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# Solving Equations by Constructing Arguments (Day 1 of 2)

Lesson 5 of 12

## Objective: SWBAT use properties of operations and equality to justify the steps of solving a linear equation.

*43 minutes*

**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.

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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.

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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.

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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.

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*Responding to James Dunseith*

James - I am going to be teaching Algebra next year and you've both inspired me and given me a kick in the pants to find ways to be more effective. I seem to have found you at the right time given that I've been curious about how a project based class would look. By reading your narrative I feel like I get it now and am starting to see how to craft my daily work with the kids into a project based format. I plan on using the projects you outline in your first 3 units. (Actually, I'm probably going to follow your daily plans for the first few units. I tend expect too much of the kids too fast and your narrative has helped me see how I can slow that down and "build a better student" over time.) I also plan on being more explicit about building a growth mindset in the kids, and again your narrative is helping me see how to integrate that thinking into the routine of what we do.

Thanks again for sharing what you do. I hope you continue to share and know that at least one guy is getting a lot out of your contributions.

| 3 years ago | Reply*Responding to Aaron Bieniek*

Aaron - I think you've identified an important continuum when you differentiate between "mathematics and mathematical reasons" versus the "tricks" that we pick up along the way. As students progress through their math education, some of the depth they get derives learning more of the mathematics behind a trick, and less of memorizing the trick itself. And when you learn the math behind the trick, it opens up all sort of new possibilities. With that in mind, I've experienced a lot of the same issues you've raised: how can I give kids all the background I want them to have without losing them? The conclusion I came to in this implementation of the project is somewhere in the middle of that continuum. Your thinking is right about the Addition Property of Equality. As you note, I've skipped straight to saying that we start with the Additive Inverse Property. Here's how I explain it my students: the Additive Inverse is what *helps us decide to* add 12 to both sides, while the Equality property is what says *we're allowed to do that*. In between the two, the Additive Identity says *we can "cancel out"* the -12 and 12. After that, we do call out the Equality property, but only to say that, if the original equation was true, then so must be n/3 = 7. It would be even more thorough to include the omitted -12+12 = 0 step, but that omission was the compromise I made in hopes of keeping kids at it. You've made me want to try it both ways next fall...if you're teaching Algebra 1 in 2014-15, I'd be into comparing notes around this idea.

James - thanks again for sharing what you do. I've found myself in the same place you are; sort of perplexed about the best way to justify each of the steps in solving an equation. I'm always trying to balance using the properties with not focusing on them too much as if to make the actual names of the properties what is important. I also think about how not to make it too much of a burden for the students to actually have to list each property and probably write more steps than necessary. I worry about tripping that alarm in their minds that says "why in the world are we doing this?" On the other hand though, focusing on the properties gets the students to focus on mathematics and mathematical reasons for what they do rather than tricks they may have picked up along the way. I have to believe that in the long run this work is important.

I know it's been a while since you did this with your kids but I'm wondering if you found that writing more steps was necessary to make sure that the properties were used accurately. Like, when you add 12 to both sides in your notes jpeg you list that as Additive Inverse Property. But isn't Addition Property of Equality a better fit? Then actually showing -12 + 12 = 0 would be the place for Additive Inverse? Then that leaves you with n/3 + 0 which is where the Identity property comes in to get you n/3 = 7. Any thoughts?

| 3 years ago | Reply

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- UNIT 1: Number Tricks, Patterns, and Abstractions
- UNIT 2: The Number Line Project
- UNIT 3: Solving Linear Equations
- UNIT 4: Creating Linear Equations
- UNIT 5: Statistics
- UNIT 6: Mini Unit: Patterns, Programs, and Math Without Words
- UNIT 7: Lines
- UNIT 8: Linear and Exponential Functions
- UNIT 9: Systems of Equations
- UNIT 10: Quadratic Functions
- UNIT 11: Functions and Modeling

- LESSON 1: Solving Linear Equations: Assessing What You Know So Far
- LESSON 2: Patterns, Progress Reports, and Practice
- LESSON 3: Linear Equations: A Subtle Note, and Choosing Your Own Adventure
- LESSON 4: Building on Our Knowledge: Intro to Inequalities
- LESSON 5: Solving Equations by Constructing Arguments (Day 1 of 2)
- LESSON 6: Solving Equations by Constructing Arguments (Day 2 of 2)
- LESSON 7: Collaborating to Level Up
- LESSON 8: Justifying the Solutions to Linear Equations
- LESSON 9: Developing Arguments and More Properties
- LESSON 10: Review: What Can You Do So Far?
- LESSON 11: Critiquing and Revising Arguments (Day 1 of 2)
- LESSON 12: Critiquing and Revising Arguments (Day 2 of 2)