To Change or Not to Change

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Objective

SWBAT distinguish constant from variable terms and combine like terms using algebra tiles.

Big Idea

Students learn that constant terms do not depend on the value of variables.

Warm Up

7 minutes

This is the second day of an activity with Algebra Tiles to model the process of combining like terms. As students enter they will see the warmup variable vs constant on the screen. The diagram is a picture of Farmer Frank's extended field (from an earlier lesson -- Extending Farmer Frank's Field). Farmer Frank's original field is labeled as 10 m by 12 m. The extended field is labeled with a width of x.

My students are asked:

  • Which dimension of Frank's field remains unchanged?
  • Which dimension changes or varies depending on the value of x?

I encourage my students to try out some different values for x and show what the area of each part of his field is.  

 

Exploration

35 minutes

Before distributing Algebra Tiles to each student I show them the three different tiles again and let them know we will be assigning new values to some of them during today's lesson. I go through a similar orientation as the last time we used Algebra Tiles, but instead of assigning variables as each dimension I include a constant as well.

Choosing a few values for x, we quickly make a table of values for each type of Algebra Tile (x^2, x, & 1) to reinforce the comparison between constant and variable. I emphasize the fact that the value of unit square ("the piece labeled 1") does not change, and, does not depend on the value of x. Asking which terms change and which remain the same no matter what value we choose for x will help students understand the concept of like and unlike terms.

After distributing Algebra Tiles to each student I will write polynomial expressions on the screen, asking my students to construct them with their tiles. I'll also write three steps for them to follow:

  • Build a representation of a polynomial with algebra tiles to model the expression
  • Combine like terms in the model by moving tiles to form clumps out of like pieces
  • Write a simplified expression for the model after like terms are combined

This is a more familiar process to them after our previous lesson with algebra tiles.

Today we will start with an expression like:

3x^2+4x+2+x^2+2x+1

After my students have written a correct simplified expression, I will ask them to look back at the original expression and circle where they found the following quantities:

  • 4x^2
  • 6x
  • 3

This helps them transition better from the physical model to the mathematical model and see the relationships and connections between mathematical models (as described in more detain in Who's Right?). Next we'll look at an expression like:

2x+x^2+5+2x^2+2 

If the students are making good progress, I might move on to some expressions that contain subtraction.

Teacher's Note: Although some of my students begin to transition away from using Algebra Tiles on their own, I still ask them to predict the simplified expression. This helps students gain experience with mental math and gain confidence in their ability to work with algebraic symbols in expressions.

 

Vocabulary Development

12 minutes

For Vocabulary Development I use a strategy I have modified from the vocabulary guru, Kate Kinsela. It is a Direct Teaching method of a single term or family of terms that gives examples of the target vocabulary in context and then has students create their own sentence using the term and practice in pairs.

The term I choose for this lesson is constant. I focus on this term partly because it has multiple meanings and is used both within mathematical context and in their daily life as well. Words with multiple, potentially relevant meanings are particularly problematic for kids. The term already has meaning for them and the meaning, though similar in math, may be different. In their daily life constant is an adjective, but in math we are using it as a noun. In addition, students are most likely not used to using it to distinguish between things that vary and things that remain unchanged, which is the point of this lesson.

When we do this type of vocabulary development in class I project the term, its definition, etc., and the sentences on the screen vocabulary constant. I read each one using a method that Kate Kinsela refers to as shared reading: 

I stop reading mid-sentence and my students read the one next word. Once we get through all the sample sentences I give students 2 silent minutes to either write their own sentence using the sentence frame provided. If within those 2 minutes they don't come up with one of their own they write down one of mine. Then they turn to a partner and take turns sharing their sentence or the one of mine they chose. Then we take a few minutes sharing out their sentences or their partners sentence. Afterwards I like to collect the sentences they wrote and post a few on the wall on sentence strips.