Opening Question: What Makes a Good Argument?
For today's opener, I simply post the question, "What makes a good argument?" on the board. Students are welcome to write it down if they choose to do so, but really I just want them to think about it for a moment as class begins.
While students are getting situated, I greet them, point out the question and tell them to think about it either alone or by talking about it, and I distribute today's two handouts to each table.
First Handout: Properties Note-Catcher
I tell students that we're getting started on a new project today, and I frame this work as different from the last project. "This project consists of a lot less work than the Number Line Project," I say. "There are just three pretty short parts to this project, but I want to encourage all of you to produce the highest quality work possible. One main purpose of this project is to think about what it means to build a great argument."
I tell students to turn their attention to the document called Note Catcher: Properties of Operations and Equality that was distributed at the start of class. "This will be a very useful document as we move through the rest of this class," I say. "I want you to keep this in a prominent place in your binder."
I say that we're going to fill in a few parts of this handout right now. I tell the class that while I provide these notes, I'll also be sharing some of my ideas about what makes a good argument. To begin I say, "We'll start with something obvious, because one way to make a great argument is to start by stating the obvious." (Here, I add that one of the secrets to making a five-page English paper feel a lot easier is to spend the whole first page saying every obvious thing you can. Then you just have four pages to go. My students seem to appreciate this.) In this case, the obvious fact we're going to state is that a + 0 = a. The identity property of addition - that the sum of any number and zero is that original number. I post the note catcher on the board, and I fill in the notes just as I expect students to do. I skip ahead to the multiplicative identity, saying that I hope this one is similarly obvious for everyone. "What important is that even though we may take these facts as pretty simple, it's still important to name them," I say. "That way, when you're making an argument, you can get the simple stuff out of the way before focusing on the more complicated points you're trying to make."
For the inverse properties of addition and multiplication, it's important to write the algebraic definitions in a few ways, and in doing so, emphasize that there's no need for inverse properties of subtraction and division, because these are wrapped up in the definitions given here. This is an idea with which we'll spend more time in the coming days, but for now the key is to show students what the algebra looks like, and to introduce the idea of how addition and subtraction are tied up in the same package, as are multiplication and division.
We skip ahead to the back of the handout, for the properties of equality. Here is what my notes look like: Properties Note Catcher Side 2. It helps to frame these properties as ideas that students already know, even if they don't often think about them. I use both arithmetic and algebraic examples to show what I mean, and I try to get students thinking in a common sense way about each of these.
Once those properties are filled in, I say that we'll use these in an example.
As I've said in my video introduction, this project comes directly from the Common Core Standard A-REI.1: 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. Construct a viable argument to justify a solution method. That's exactly what we're going to set out to do now.
Today's class is one of my most lecture-heavy classes of the year, but it moves quickly because it's so highly structured. There are a lot of notes: first with the properties that I've described in the previous section, and now with an example of what I'll ask students to do on the project.
I draw everyone's attention to today's other handout: Part 1 of the Linear Equation Project. "Here are four equations that are pretty much solved," I say. "What you will need to do for each is fill in some of the missing steps and describe the properties that can be used to justify each step. To help you get started here, I'm going to give you an example. You should copy my example into your notes."
I project this file on the screen: LEP Part 1 Example for Projector, and I move through the steps to fill it in. First, I simply elicit what moves were made to move from one step to the next. The solver had to add 12 to both sides and then multiply by 3. The majority of students are fine with that part. Now, it's time to justify each of these steps with properties. The results look like this or this. As I implement this project for the first time, I'm still playing with the wording and level of verbosity that I choose to use with students.
Whatever the exact word choice, the first step is, as the standard states, to start from the assumption that the equation has a solution. This, I tell my students, is another key tenet of constructing a great argument: you have to assume that you're going to be able to win.
As we dig into this work, the only way to really get it is for kids to practice. The distinction between how the inverse property is used and how the identity property is used is fine point, and here's how I frame it for kids: the inverse properties are what we use to inform our decisions about what to do; to add, subtract, multiply, or divide to undo their opposites. The identity properties are what we invoke when we cancel those opposite operations. If inverses pair to make the identity element, then the idea of identity is why we're able to not write that value and simply cross it out.
The next equality follows from the properties of equality. Note that there are four properties of equality, but just two properties each of inverse and identity. That's another discussion that begins now, but will continue to develop over the next few days.
Once we work through that first example, I circulate to make sure that all students have it in their notes. "It's a lot of writing," I acknowledge, "but I'm not going to ask you do this very many times. I'd like for you to have this example, and then you're going to try this four times on your own for Part 1 of the project."
In the time remaining, I allow students to get started on that handout. I encourage everyone to refer heavily to their notes, and to make sure that they're referencing the properties just like I've done. The first two equations are very similar to the example I've shared. On the back are two approaches to the same equation, and I allow students to discover this for themselves before making it a focal point next week.
Students have kept these throughout the week. I'm curious to see what they're writing, who looked up vocabulary in the textbook, and what I can learn about their perceptions of their own progress. This is not for a grade, and for now, I don't penalize students if they haven't kept up. It's still early in the year, I'm still learning about my kids and their work habits.