Making Sense of Surface Area (day 1 of 2)
Lesson 3 of 6
Objective: SWBAT understand and compute the surface area of a 3 dimensional object.
This lesson introduces the idea of surface area without providing the new terminology. I don't want students to think that surface area is a completely new topic, separate from area. But I also want them to notice the subtle differences as well without me having to tell them. The tricky part about surface area is that one is finding a 2 dimensional measurement of a 3 dimensional object. Many times students have become confused by the adding component, because they have learned that area IS multiplying and perimeter IS adding. This happens a lot when students have learned in a very rote fashion without a focus on understanding. Having students struggle and figure out what's tricky about surface area with their partners helps them better understand the definition of surface area as well as its relationship to area in general. When they have a strong understanding of what it is they don't get confused about which operation to do.
Working with a physical model helps all students, including ELL students, demonstrate and build meaning. Asking students to represent their ideas in diagrams is also really helpful when developing new vocabulary and expressing mathematical ideas with each other. Having students make decisions and plan together helps them construct and critique arguments and also develop more precision in their vocabulary.
Warm up exploration
As a warm up task students in small "math family" groups are given a 3 dimensional object built out of connect cubes. They are asked to find the total area of the object. While they are trying to figure it out in small groups I circulate to check their thinking and to help them progress. I purposely do not introduce the new terminology of "surface" area. I have found that when I make such a distinction up front that students think that surface area is a new topic unrelated to the area they have already come to understand. I think it is important for them to struggle with making sense of the question to gain a better understanding of the concept and its connections to other math topics. Once they understand the concept the calculations fall into place. So, my job is to help students investigate the task and engage in some discussions that will help them build the idea of surface area.
There are several points where students are likely to become confused and several of the videos show how you can help them navigate:
- Students may find the volume volume instead by counting or calculating the number of "cubes" rather than flat surfaces.
- Some may be confused, but not be able to make sense of what is confusing them. Some of them may be able to articulate what is confusing about the question is that they are asked to find a 2 dimensional measurement of a 3D object. If they can then I ask what is 2 dimensional about the object so they start looking at the surfaces. Once they start identifying the surfaces I suggest they might want to draw draw them out.
- Someof my students have trouble representing the side with the steps and may need help integrating different ways to look at the steps. Asking students to show me helps them discover mistakes and can help them make adjustments to their thinking.
Moving on to this next stage does not guarantee that all students have a clear understanding of what surface area means. Planning their poster helps students continue to think about, ask clarifying questions, and develop their understanding. My job is still just to facilitate their group conversation and investigation. Asking students about their diagrams helps them think about and engage with each others ideas. Students are given a large piece of butcher paper and are told to make a poster that they can use to help show the class how they figured out the total area for the object. I tell them that each group is thinking a little differently from the others and I want them to be able to clearly express how they reached their solution. They should put on their poster anything that helps the class understand where their numbers came from and how they arrived at their final answer. They might include diagrams, labels to explain what the diagrams represent, and numbers that the group can refer to as they explain. Students have access to other supplies in the room as well like markers, rulers, etc. I didn't introduce the poster idea earlier, because I was afraid they would spend too much time making pretty pictures rather than representing their thinking.
As I circulate I am looking for a couple of things. I first want to make sure each group is arriving at the correct solution of 36 square units. Some groups are so focused on making their process clear that I may need to remind them to write their final answer on the poster. I first look for incorrect answers, because I don't want them to get too far along in their poster and feel like they wasted their time. For those that have the wrong answer I ask them to explain so that I can figure out where their thinking went astray to help me focus my questioning to get them back on track. Even for groups with the right answer I find that asking them to explain their process helps them decide how to represent it on their poster. I am also looking for what is unique at each group. I will tell the group that no one else saw it this way and I want them to make that part really obvious, because it will be new to the class and they will need help understanding it. I may ask questions like:
- "what made you realize that?"
- "what would that look like?"
- "how could you make that clear to your audience?"
- "show me where that is on the figure."
- "what does this drawing represent?"
Students will have more time to finish their poster and practice their presentation in the following lesson.
Before students leave I want to know how they have come to understand the idea of surface area. I tell them that what they have been calculating is the object's surface area and ask them to do a brief silent write to explain what they think surface area is. Then I ask them to share with their group and make any changes or additions to their original definition. Then I ask each group to share what they have come up with. Students have many different ways of explaining it and will use interesting vocabulary to express their ideas. Some responses are very mathematical in nature and some appeal to the visual/special sense, etc. Sharing these with the class can help students gain a richer more complete, nuanced definition. In addition, it helps me see if we need any redirection or further exploration before we move on the presentations in the next lesson.