Who's Right?

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Objective

SWBAT distinguish between unlike and like variable terms and combine like variable terms.

Big Idea

Students will learn that variable terms with different characteristics represent different values and can not be combined.

Intro & Rationale

This activity utilizes Algebra Tiles as a concrete way for students to experience how to combine like terms and how to distinguish unlike terms. The Tiles provide a physical model that helps students see that variable terms with unlike characteristics may well represent different values. Thus, they cannot be combined. The use of a visual model makes operations that can and cannot combine terms more obvious to my students. My role is to help them connect the concrete model provided by the tiles with the algebraic expressions the students are using.

In my classroom Algebra Tiles allow ELL students a way to share their thinking while they are developing the new terminology. They can indicate with an algebra tile if they are unsure of the correct vocabulary. I also attach some magnetic pieces to the board and label them x^2, xy, and y^2 and encourage them to refer to these when they are searching for the variable term names. We do a lot of naming as a whole group for this lesson to help ELL students find the right words.

 

Warmup

15 minutes

As my students enter class today, they take out the Farmer Frank homework from last night. The data table from the homework is needed for today's warm up:

Jayden says he did 120 + 10x to find the total area of Farmer Frank’s field after extending it (x) meters. Yazmin says 130x will also give the total area. Who’s right?

For problems like this one I want my students to take time to make sense of the task. When they do not, I expect them to miss important connections between the values and variables in the Homework and Warm Up. The first advice that I give them today is, "Take 2 minutes to look for connections between the homework and the warm up task." As I share this request, I circle the "x" and "total area" in the Warm Up. After a pause, I'll ask:

  • Where do these show up in your homework?
  • Did any of the variables and numbers show up in their homework? What did they represent?
  • Can you use the data from the homework to test which expression (Jayden's or Yazmin's) gives the correct total area?

If I notice that students do not realize that "x" represents the possible lengths Farmer Frank might extend his field and that 120 is the area of the original pumpkin patch, I will observe carefully and ask probing questions. Students who do not make these connections will have trouble figuring out the task. I may go as far as asking students to point to where they see these in the table, the task, the diagram. 

When the class is done with the Warm Up, I will make a big deal of telling them that they have discovered an important fact:

120+10x does not equal 130x

Today, we are going to do an exploration to explore why that makes sense.

Modeling with Algebra tiles

15 minutes

Before handing out the Algebra Tiles today I need to introduce student to the use of these manipulatives (see students to algebra tiles). 

Next I tell students that we can tell each piece is different from the others visually, but let's choose some values for x and y just to make sure they have different values and can't be counted together as the same. We choose a couple of values for x and y and calculate the values to reinforce the meaning of the concept of like and unlike terms.

Now I distribute the Algebra Tiles. As I do I ask my students to hold up the tile that represents x squared, xy, etc. Then I tell the class we are going to use these pieces to build expressions. We will complete the first expression together:

 x^2+2xy+3y^2+2x^2+xy+y^2

As we do I point to each term and ask my students to hold up the piece we will use to represent this term. When they hold up the piece I ask them how many of the piece we we will use. Then, we build a collection of the pieces representing each term. My students are working on the table. I am working on the overhead projector.  Finally, I ask students to count up how many of each type of tile (like terms) they have and put the like terms together. I ask a student who finishes quickly to combine mine on the overhead.

Now we are going to use this model to help us write an expression for what we have after combining like terms and we count up the number of each piece we have. Once we have the expression I circle each term (i.e. 3x^2) and I ask where they can be found in the original expression. I have students come up and circle them in the original expression and I connect them. I do this so they can see the relationship of the concrete model to the variable expression and can transition more easily to combining like terms without the manipulatives. I go through this process for each of the expressions below after they build, combine, and write.

Exploration

35 minutes

Once they become comfortable using Algebra Tiles and writing expressions, I give them three more examples to build, combine, and write on white boards. Algebra tiles are a great visual model for like terms. I make sure to include examples that may help to surface and prevent common misconceptions:

  1. Terms without a coefficient so they recognize that there is ONE to be added.
  2. Terms with the same coefficient for unlike terms so students recognize that it is NOT the coefficient that make the terms "like" terms.
  3. Expressions that have NO like terms, because sometimes students will leave them out entirely if there is not a term to combine it with.
  4. Expressions with the terms not alternating, but in different proximity to each other.

This helps them both focus on the "like" characteristics and also allows me to give them experience working with the Standard Form of a Polynomial Expression.

  1. 2x^2+4xy+y^2+2x^2+xy+2y^2
  2. x^2+2xy+3y^2+2x^2+y^2
  3. 4xy+3y^2+2y^2+2x^2

Since this is new, interesting, challenging work, I like to pose a challenge. I'll give them a choice. They can do x^2+3x^2+2xy+4y^2+4xy if they don't want a challenge or they can do x^2+2xy+3y^2+2x^2+xy-y^2 if they do want a challenge. Some may not see the difference (no pun intended), but then they notice that there is a term being subtracted. When I go over this one and I am asking where they were in the original expression I make sure when I circle the term that I also circle the operation.

If students still don't understand why x^2 terms can't be combined with xy or y^2 terms you can do a couple of things. You can either choose new values and show that they are different amounts again and that some number of one is not the same as some number of the other. Context is another way to emphasize what makes terms like or unlike.