Our 6th graders came to middle school with varying levels of familiarity applying the order of operations. So far this semester, they have built on their experience to evaluate increasingly complex expressions. Now, they are folding in more and more negative numbers to the problems they tackle. These skills are foundational to all subsequent math courses and the aim is for students to work toward “mastery.” When we go over a problem together in class, we want to arrive at the answer; however, an even more important goal is to analyze all the different pathways that can take us to this same destination.
In 6th and 7th grade, Explorer West students solved equations with a single variable (equations like: x + 5 = 10). In 8th grade, they have worked with equations that have two variables (equations like: x + y = 10). This sort of equation is interesting because instead of having a single solution, there is an infinite number of solutions — the 8th graders have learned how to show all these solutions with a line on a coordinate plane. Now, they are starting to look at situations where two such equations are given. Their job is to figure out what values for “x” and “y” make both equations true. And, as with many topics in mathematics, there are several strategies for doing this. After they learn how to solve these “systems of linear equations” by graphing, by substitution, and by combination, the 8th grade can examine each problem and identify the most efficient and elegant way to arrive at the solution. In addition, they will gain more and more experience using linear equations to represent real-world scenarios!