This lesson is a modification of an idea from The Book of Perfectly Perilous Math by Sean Connolly. My goal is for my students to find unlabeled measurements using the given measures. The unknown measurements are necessary to calculate the area of some of the rooms in the diagram.
A key teaching objective in this lesson is enabling students to apply the knowledge that the opposite sides of rectangles are congruent. They have seen this relationship many times, but it does not always come to mind. I suspect that they are too used to being given all of the necessary information. To accomplish this goal, I encourage my students to think about the components in the diagrams as pieces of a puzzle. I ask, "How can we determine the missing pieces?" My job in this lesson is to keep revoicing disagreements until the students resolve them mathematically. I will try to motivate them to respectfully argue with evidence until all are convinced (MP3).
The For or Against warmup asks my students to make an argument for or against my statement about how much flooring I need for a room. They are given the dimensions for the room and a suggestion for how much flooring to order. My goal for this warmup is to have a rich discussion around a common mistake that I see during this unit. In this case, the mistake is a hypothetical, of my own making.
I expect many of my students will determine that I have ordered too little flooring. For some, it will be obvious that the area of the room is much larger. I want to dig deeper, though. I am hoping that some of the students will determine that I probably calculated the perimeter instead of the area. Taking the conversation in this direction helps to surface the common mistake that I see in my students work. It also gives me the opportunity to encourage students to reflect on the idea that it is not always enough to know that one answer is wrong. Sometimes, the important work is to figure out why it is wrong.
So, during this activity I want students to explain their reasoning clearly and completely. Once again, we'll work until everyone is convinced. In order to encourage students to connect the dots I may ask what questions I, as the person who made the mistake, might still have that need to be answered. This is a good way to get students to learn to assess or critique the strength of an argument as well as help strengthen their own arguments (MP3). I'll also "play dumb" and ask why I should have determined the area, instead of the perimeter. This task is also a good opportunity to focus on the importance of labeling the units on a measurement.
Next I will read my version of the The Punk Prank Payback story from the book Perfectly Perilous Math. I use this story to help set the scene. I want to motivate a playful tone for their team work, when students begin to figure out the amount of flooring needed. I want to make sure that the task is fun and engaging. My students have calculated area before, but in this unit we are learning to solve problems where area calculation is a means rather than the end.
Teacher's Note: In my experience middle school students love to be read to. And, the sillier and more animated the storytelling, the more that they enjoy it. Punk Prank Payback provides a somewhat realistic, although fanciful, situation in which the skills we are learning at the moment may prove very useful.
Here'e the gist of the story: You and a group of friends are vacationing at your uncle's house, but a prank-turned-bad ruins all the flooring. In order to replace the flooring, you need to determine all the measurements and get to the store before it closes. The store closes in 30 minutes and it takes 20 minutes to drive there!
After the Story, I will hand out the Punk Prank Payback worksheet. The worksheet includes a floorplan with some of the quick measurements referenced in the story. I tell my students, "Okay, the car ride is 20 minutes, so that is all the time you have to complete your back seat calculations." I encourage students to work cooperatively. As they do, I circulate and point out when I see a disagreement in their tables, which motivates more careful work and engagement.
On the worksheet, the length and width for the bedroom and kitchen are given. The bathroom appears to be unmeasured. Only one dimension is given for the living room. My students need to recognize that opposite sides of the rectangular rooms are equal. Then, they need to add or subtract the lengths on those sides to find the missing measurements. The width of the bathroom is the trickiest for them.
After the 20 minutes are up, I will use the document camera to display the table from the worksheet. We will go over the measurements and discuss how they were derived. I will be careful to ask students to give their answers using appropriate units.
Usually, the kitchen and bedroom are very straight forward and we can discuss the answers. They only take a little extra time if students are unsure of how and why to find the area. For the bathroom and the living room, I plan to have students come up and show how they found the missing lengths and widths using the other given measures. I may have to play dumb to get them to explain as completely and convincingly.
For homework, I ask my students to complete some more Perimeter and Area Arguments. We began today's lesson with a problem like this. I find that my students enjoy this type of problem, so this is an effective homework assignment for engaging students in mathematical practices. In the resources there are two examples of student work to explore.