Solving systems of equations using the Substitution Method may be the most task intensive algorithm the students learn. It is easy to overlook concepts as the students attempt the numerous steps. This post provides details about a scaffolded handout that guides students through the mental and conceptual steps, as well as the traditional written steps.
The focus is on systems with a variable that has a coefficient of 1. The variable is not isolated. The entry point is typically with the variable already isolated with the written step being substitution. This handout works for both situations. For the variable already isolated, step 2 can be skipped.
The Scaffolded Handout
The scaffolding addresses the skills used the method but also reinforces vocabulary and concepts.
The first step is a mental step but the students are asked to circle as a means of focusing attention on this step. It is useful for a think aloud as the teacher talks through the process.
Substituting an algebraic expression in for a variable allows for focus on the terms variable, expression, equation and the idea that a variable is a simple expression being replaced by another one.
The solution is an ordered pair. Often, students may stop once they find the value for the first variable.
Page 2 of the handout is scaffolded to guide the students through the steps I found to be problematic. Replacing a variable with an algebraic expression is the central component of this method and is new to students at this point. previously, they would have substituted in only numeric values for variables.
Classroom management is a common, if not the most common, issue that arises among teacher candidates and new teachers. It is challenging to some degree for almost all teachers. As a long time teacher trainer for the states of Connecticut and South Carolina, I had compiled presentation ideas about classroom management I used to share with the candidates I was helping. I hope this post can serve as a starting point for new teachers.
First, I identify 3 stages of classroom management:
Prevention – actions taken to avoid common undesired behaviors, and shaping desired behaviors
Intervention – actions taken to address problems as soon as they arise
Remediation – actions taken to address full blown problems.
The use of the image below left was inspired by a teacher candidate who lamented that during a school observation of a class there were no behavioral issues that would allow her to see classroom management in action. I explained that in fact she saw effective classroom management because of a preventative process implemented. Similarly, there are a couple retail chain stores I have observed with parallel issues with some process, such as how used dishes are cleared out or how bathrooms are maintained (at least the men’s rooms).
I will focus on prevention, which involves being proactive as opposed to being reactive. Like the man waiting for the leaf to fall, some teachers fall into the trap of waiting for a problem to occur. There are two books I recommend that help with being proactive: The First Days of School and Every Minute Counts.
Being proactive involves helping students understand what they are to do. This involves creating positive “Norms!” which is short for what normally happens in class. These can be positive or negative (often assumed to be positive). An example of a negative norm could be students standing at the door the last couple minutes waiting to leave. To establish positive norms, a teacher can set expectations and procedures for various situations in daily class functioning: how to enter and leave the classroom, ask a question, sharpen a pencil, work through problems in classwork etc. For example, when asking a question, are students allowed to blurt out answers or do they raise their hand and wait to be called upon?
Norms will NOT be established by simply posting rules or by verbally explaining our beloved syllabi. Posting rules (bottom left) is akin to a speed limit sign. Many if not most cars routinely drive over the speed limit on an interstate. The drivers do not respect the sign because they know there is an unwritten actual limit that one must cross before getting a ticket. Similarly, students know that often classroom rules are malleable as well.
Here is an example. Years ago, during the first day of class for my freshmen, one student was barely doing any work, despite my prompts. I pulled him aside to ask why he wasn’t working. He replied, “If I do all of my work now, you will expect me to do all my work all year.” He understood that often classroom expectations are conditional.
This Garfield cartoon (below) speaks to such expectations. Most students will toe the line once they understand what the line is. Again, not by what is posted but the teacher’s actions, day in and day out. In other words, their behaviors are shaped.
The image, bottom left, shows one of my classes at the start of a class during a first week. I was projecting a daily point sheet I use to provide them immediate feedback on how they are meeting expectations. The first expectation is that they are to work on the Do Now within 2 minutes of the bell. I have that part enlarged on the projection in the photo. To help shape behavior, I use a “Gotcha” ticket in which I am catching students doing something right. I do this to provide feedback on my 3 classroom expectations by writing a short blurb about what they did RIGHT. Every Monday I would start class by collecting the tickets I gave them and conduct a lottery with a handful of small prizes from Dollar Tree type stores. Some students would not turn in their tickets because the written praise was reward enough!
Classroom Set Up
An organized classroom can support establishing norms. The images below are examples from my classrooms. (Elementary school teachers are routinely doing this!)
Left: I color code each course (e.g., green for Algebra 2). Each class has a file box in which each student has a folder in which I return papers and share individual information as necessary. (e.g., I researched some content on welding for a student interested in that vocation.) The white paper next to some file boxes is the Do Now for the day.
Middle: Each student is assigned a number and I use a shoe caddie to keep track of their respective calculators.
Right: This is a learning wall with notes for the current topics. When students would ask a question about a problem, I would direct them to find the matching problem on the wall and explain the first step (8th grade is coded red and 7th grade was coded blue.)
Bottom: On the right is a board with the daily objective and do now for each class – again, color coded. The posted photos of leaders like MLK Jr. are representations of my expectations: Responsible, Respectful, Resourceful as seen in the Gotcha Ticket. They learn these expectations by our daily activities and feedback – shaping.
As a complement to a set of classroom norms for effective classroom management is engaging instruction. Math teachers are often guilty of standing in front of the class going through multiple examples. To many students, this sounds like the teachers in the Peanuts cartoons, “wah waaah wah ….” Oral communication of information is far less effective than student centered learning. If students are not engaged by your instruction they will find something else to engage them – often an undesired behavior.
It is very challenging to differentiate while standing at the front of the room. Engage students with hands on work and walk around to differentiate. Parents do this with their children. Below my son is having a snack and watching a fire truck parade on YouTube and I was free to attend to other matters, like helping his brother.
Universal Design for Learning
Also, make the initial instruction accessible to all students using a Universal Design for Learning (UDL). The cartoon explains this approach. I use scaffolded handouts, color coding, manipulatives, meaning making strategies, and multiple representations. I refer to this as reverse differentiation. In lieu of waiting to provide specialized instructional strategies after an issue arises, I provide the strategies to all in the initial instruction and allow them to opt out, e.g., one student would repeatedly tell me, “Mr. E, you know I don’t use highlighters.” and he didn’t need them. An example is a lesson on perimeter and area (below) I taught to a group of elementary students (image below). I started the lesson with them building a rectangular pen for their animals and then they counted the number of pieces to determine perimeter. The formula was the LAST thing I showed them.
Finally, we can be responsive to student needs. In the image below, the female circled in red was having serious family issues. She could not focus so I offered her an alternative of playing math games on the computer instead. Often, I hear teachers warn that other students will complain about different treatment. I have almost never encountered this (see all the students on task in the photo). If the students see a teacher differentiating and attempting to meet a variety of needs for all students, they are very accepting.