Developing Arguments and More Properties

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SWBAT choose a linear equation at a level appropriate to their abilities, and construct a viable argument to justify its solution method.

Big Idea

Constructing a justification is hard work! It's important to show students the value in the process of that work.

Opener: What Properties are In Play?

5 minutes

Today students will dig into justifying the steps in solutions to linear equations. The purpose of today's opener is to serve as a warm-up for that.  Each of these three questions asks students to identify the properties that are in play when someone makes a given algebraic maneuver.  Each of these questions is an example of what kids are doing on the Linear Equation Project, on which their work will continue today.  

A brief discussion of these problems should transition seamlessly to project work time, so if students can confidently answer each question, then there won't be too much more to say.  If there's doubt about any of answers, then I'm willing to slow down, refer back to the work kids have done so far, and give them some notes.

For example, we can note that the first step in solving the equation 

3x + 7 = 19

is to subtract 7 from both sides of the equation.  As we've studied this week, there are actually three properties that are used to fully justify that move: the inverse, identity, and equality properties.  In conversations with my students, here is how I distinguish between them:

  • The additive inverse is the property that helps us decide that the move we're going to make is to subtract 7.  
  • The additive identity is the property that allows us to cancel +7 and -7 from the left side of the equation.
  • After 7 is subtracted from both sides, the additive property of equality is the property by which we can say that the new equation (3x = 12) is still true.

Linear Equation Project Part 2: Check In and Work Time

35 minutes

Yesterday, when I initially introduced Part 2 of the Linear Equation Project, I only did so verbally.  Today, I'll check to see what students have so far, and I will help everyone in any way I can.  Around two-thirds are started on this work, but the majority need help and time.

First, for any students who aren't started on this part of the project, I explicitly set them up.  I help find an equation that works, and I help them organize the work they've done so far:

Most importantly, for everyone who feels stuck, I say that it's ok - you're going to try to do your best work now. I explain, "This is a difficult, detailed task I've given you, and all I'm asking for is whatever you can do. Our next step will be to revise and refine it."  What this entails for today's lesson is circulating around looking for opportunities to have the best kind of conversations:

  • When exactly do we use the inverse property?
  • What does the identity allow us to do?
  • How can we properly utilize the properties of equality?

At some point, we return to the Properties Note Catcher and fill in more blanks.  Examples should be shared on the board to help students choose illustrative equations.

Upcoming Review is Ready When Kids Are Ready

There are approximately two weeks left in the first marking period (my school is on a quarterly grading schedule), so I want to gently begin to prepare students for the quarterly exam that will be here soon.  During tomorrow's class, we will begin to review by looking at some problems related to the work we've done so far.

Today, the review sheet is ready.  I distribute it as kids are ready for it, when they show me that Part 2 is done.  I'm looking for ways to make sure that everyone has something to look at when we engage in critique.  For that reason, today and tomorrow are good days for beginning to review as the end of the unit and marking period near.

Closing: Making a Plan

3 minutes

With a few minutes left in class, I acknowledge the good work that students have done today, and I note that different students are at different places in their work.  I encourage everyone to assess their goals for this evening, and to make a plan for what they'd like to accomplish for homework.  By now, it's really starting to make sense to kids that homework is not usually something I assign, but an extension of the work they're already doing in class.