Invisible Boxes

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SWBAT calculate the surface area of prisms when given only the dimensions.

Big Idea

The dimensions of the individual faces is related to the dimensions of the rectangular prism.

Intro & Rationale

After using physical objects and working with flat pattern nets and drawings of 3D figures my students need to transition to calculating surface area when only the dimensions are given. This lesson has them work backwards from a physical description of each face and askes them to figure out the dimensions of length, width, and height. The key to helping them transition is to help them visualize and all the time they spent drawing and building really paid off. In this way I hope they will come to generalize that there are 2 faces each for each pair of dimensions. But even if they don't all make this discovery they will still be able to calculate the surface area using an alternative strategy. Another key element of this lesson is generating multiple representations. The individual quiet writing time is crucial for allowing students to gather their thoughts to produce more diverse ideas.

Warm up

15 minutes

Students are asked to spend 2 or 3 silent minutes writing a response to the following prompt:

If a rectangular box has two faces that are 6 square units, two faces that are 2 square units, and two faces that are 3 square units what is the length, width, and height of the box? 

Then students share what they come up with in their math family groups. As they discuss in their groups I circulate to note what different ideas and representations they came up with to make sure they all get shared during the whole group discussion. To make this section move a little faster I may just grab several different representations to show the whole group rather than open the discussion to volunteers. I expect some students to draw a 3D representation of the box, some will draw separate faces, and others may draw a flat pattern

When I display each example I will ask the groups to discuss where they see the dimensions in each representation. I also expect to have to make the distinction between the dimensions of each face and the dimensions of the box. I purposely chose to name each dimension in the prompt rather than just ask for the "dimensions" in order to make the distinction clearer. If I had asked for the dimensions I think many of my students might have given the dimensions of each face. If they still do that I can ask "so what is the length of the box?" 



20 minutes

For their second task I give them the dimensions of a rectangular prism 3x7x5 and ask them to find the shape and dimensions of each face as well as the total surface area. Again I want to ask about the dimensions of each face so they begin to see the connection between the three dimensions and the 3 pairs of dimensions. 

Students may draw any combination of 3D figures,  flat nets, individual faces, etc. As I circulate I ask students to explain their solution. I am looking to see what visual supports they are using and listening for signs of generalizing, like the idea of doubling. When a student shares a "doubling" strategy I engage the class in a discussion of why it makes sense that we can double.

Exploration 2

19 minutes

In the last prompt I ask students to work in their math family groups to find the surface area of a 6x7x4 rectangular prism in any way they like. Then I ask the group how they could figure out what the areas of each face would be just by looking at the dimensions. 

As students are working together I circulate to see if anyone records their work without drawing any pictures. If they do I will address the whole class and say "Angelina, Cristina, and Molly didn't draw any diagrams and have figured out the surface area just by looking at the dimensions. Think in your groups how they might have done that." Then I might share their work with the class and ask questions like:

  • "how did they know that two faces would have dimensions of 6x7? 6x4? etc"
  • "how did they know there would be two of those?"
  • "how did they know that there would be faces with areas of 42, 24, and 28?"
  • "where did they see that in the dimensions of the prism?"
  • "how does this relate to that idea of doubling from earlier?"
  • "how do the dimensions of the prism relate to the dimensions of each face of the prism?"


If all of the students draw diagrams then I might ask these same questions while referring to their diagrams. For example if a student drew each face separately I might point to one of the faces and ask what it's dimensions are and where we see that in the dimensions of the prism or how could we tell from the dimensions that there would be faces with these areas?