly, or even while reflecting, whether each and every student grasped the topic, especially one with which so many struggle, especially if they lack the requisite knowledge upon which understanding is built.
I am not giving up on the use of applications like GeoGebra. My CT suggested that a better approach might be to plan a lab day for students to experience GeoGebra directly in a hands on way while completing a worksheet that I scaffold to help them identify what I believe is important for them to understand. Her idea is a good one. I plan to try soon while this topic is fresh although since we need to move on to our next learning segment, I may try some other mathematical concept first.
It truly is a process of “trial and error” in teaching, not only with technology, but in almost all pedagogical practices since there are so many variables in play at any one time. That appeals to me intellectually, but it also can be quite draining emotionally. I am glad I have the opportunity to recharge my batteries every weekend!