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

Lesson 6 of 12

## Objective: SWBAT continue to develop strategies for constructing viable arguments and critiquing the reasoning of others.

## Big Idea: It may just be a two step equation, but there are all sorts of properties in play when you solve it!

*41 minutes*

I actually meant to do today's opener a few weeks ago, but there hasn't really been time until now! This is a brief technology survey, that will provide me with an overview of what kinds of technology my students can access both in my classroom and outside of school. What I gather from this survey will inform some of the decisions I make as the school year continues.

As students enter the room, I hand each an index card. I give them a few minutes to answer each of these questions, and while they're doing so, I distribute this week's homework sheet. After that, I collect completed cards and we move on with the day's class.

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This week's homework sheet gives individual students some space to record what they'll work on for the first two nights of the week. There are a variety of possibilities: whether it's practice solving a particular level of linear equation, finishing up the first part of the Linear Equation Project, or further solidifying prerequisite background knowledge. In the latter half the week, students will work on the second part of the Linear Equation Project.

On the back (the second page) of this homework sheet is the same chart as last week: a list of the levels of linear equations to be solved, and a place for students to record their progress.

I take just a minute to give students the overview of this handout. By now, students should expect to receive this every Monday, and everyone should have a plan for where to file it.

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Today's lesson is a continuation of the work we started during the previous class. Please see the previous lesson, Solving Equations by Constructing Arguments (Day 1 of 2), for an introduction to the work we're doing today.

To get started on today's work, I draw the attention of my class to the agenda. I say that just like last week, everyone is going to continue to learn how to solve linear equations at the highest level possible. Our focus is going to shift, however, not just to solving equations but to being able to justify all the steps we take using the properties of operations and equality. In doing so, we're going to pay a lot of attention the third Mathematical Practice, which is written atop the agenda:

**I can construct viable arguments and critique the reasoning of others.**

I make the point that this is a very social standard, and that * collaboration* is going to be key. I write that word on the board, and ask, "How many people does it take to have a great argument?" Although we usually acknowledge the benefits of arguing with oneself, everyone also agrees that a real argument takes at least two people. Critique is the same: to really work, it must be a collaboration between two or more people.

As the week continues and we move on to the second part of the project, I will introduce some formal (and assessable) structures that students will use to record the critique they exchange. For today, we'll build capacity by keeping it informal. As the agenda says, I prompt students to take out Part 1 of the Linear Equation project, which was their homework over the weekend, and to work with a partner to make sure their work is perfect. I don't group formally, because I'm trying to foster independent decision making in my students, but it would certainly be reasonable to group purposefully here. I just make sure that there are an even number of students at each table (if that's possible), and that I know who is working with who.

Now, how it actually pans out may differ depending on the class. In the highest achieving classes, everyone has tried the work and may need help filling in the details as best as they can. For these students, this class structure works perfectly: the conversations begin, students have great conversations and great ideas, and the class passes so quickly that everyone can't wait until tomorrow.

Of course, it's not always like that. Often, I'll have a number of students who really struggled to get started on this assignment. With these students, I take a few steps. First, I always ask everyone to find their notes from the previous class. I want students to be able to look at what they have there, and to use it to complete Part 1. Then, I try to see if I can get students paired up for tutoring. If the class is split into half successful students and half who need help, then this might be a good option. Other times, I may need to use the first equation on Part 1 as another example. If this is the case - and it is, about have the time - my notes will look like this example. Even as I provide this example, I ask students to look at this alongside Friday's notes. When students don't have those notes, either because they slacked off in the previous class or lost them, I firmly make the point that success begins with students holding up their end of the bargain.

If the class follows this latter scenario, sharing this second example will comprise the majority of the class. When we're done, there should be just enough time for students to try to follow the same procedure on the second equation. This is highly formulaic, and may feel a bit prescribed or like students are just filling in blanks. But using highly structured language like this is new to most of my 9th graders, and the way to get started is to model it, then to allow them to try it out.

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If class has gone well so far, we will get to this part of class today. If students needed a lot of help on the first two equations in Part 1, this section may have to wait for tomorrow.

When students try the 3rd and 4th equations on Part 1 of the Linear Equation Project, they should notice that they're the same. More specifically, the initial equation is the same and the solution is the same, but the steps in between are different: for #3, the distributive property is used, while for #4 the equation is solved by first dividing through by -3. This distinction was a focal point of one of last week's lessons.

Once everyone is comfortable with the idea that we'll name the distributive property in our justification of the solution to problem #3, I ask everyone to look at the Properties Note Catcher that we started during the previous lesson. We find the property by it's full name: *the distributive property of multiplication over addition*, and fill in the algebraic definition and a few examples. It's fun to make sure that students are comfortable with the full name of what we almost always refer to as just the "distributive property", because it gives them a chance to think about what's really happening when they distribute.

If there's more time, I'll ask students if there are other properties they'd like to fill in today, and we'll do what we can as time allows.

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This week's homework sheet begins with blank spaces for the first few nights of the week. This is because I don't know exactly what my students will be ready for at the end of today's class. Ideally, all students will work on a set of equations - generated by Kuta Software - on levels 4 through 7. When I create this "Levels 4 through 7" worksheet, I include six equations at each level. I use the "regenerate" feature on Infinite Algebra to print a different worksheet for each student. I explain that what I'm really looking for here is for all students to be able to show their steps clearly on every equation they solve. In a day or two, students will select one equation from here to explain fully on Part 2 of the Linear Equation Project.

I know that many students may need another night on Part 1 of the project, so if that's the case, then that will be their homework tonight.

Other students may need a little remediation on one-step equations, so that's an option to have in my back pocket as well.

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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)