A significant portion of our seventh graders come to the middle school without a solid understanding of what area and perimeter represent, how to calculate them, and when to use them. This lesson is meant to help students understand when it is appropriate to use area and when to use perimeter. Working together in their math family groups surfaces disagreement, promotes discussion of misconceptions and encourages students to explain their reasoning. My hope is that weaker students will be convinced by the explanations of their peers. I also anticipate that if students are not convinced, it forces the other students to strengthen their argument and find new ways of explaining their reasoning.
The Warm Up tells students that I am putting carpet in a rectangular bedroom measuring 12 feet by 10 feet and that I asked the sales manager for 44 square feet of carpeting. Students must decide if I ordered too much, too little, or just the right amount. Students are asked to explain why they agree or disagree with the amount I ordered. This type of question which forces students to take a stand and support it promotes argumentation that helps them clarify and correct misconceptions. (mp3)
I expect students working in their groups to figure out that it is the area that needs to be calculated and that I needed to multiply the length and width and order 120 square feet of carpeting. Some students have completely mixed up the area and perimeter concepts and formulas, so having a group discussion about it helps them sort out misconceptions. Some students may also recognize that I have found the perimeter instead of the area, which may generate a discussion of why the area is needed instead of the perimeter.
I also leave Warm Up notes for my sub so that he/she is able to help students explain their reasoning.
In this area or perimeter, students are given several scenarios in which they must decide whether to use area or perimeter. They work together in pairs to decide. Students have access to the visual representations of area and perimeter in the form of posters on the board. The substitute is asked to encourage students to draw a diagram to help them decide.
The second part of the activity gives students statements similar to the one from the warm up and asks them to agree or disagree and explain their reasoning. This section will be used in a later lesson to work on strengthening weak arguments.
The substitute is instructed to go over the work with the area or perimeter notes before handing out the homework. He/she is asked to encourage students to explain why they made the choices they did. For the argumentation piece the sub is asked to continue asking for more until all components of the explanation (from my answer document) are given. This gives them a frame of reference for the work we will do later.
The homework perimeter or area requires students to calculate areas and perimeters of rectanglular figures. However the work is set in a more realistic scenario which doesn't ask for area and perimeter. Instead it tells students that the city planners are creating several "green belt" grassy areas in town and asks students to calculate how much grass sod is needed to cover the entire area and how long a walkway would be around the "park". Not only does this show students how and when these skills might be required in the real world, the context helps them decide which is the appropriate calculation.
Since some students are more clear than others about the concepts of area and perimeter it is beneficial for the weaker students to use the stronger ones as a resource. Because they so recently worked on explanations my hope is that they will be more forthcoming with offerring help. The sub is asked to circulate and look for groups that have different answers. The sub is then asked to tell the group to have a discussion in which students make arguments for or against one of the answers.