The File Cabinet problem was designed by  Andrew Stadel and shared on his website, Estimation180. I have wanted to teach it for a while, and last week I got my chance when I had the opportunity  to cover for a teacher who was out of town.

The problem explores two different geometry standards:

• Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
• Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (The examinee will demonstrate an understanding by applying geometric methods in modeling situations.)

Here are the steps to my lesson:

1. Warm-Up (included in the Materials doc above)
   • Give students the warm-up picture and ask them to write down as many things as they can about the figure. (It is a rectangular prism).
   • Ask them to do it solo and then share out as a class while you record their ideas.
   • Give them some numbers – two dimensions (the length and width of the front face – and ask them what else they could say about the figure.
   • If possible, use a rectangular prism (garbage receptacle, cereal box, etc.) and have students describe different ways to measure it (what does it mean to measure its length, its area, its volume, etc.)
   • If it doesn’t come up, ask:
     • Does anyone know the name of this figure?
     • How many sides does it have?
     • Say, “I found an interesting video that had a similar shape in it and I would like to share it with you”
2. Watch Video (Act One)
   1. Ask students to write down one question they have and are curious about.
3. Notice/Wonder
   • Ask students what they noticed. Have them write a few things solo. Then share with partner.
   • Ask students what they wonder. Have them share their initial question with a partner. See if they were wondering the same thing. Come up with some additional questions.
   • If you need to, you can focus student questions by asking, “What math questions can you ask about this situation?”
   • The question you want them to work on is
     

     How many post-its will it take to cover the entire file cabinet? 

   • If they ask it (or something close), great, if not, have no worries. Just say, “These are all great questions, and I hope we get to answer all of them. To begin, I’d like your help with one question in particular – How many post-it notes will it take to cover the entire file cabinet?”
4. Estimations
   • Ask students to write down three estimates. One they know is too high, one that is too low and one that is their best guess.
   • Write down their best guesses and names on the board.
5. What information would be useful to know? (Act Two)
   • Clarify the question again. Tell them you have some more information about the situation. Ask them what information would be useful to know and give them some time to discuss in groups.
   • Bring the class together and record the questions they have/information they would like.
   • What you actually have to offer is:
     • The dimensions of the file cabinet (height, width and depth in inches)
     • The dimensions of the post-its.
   • If you are able to project the image, I would recommend it (see Google Slide presentation below).
   • If not the Materials Doc has a handout with the photos of the file cabinet and post-it notes with the dimensions.
6. Problem-Solving
   • Have students work individually first.
     • Work on your plan.
     • Test your ideas.
     • Make changes to your plan (if you need to)
   • Then let students come together in groups of 2-3.
     • Write your names on the paper
     • Collaborate to solve the task together
     • Organize your solution so someone could read your work and understand what you did.
   • Potential Student Approaches ( I would not push any one of these, not limit your students creativity, but I offer them here because I think it is helpful to know some different approaches.)
     • Some students might figure out the surface area of the file cabinet in inches and divide that number by 9 square inches (the surface area of the post-it notes)
     • Some students might use the dimensions in inches to figure out the number of post-it notes that could fit along the height and width of each side and multiply those two numbers together,
     • Some students might notice from the video that there are 12 post-its across the front. They might also notice that the height is twice the width (when given the dimensions in inches). And if the height is twice the width, that would mean there are 24 post its down the front of the cabinet. 12×24 is 288. There are also that many on the opposite side (the back). Then for the sides, they would have the same height (24 post-its) times half the width (since the depth is half the width) which would be 6 post-its. So the number of post-its on each of the shorter sides would be 144. For the top, we know the depth is 6 post-its and the width is 12 post its, so there would be 72 additional post-its to cover the top.
   • Support and Push questions: These questions are included in the Materials doc above. I usually print out a couple of copies of the page and cut out the individual questions. Then I walk around looking and listening to students work. If students seem like they need a little support or an extension, I drop a question that builds off of what they are doing.
     • Support Questions: these are questions to ask students you are struggling. These questions allow you to help get students unstuck, without doing too much of the work yourself
       • Draw a picture of one face of the file cabinet. How many sticky notes would fit across? How many sticky notes would fit going down?
       • Draw and label a picture of one face of the file cabinet. What is the area of that face? How many sticky notes would you need to cover that area?
       • How many square inches does it take to cover one Post-It note?
       • Check-in – included in the Materials packet is a photo of the front of the file cabinet covered with the numbered post-it notes (288).
     • Push Questions: these are questions to ask students who have answered the first question and are ready for further explorations within the same situation. These questions allow finished students to remain engaged without disturbing the students who are still working
       • What is the surface area of the file cabinet in square inches? What is the surface area of the file cabinet in square feet?
       • If the DEPTH of the cabinet was doubled, how many more post-its would be needed?
       • If the WIDTH of the cabinet was doubled, how many more post-its would be needed?
       • If the HEIGHT of the cabinet was doubled, how many more post-its would be needed?
       • How many sticky notes would it take to fill the cabinet?
       • What formula (process) could you come up with to calculate the surface area of any rectangular prism?
7. Sharing Work
   • Record each groups answer on the board (correct and incorrect, in any form students give it to you)
   • If you have a lot of time, have each group present their work. Ask other students to share one thing they appreciate and one question they have about each presentation.
   • If you have less time, let your students do a gallery walk. Have them walk about and view each poster. Write one thing you like about this method and one question you have (either on post-its or on the poster itself)
   • If even less time, have one group share their strategy (maybe the most common or most unique)
   • If no time, just go with the list of possible answers you recorded.
8. Watch Act Three video 
   • Whose guess was closest?
   • If we weren’t exactly right, what could account for the discrepancy?
   • What are some easy to make mistakes that we can all learn from?
9. Titles
   • Ask students, “How could we title this lesson so that it captures the math we used and where we used it?”
   • Using the Math of ________ and ________ to cover a _______ with _______
   • One potential title is “Using the math of surface area and rectangular prisms to cover a file cabinet with post-it notes.
10. Reflection
    • Have students complete the following statements:
      • In today’s class, I am proud of how I…
      • I could do an even better job next time by…
11. Further Exploration/Sequels (included in the Materials doc above)
    • You have 1,000,000 sticky notes. Design a file cabinet that would be covered by all 1,000,000 sticky notes.
    • Find a rectangular prism in your house. How many 3″ by 3″ post-it notes would it take to cover it?

If you have the ability to project, here’s the link to my Google Slide presentation slides (which can be downloaded if you do not have internet access. Please note that if you do not have access to Internet, the two videos would also need to be downloaded from the URL link above).

Here are a few photos of student posters:

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