##
* *Reflection: Developing a Conceptual Understanding
The Many Faces of Surface Area - Section 3: Exploration

Even though the physical model, the 3D drawing, the 2D net, and the verbal descriptions all represent the same thing students have trouble transitioning between them. One tricky part of the 3D drawing is that students have trouble visualizing each face because they are not all visible. They may think there are only 3 "sides" or "faces" when there are actually 6. Asking them to label the faces "front", "top", "right side" helps a lot because these terms naturally imply an opposing but equal face (back, bottom, left side). Also, asking them to think about the shape and dimension of each face as well as attend to which other faces each is connected to can help them visually "turn" the figure in their heads.

They also have trouble with the "side" and "top" views of the faces in a 3D drawing. Because they do not appear rectangular in shape they have trouble "seeing" the dimensions. At one point I asked if it would help them if the dimensions were labeled. This was a nice way of letting them think that it was their idea, when actually it supports my aim of weaning them from the visuals. Seeing how useful the dimensions are will help them see how the calculations relate to them.

*Visualizing helps spacial reasoning*

*Developing a Conceptual Understanding: Visualizing helps spacial reasoning*

# The Many Faces of Surface Area

Lesson 5 of 6

## Objective: SWBAT calculate the surface area of a rectangular prism from a drawing.

## Big Idea: Students discover the relationship between the dimensions of each face of a prism and the dimensions of the prism.

*55 minutes*

This lesson allows students to apply and cement what they have learned from their earlier explorations of surface area (Making Sense of Surface Area). It helps them start to notice**connections between the dimensions of the physical model and the dimensions of each face.**One problem that students have when calculating the surface area when they are given only the dimensions of a 3D prism is understanding what the dimensions tell us about the number and dimensions of the individual faces. They don't automatically know that if the dimensions are 2x3x4 that there will be two 2x3 faces, two 3x4 faces and two 2x4 faces. Rather than the teacher teaching students a cute trick like the The SAD smiley face trick I think it is better for students to notice for themselves where how the algorithm relates to the dimensions. Students persevere and teacher does no direct teaching of an algorithmic process.

Most of my students come with a lack of experience with geometric figures. Asking students to work with and explain their work with physical models and visual drawings helps their prior knowledge and vocabulary come together. The key to this lesson is focusing on **multiple representations** and multiple methods. It is also very important for the teacher to really listen to student thinking and help make their** thinking visible** to others. This can help the teacher identify gaps and intervene and, at the same time gives students exposure to different explanations in search of the one that clicks for them.

This lesson is the **perfect storm of knowledge gaps**. Students need remediation with respect to prior knowledge about shape, area, 3 dimensional figures, and geometric vocabulary. Be ready to address a lot of confusion with clarifying and defining terms, referring back to the models, asking student to explain using evidence from the models. Much of their prior knowledge may be there, but is disjointed or unclear. Taking time to clarify terminology like side, edge, face, dimension, prism, rectangle, parallel, equal can really help clear up some confusion.

#### Resources

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#### Warm up

*15 min*

Students were provided snap cubes and asked to Build the figure that has two square faces measuring 2 by 2 units and four rectangular faces measuring 2 by 4 units. Then they were asked to calculate it's total surface area. Because they had all just finished a visual presentation of their initial exploration of surface area (Making Sense of Surface Area) many of them began drawing different representations of the figure. Some represented the figure as a flat pattern, others drew a 3d prism with three visible faces, while still others showed individual squares.

As I circulate I am assessing how well they have come to understand the meaning and calculation of surface area. I expect them to understand that the surface area consists of the sum of the areas of each individual face. Students will have different ways of organizing their calculations, but I want the **concept definition to be solid** before moving on without the physical model. I will not take away access to the snap cubes, they will always be out and available to students, because it is important to recognize that students progress at different paces. Also, the physical cubes are a great tool for **ELL students** to express their understanding and clarify vocabulary.

If students seem to get this I would not spend more than 10 or 15 minutes on this section. If not, I would spend a little time having a group discussion to help refine their definition of surface area.

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#### Exploration

*25 min*

This section gradually eases students away from the physical model, but they still have the snap blocks at their tables to refer to. They are asked to describe the 3d prism drawn on the board on the board using shape and dimension and then calculate the total surface area. Students use many different ways to describe the figure. Some will use a drawing with labels, some may describe with words. It is important to let them choose their own method and share them with each other. Having multiple representations and multiple methods allows students to make connections and gives them a foundation for the algorithm. They need to see where the numbers for the calculation come from and what they refer to.

As I circulate I am looking for multiple methods of calculating the surface area. Strategies range in complexity from counting to multiplying to find surface area. I ask students to explain their strategy and their reasoning to get an idea of who to ask to share with the class. They may express their reasoning differently than I would so I sometimes have to really listen or probe for clarification so I can follow their thinking. I also look for opportunities to help them clarify definitions and use appropriate terms . Forcing them to be clear supports their efforts in **constructing arguments**.

##### Resources (9)

#### Resources

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#### Whole class share

*15 min*

I ask students to share with the class different methods of calculating the surface area. My goal here is twofold:

Firstly, I want them to compare the methods and think about which might be the most efficient. I ask them to decide which method they might like to use and why. Some liked the fastest method, but some preferred the longer method because it was easier to relate to the figure. Either way I want students to see how the methods relate to each other. This makes them more flexible, but also gives them some understanding of how the standard algorithm was derived. While students may still want to add each individual area together they can see where 2(LxW)+2(WxH)+2(LxH) comes from and it makes sense to them.

Secondly, In order for students to calculate surface area when they are only given the dimensions they need to be able to relate the dimensions of each face to the dimensions of the 3D prism. To get at this relationship I ask a few different questions. I may refer to one particular method on the board and ask them where the numbers came from, or where do they see these numbers in the original 3D version. I may need to break it down for them and ask "what are the dimensions of each face?", " how many of each face are there?", "what are the dimensions of the 3D figure?", "how do the dimensions of each face relate to the dimensions of the 3D figure?"

Once students can make this connection I like to solidify and extend their thinking with some deeper questions. The more open ended questions really lend themselves more to sense-making, argumentation, and collaborative groups. A whole other lesson could be devoted to these or you could choose a few for homework. I might send them all home with them and have them choose 1 or 2 to work on. This gives them an element of choice and allows for some natural differentiation.

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- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter