At the start of class we will work from Connecting Expressions and Area. (These slides are used throughout the lesson.) When I project Page 2 I plan to ask students to read the two verbal descriptions, but not write anything down. I will use a Non-Verbal Cue to survey the students about whether these two expression say the same thing, or not. After making the students ideas observable by the entire class, I will then ask the students to write their algebraic representation of each description. Once they write two expressions, they can compare with a partner. If the students agree, each should explain how they came up with the expressions. If they disagree, they should work together to describe the commonalities and discrepencies. I want to give students a chance to make an argument in support of their expression and to hear another student's argument (MP3). As students share, I will circulate and encourage them to use precise mathematical language and vocabulary. After reviewing the correct expressions we progress into the main content for today's lesson, beginning on Slide 3.
To begin this section of the class, I will have students Think-Pair-Share around the question on Slide #3. Whether students choose true or false, I will encourage them to justify their answer mathematically. I encourage partners to listen and to help their partner form a good argument (MP3). After a few minutes, I will poll the class by show of hands to see how many students thought that the expression was true. Then, I will have a few students offer their argument and then see if others who thought the same way can build on that argument. Some students may try to solve the equation which will lead to no solution. I will listen carefully to see if they realize that this proves the equation is false. I will also listen for students who plug values in for x. The reasoning behind this strategy provides a good introduction to the next task.
On Page 4 of Connecting Expressions and Area we get to the practical goal for the lesson. I want to demonstrate for students that considering an abstract situation from a more concrete perspective can often offer valuable insight. To scaffold this, I ask students to construct and fill in table like the one on Slide 4. Once students have done so, I ask for students to share what they have noticed. I will eventually ask, "Did any of you change your mind about whether the original equation was true or false?"
Slide 5 extends the concrete investigation of the equation 4(x + 5) = 4x + 5. Some direct instruction will most likely be necessary to help all of my students interpret the representations. First, I will ask if students can explain why the red figure is representative of the red expression and why the blue figure is representative of the blue expression. Students who are having difficulty making the connection may be helped by actually writing the areas in each of the boxes and then seeing what the red expression is actually equal to (4x + 20). This is showing, in another way, that because the two figures clearly have different areas the two expressions cannot possibly be equal.
Using the cards from the previous lesson, students will now add in the table cards and the area cards to make matches. As I discuss in the video_narrative, I will have my students start with the tables. Some numbers are missing from the tables. These can be written in once students match the appropriate tables to the correct expression. Notice that there are fewer tables than expressions in the connecting_expression_to_area cards. Some tables match with two different expressions (see connecting_expressions_to_area_key). I will not point this out to students. It is better for one or more students to discover this fact. When they become aware of this, they have an interesting opportunity to reflect on the concept of equivalence.
To conclude this section of class, I will have students match the area models to the other three cards. Once again, it is possible for students to observe how equivalent expressions can be represented by the same area model. Once students complete the matching activity, I will ask them to paste or tape their cards onto chart paper. I will ask them to present the cards in a way that highlights equivalence.
When students worked on this portion of the lesson I made some really interesting observations. First, I noticed that some students struggled with the area model because they didn't see the area as being the product of the length and width. Once students were able to make this connection they could connect the area models to the various algebraic expressions. My second noticing was about the tables. I was thrilled how quickly students were able to make meaning of the tables and how they connected to the expressions. The whole idea of "input and output" is extremely important to building the concept of functions. Students very quickly noticed that the independent values could be plugged in for the variable in each expression and the result was the dependent variable. Some students even commented that "those were the easiest to match up." This also gave me some time to observe students arithmetic skills when substituting values into each expression.
For today's Ticket out the Door, students will demonstrate their understanding of the area model. The Close for this lesson is on Page 6 of the presentation. While students may not be familiar with the process of multiplying two binomials, they can use the area model to think about the process by using the line of thought developed in this lesson.
I expect some of my students to struggle with determining where the term 12*n comes from in the expression. If this is the case, I will encourage students to write the area of each piece inside the rectangle. This will enable them to visualize the meaning of the expression, and its origin. Some learners may benefit from transferring the measurements from the left hand side of the diagram to the right hand side, giving each sub-rectangle both a length and a width. I will make this suggestion individually.