The reason I use floor-plans to teach area of composite figures is to provide context. Not only does it illustrate a real world application, but it makes it easier for students to access and apply their prior knowledge about rectangles. Without context students don't automatically look to opposite (or facing) sides of a rectangle in order to calculate unknown lengths. But when the measurements are related to walls make the connection more easily. This helps students calculate unknown lengths by adding or subtracting the given lengths on the opposite wall. Having the rooms labeled (living, dining, etc.) also helps them consider different ways to partition the composite shapes into rectangles. Separate room labels implies separate rooms which helps students visualize the "invisible" partitions.
The floor-plan for Minion Manor is distributed to each student. Although students work in small "math family" groups I find it helpful for each student to have their own copy to work with. I display one under the document camera with the living, dining, and kitchen area outlined and highlighted. I tell students we want to install the same flooring throughout these three rooms and ask how much we need to order.
As I am circulating I am looking to see that groups are drawing on their floor-plans. I am looking to see that they are separating the area into rectangular shapes and using the known measurements to calculate the unknown measurements. When I stop at each group, some will immediately have a question for me while others will just continue explaining. If the group is stuck and asks a question it is important for me to have them go through their process with me so I can guide them. In one case two students got different answers and asked: What did we do wrong? Whether students ask a question or not it is important for me to ask them to explain their process. This is helpful to them in many ways. It can help them practice articulating their thinking in preparation for argumentation. It can also help me model their thinking for them and make it more visible to the group so they can find a mistake or next steps. Here is a case where I modeled student thinking to help the group use the given lengths to find the unknowns.
Students then come up to board (where the floor-plan is projected) and explain their solutions. If a student makes a mistake I want it to be critiqued & corrected by another student. It is critical that a supportive and safe community has been established within the classroom so that this is non-threatening. I also encourage students to share multiple methods (Ruben & Dale). This helps them develop flexibility as well as perseverance.
Once the class has been exposed to multiple strategies for separating the composite "room" shapes into rectangles and calculating unknown lengths I move on to the Black Pearl floorplan. It is not necessary for all students to have arrived at the correct solution to the warm up, because this next problem gives them additional practice. I expect collaborative groups to share their thinking and try out and critique different strategies. Collaborating with the group also gives ELL students more chances to practice the academic language.
As I circulate I am still looking for groups that need scaffolding. My main tools are modeling student thinking and questioning as the least intrusive means of scaffolding. As the student explains I will diagram or highlight to model their thinking for the group and turn the question back to the group. This helps me guide their thinking without giving too much information. They gain confidence when they see that the ideas are really coming from them and not me. When they gain confidence they are more likely to persevere.
The last few minutes of class I ask students to do a quick silent write (about 2 minutes) to explain what was tricky about these problems and what steps they had to take to calculate the amount of flooring needed in both cases. I then ask them to share within their groups and then with the class. Student ideas are expressed in many different ways using varying levels of terminology which is sometimes hard to decipher. When students are summing it up it is important to model questions like "what does he/she mean by....?" and "how is what Jake said similar to what Austin said?", "who explained it in a different way?", etc. This helps students (especially ELL) develop their academic vocabulary, make mathematical connections, and conduct academic conversations on their own.