This lesson is a continuation of the previous one in which students developed a working definition of surface area. Students continue to figure out how to best represent their strategies and present them to the class. Working in small "math family" groups forces students to attend to precision as they try to articulate their mathematical thinking. This is a good way to engage them in constructing and critiquing arguments. It is also particularly helpful for ELL students as they practice new vocabulary. Referring to physical models and visual representations is also useful for ELL students as they attempt to make their thinking visible.
Each group may have a slightly different method for calculating the surface area of our 3D object, but they all come to understand the basic idea of adding the areas of all the flat surfaces on the object. I think it is important to focus on this underlying concept rather than on a single algorithm, because once they know what they are trying to find they won't get confused about what operation to do. Once they have this concept students will find a strategy. There is no need for teaching a step by step process or a trick for remembering. The key role for the teacher is to really listen and help students articulate their thinking clearly.
At the end of the previous lesson (day 1) students shared their definitions of surface area. Today we revisit this to help cement the idea and to give more students a chance to share. This is a very quick warm up activity just to reengage students. I ask students to do a quick silent write explaining what they think surface area means. They then share in their small groups and then with the whole class. It is important to let students first share in small groups when they are learning new ideas and new vocabulary. This is especially important for ELL students to gain more confidence in their speaking ability as well as to get help from their peers on pronunciation and word usage. You will find more volunteers for sharing with the class after allowing small group sharing first.
Students started working on posters in the previous lesson and they will spend some time finishing them in this lesson. I return their posters and remind them where the supplies are if they were using rulers or markers, etc. Before they get to work I also remind them to focus on making the poster a useful resource that will help the class understand their strategy for calculating the surface area of our object.
The teachers role is to circulate and help students decide how they can make their thinking visible on the poster. I do this in a couple of different ways. I expect to see a lot of drawings on the posters. I ask students what their drawings represent and ask if there are any labels or numbers that would make their strategy clearer and more apparent. I am also looking for groups who seem to be stuck in which there may be a lot of blank space on the poster or they may be arguing or only one person is working on the poster. For these groups I ask them what the total surface area is and how they figured it out. As they begin to describe their process I may ask questions like:
It is also important to keep the class informed about how much time they have left.
One thing I noticed is that some groups get really involved in drawing and labeling that they forget to put the final answer on their poster. Since there are no units provided I noticed that some of my students forgot to represent these as well so reminding them will be helpful. This might bring up an interesting conversation reinforcing the idea of a 2 dimensional measurement of a 3 dimensional object.
Each group uses their poster to present their solution process to the class.
One group did a flat pattern for the entire figure. Another represented just the stairs in a flat pattern while a third thought about the stairs as sets of two. A couple of groups shared the idea of missing cubes while trying to use the area formula while another group combined two sides into a rectangle so they could use the area formula. Another group made a case for the most efficient method. Sharing and discussing multiple methods allow my students to make connections to foundational or related mathematics and they help students gain a more nuanced understanding of surface area which allows them to think more flexibly.
I also encourage students to talk about the problem solving process and any points when they changed their minds. This helps them see mistakes as a starting point.
Because many of the groups had unique elements in their solutions I want to make sure to highlight those. I may ask them to explain a little more about that aspect or I may point something out on the poster and ask that they talk a little more about it and answer some questions about their strategies. I will point out that they may be the only ones that did it this way or had this idea. I remind the class that it is hard to follow someone else's thinking and they would benefit from a little more in depth description. Not only does this boost the presenters' confidence, but it also makes the audience pay a little closer attention. It gives students permission to not fully comprehend at first and gives them a second change to follow along. So many times students give up when they don't understand the first time because they think they are not smart enough.