This lesson focuses on the real world applications of area and perimeter measurement. As we begin the lesson I display three apartment floor plans:
We will use these plans to begin our work with area and perimeter. Our topic for discussion is how much flooring must be ordered to replace the existing floors in each apartment. During the Introduction we will brainstorm strategies for calculating the area of each apartment.
This initial work will help me to assess my students' ability to determine the area and perimeter of regular and irregular shapes. Students have already had some review of calculating area in earlier lessons (Checkerboard series and Farmer Frank lessons in the Writing Expressions and Equivalent Expressions units). At this point I expect that some of my students will still be mixing up the area and perimeter definitions (and formulas). So, I am looking to see who is able to model a problem correctly and who cannot (MP4).
As we work determining how much flooring is required for each apartment, I am asking my students to explain what they see in each floor plan. As they explain their observations I will ask questions to guide them towards determining which calculations they need to make. If necessary, we will do a brief review of why area and perimeter are calculated the way they are.
For the most part, after this introduction I will allow my students to work together in their Math Family Groups. I think it is the best way to surface all the different mistakes and misunderstandings. This not only engages students in mathematical conversations, it helps students gain clarification and correction from each other (MP3). I want my students to learn that they can use their peers as a resource. I also want them to gain the confidence that they themselves have valuable math knowledge and expertise to offer.
Next we will apply the ideas that were discussed during the Introduction. I tell my students that they will be choosing new flooring for the living area of their first apartment, and they need to know how much to order. I pass out the Warm up which has a diagram of a three room apartment. I'll say, "Here's the floor plan of your first apartment!"
My students will now work with their groups to calculate the area of the living room, dining room, and kitchen of this apartment. There are no measurements given. I tell my students to work with their groups to identify:
After a few minutes I will ask groups to share their ideas. As students share I project a copy of the warm up and scribe student responses so that all of the ideas are visible to the class.
My students usually know they need to measure the length and the width of each room. When a student makes this suggestion, I will ask him/her to come up and highlight the necessary measurements on the overhead. I want my students to observe other students selecting measurements that are needed. It shows everyone how they are seeing the shapes in the floor plan. I've found that my students often see the shapes in very different ways.
Teacher's Note: One thing that usually needs clarifying is that the hypotenuse of the triangular kitchen is not a useful dimension for calculating area. If my students are confused by this, I use Auxiliary Lines to complete the rectangle. Then, I will ask a student to identify the the length and width of the rectangle. If many of them don't realize it is also half the area, then I know we will need to spend some time on the area of a triangle during this unit.
Once we have discussed the visual representation of each apartment on the floor plans, I like to introduce two different visual scaffolds to help my students gain a deeper understanding of the meaning of Area and Perimeter as measurement concepts.
How many small squares are in the bottom row? How many are in the row above it? How many in row 3? Row 4? ... Row 8?
What mathematical operation can we use to model counting 8 rows of 8 squares?
On the back side of the Warmup is the Fancy Flooring Activity. On this page the students are given some of the measurements for the floor plan of the apartment, but not necessarily the ones the students identified during our discussion. As students look over the given information, I will say:
I'd like you to work together to figure out the length, width, and area of each room. We need to know how much flooring to order.
As they work, I will circulate and make sure that my students are using the table that is provided to record their measurements and area calculations.
As students work together on this task they will usually need to resolve misunderstandings. For example, the measurement that one student lists for width of a room, another may have recorded as the length. In a situation like this I will try to encourage a discussion like, "What's the difference between length and width? Why do we have two different names for these measurements?" Of course, the choice is a relative one. My students usually recognize this, but I like to ask them to check their area calculations.
Two other discussion points that I expect to arise are:
As I circulate I ask students to explain what they think and why. Then, I ask other group members to respond. This helps students gain clarification, support their arguments, and critique each other (MP3).
During our closing discussion today I want my students to come up and show how they figured out the measurements they needed and how they used them to find the areas. I project warm up for them to use as they share their explanations. I am listening for complete explanations with evidence and an explanation for how their evidence supports their claim. I may ask a lot of questions like:
I may also ask the rest of the class, "Who is not yet convinced yet? Does anyone have anything to add?"
I find that giving students the opportunity to present and challenging them to be convincing helps them to acquire self confidence in their ability to explain mathematical ideas and problem solving strategies. I think both the presenter and the audience benefit, as everyone sees a model of a good explanation.
I also like to go around with my ipad and videotape some of the conversations during the exploration to play back to the class. I've included a very short video (Pedro and Ruben) I shot of two students arguing about the length of the dining room. One of them makes the common error of assuming that it is a square, because it looks like one. However, he is eventually convinced by his partner's evidence.