Justifying the Solutions to Linear Equations
Lesson 8 of 12
Objective: SWBAT explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution, and construct a viable argument to justify a solution method.
Over the next few lessons, I really leverage the work that was done on the Number Trick Project during the first weeks of school. I use the wording that we developed on that project to help students make sense of solving linear equations. In that project, students saw that by working backwards it was easier to write a new trick. Now, students see that by working backwards, we can "undo" the operations laid out in an equation.
Here is today's opener. It's a set of linear equations. I give students at least five minutes to try these on their own and to record the work in their notes. When the majority of the class is either done or stuck (there's a mix of abilities), I call them to attention and show them how to make a number trick out of the first equation. At the end my notes look like this: Make it a Number Trick 1. Here is what happened as I wrote these notes. First, I followed the advice of some students and used the distributive property as my first step. Then, I said that I'd like everyone to remember an idea that I've shared previously: that sometimes it's possible to solve an equation like this without distributing. Before we actually do that, I ask for a student to read this equation to me. Inevitably, someone says, "Negative five parentheses x plus 2 equals 40." I say that I'd like to try reading it again as a series of steps that happen to the unknown number. I point to x and say, "What's the first thing that happens to x?" We see that the parentheses tell us to add 2. I write these steps, which you can see in the upper right of the photo. After that, we multiply by -5, then the solution is 40. In green, I've written the reverse of these steps, and when I record this on the board, I move from bottom to top. Once these steps are written, we see that this is precisely what we follow when we solve the equation algebraically.
Now, regarding the other two equations. The third one can be solved by the same process, and you can see the notes here: Make it a Number Trick 2. The middle equation is different, however, and it gives us the chance to see that, as powerful as this method is, it has its limits. We talk through the steps of combining like terms, and if the class is really tuned in, we might discuss some of the properties that are in play when we combine like terms (commutative property for re-ordering, and the distributive property when we add those coefficients). Independent of the depth with which we treat equation #2, we also see that it has no solution. Students have been encountering such equations on their own, and this is chance to solidify some of this knowledge as a group.
At this point, we've been solving equations together and the practice part of class is already underway. I tell students that the next 15 minutes are theirs for working on whatever they need to do on Levels 4 through 7. Some get right to work, others look for another opportunity to collaborate like yesterday, and I'm happy to oblige them.
We might work on an example of a level 7 equation, for example, or we might work on some finer details of rearranging equations, combining like terms, or being more efficient. I hope that students are willing to share their work at the board, but if they're not, I take suggestions. In general, there's a nice vibe that we're all in this together, and we're getting stuff done.