Students will be able to build models wherein area remains the same, while changing perimeters.

Students need time to work on how area and perimeter are connected. In our second day of this two day exploration students apply their learning.

15 minutes

Today I am using the book *Spaghetti and Meatballs for All* with a different task in mind for my students. We will discuss the fact that the area never changed for the Comforts. In other words, the amount of actual “space” taken up by the tables did not change. There were always 8 tables. However, the perimeter, or space around the edges for guests to sit, changed. Since the children explored this concept in the first lesson, I am now going to attach the correct terms to what they were doing and go a little deeper.

*Yesterday Mathematicians, you explored ways to arrange the “tables” so that more people could sit down. Please bring your reflection journals to the community area and open to your work. Let’s look back at Spaghetti and Meatballs for All and see what the Comforts did as more of their guests arrived. Maybe you will see one of your solutions.*

As I read, I stop and allow students time to interact, and celebrate if they discovered their configurations matched those found in the book. I also keep bringing students attention back to the fact that the number of tables (area) never changes.

*Boys and girls, notice that there are never more tables brought into the room. There are only chairs taken away or added, depending on the configuration of the tables. The tables represent the area of the room filled and the chairs are like the perimeter, or the space around the outer edge of the tables. *

20 minutes

Next, I prepare the students to use the area of 12, and find as many configurations as they can. After they label those area models, they will begin to explore the perimeter of each of the models (MP1). The goal here is to not force a procedure for this, but to have the children work to understand that area may remain the same and that the perimeter increases or decreases depending on the model (MP4). They will also begin to develop their own strategies for figuring out the perimeter, and hopefully will see the patterns in the rectangles created by the arrays. Using these blocks helps the students conceptualize the problem and then they can transfer the model to paper.

*Students, you will now work to build table configurations with an area of 12. It is exactly like finding all the arrays you can for 12. Once you do that, I would like you to find the length around those rectangles - that's called the perimeter. Let’s do one together here on the board with the area of 8 as a model. *

As I model, I take particular care to use the words area and perimeter in context, but always make a strong visual and spoken connection to the words students are using for their current understanding. Following my model using the area of 8, I send the students off to work with the area of 12 at their tables. I don't say this is a group project, or an independent one. Rather, I allow the students to explore the task in the best way for them.

While students work, listen to the conversations they have with each other or those near them. Also, I ask if they notice anything. My students are so used to this nonthreatening dialogue that they are constantly saying, “I notice something”. So I always rush there!

Listen in as this student explains how he sees the commutative property working to help him. Even though he got an incorrect perimeter, he is seeing an important pattern. This will help him understand the algorithm much deeper when it is introduced. It made my day! Also, I saved correcting his mistake for the closing when he shares. I always like to have peers help each other with these sorts of corrections.

15 minutes

In closing, I have the students share their configurations with area and perimeter. Many students have the same area shapes, but different perimeter amounts. This is when I ask them to “defend” their thinking and teach each other their strategies (MP3).

The student in the video in the previous section put up his work to discuss his observtions of the commutative property. When he was done, another student raised his hand and said, "I disagree with your perimeter because I have the same array with a different amount. This is how I did it…." This is clear evidence that the talking moves and the safe environment in the classroom is allowing everyone to grow.

In many ways, the close of a lesson is the most powerful portion of the day, as this is when the kids really grapple with verbalizing their thinking and that of other's.