True & False Equations (Day 1 of 2)

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Objective

SWBAT determine when an equation is sometimes, always or never true.

Big Idea

Let's take an in-depth look at what it means for mathematical sentences to be equal.

Launch

15 minutes

To begin today's lesson I will have my students work with a partner on the  true_false_equations_opener. As I distribute the papers, I will explain to my students that a numerical expression is made up of only numerical values and no variables.  Other than this brief explanation, I do not plan to offer much instruction. I want my students to think about what differentiates a true equation from a false equation using their own ideas.

Once students get to work, if they ask for help, I try to guide them in an appropriate direction, without telling them how to simplify the expressions. For this activity I encourage my students to think about each equation without the aid of a calculator. (Later, once they arrive at an answer, I will let them use a calculator to verify their work.)

I have selected many of the questions in this opener because they touch on common misunderstandings that students have acquired through school (e.g., Problems 8, 9, 11 and 13). So, while I monitor my students' progress, I keep an ear open for some of these misunderstandings. I make a note of them so that they can be corrected or used as a focus of discussion at some point in time.  

After a few minutes, I plan to ask my students to discuss their work. I will say something like, "Take a moment to discuss with a partner your reasons why the equations are true or false (MP3)."

Once all of the students are discussing their work, I plan to visit each group and ask them to share whether the equations were true or false. In order to help students to make necessary connections, I will remind my students of our prior conversations about the Associative and Commutative properties. When appropriate, I will guide students to look more deeply at the structure of the equations (MP7).  I think that this is an important, sometimes overlooked, strategy for determining if an equation is true or false (consider Problems 3, 4, 12, 14 in this regard).

 

Direct Instruction & Guided Practice

10 minutes

During today's Direct Instruction I will be projecting true false equations slideshow as my students work together in pairs and we join in whole class discussions. 

Slide 2: I ask students to do a Think-Pair-Share around the questions in this slide. As I listen to their conversations, I will try to guide students to the understanding that because each side of the equal sign will simplify to a single value, either the values on the left and right are the same, or, they are different.  There is no way the values can be both the same and different at the same time.

Slide 3: Then, I take a couple of minutes to discuss the difference between algebraic equations and numeric equations (namely, algebraic equations typically contain variables whose values are not known, where numeric equations do not.  That said, an algebraic equation can be a numeric equation if both expressions are numeric). I allow students to make a conjecture about difference before putting up this slide.  If necessary, I plan to show some examples of algebraic equations before going on to the next slide.

Before moving on from this slide, I will remind my students that algebraic equations are not simply true or false.  They can be always true, sometimes true, or never true.  Rather than explaining these three meanings to students at this point in time, however, I am going to let them try to discover the meaning through the sorting activity on the next slide.

Slide 4: I give students about five minutes to complete this sorting activity.  They will make three columns in their notebooks (or on dry-erase boards) and try to sort the six equations into the three columns by reasoning abstractly about their values (MP2).  I inform my students that each column will contain two equations.  I will not always give them information like this in preparation for a sorting activity, but my objective here is for them to make meaning around why some equations are never true (no solutions) or always true (infinite solutions)

Slide 5: When we discuss the solutions for this activity, I plan to take the time to help my students build meaning. These concepts will be important in many of the lessons to come in this unit, and, when we study Systems of Equations later in the year. 

Independent Practice

10 minutes

I will be projecting Slide 6 from true_false equations as we begin this segment of the class. The instruction on this slide asks students to try to come up with one more equation for each of the columns in their table. I will ask my students to work together on this task with a partner. I will also encourage them to try new things, even if they do not work!

I fully expect my students will need to brainstorm, design and revise their new equations. So, I will encourage students to keep working and I will respond as positively as possible to all of the ideas that they come up with. Once all of the groups have come up with examples of new equations that meet the given criteria, I plan to assign roles to the members of each partnership:

  • Person #1 is on the left is #1, Person #2 is on the right.
  • Person #2 needs to get up and find a partner among the #1's in the room. Take your work with you and be ready to explain your equations to your new partner.
  • Person # 1 needs to be ready to ask: Why is the equation always, never, or sometimes true?

 

Closing

5 minutes

Today's ticket out the door will help me to gauge student understanding of today's lesson. For this reason, I will ask my students to work on true_fales_equations_close individually. 

I designed this Exit Ticket so that at a quick glance I can judge student understanding of the content. If time permits (or as an extension), I plan to have students explain why the value they substituted for the variable in Questions 1 and 2 makes the equation always true, or never true, respectively.

I have included some examples of student work to show how some of my students respond to the prompts.