You may have heard me or my students mention the Box Factory lately, and wondered what in the world we were talking about.  Let me tell you about this fabulous math work we’ve been doing lately.
In 5th grade, we have a unit on 3D geometry, focused around finding volume of different kinds of rectangular prisms and figuring out a formula for how to do this (l x w x h or b x h).  This year we incorporated a unit by Cathy Fosnot, which created a context for this learning.  Enter the “Box Factory.”

The basic premise of the investigation is that kids work in a box factory and have to figure out certain things related to volume and surface area (although these things are not specifically named until later in the unit).  There were three parts, and kids worked small groups to investigate the answers to these questions:
1.  If the box factory wanted to create boxes that held 24 items, how many different boxes could they create?  What would the dimensions be of those boxes? Which box would be the cheapest one to produce? There were 16 possible answers to this question, and the students used cubes, graph paper, equations, drawings, or whatever necessary to figure it out.  They had to then create a poster to show their strategies and explain their thinking to show to the other groups.

2. How much cardboard would you need to cover each of these boxes? This one extended the conversation into surface area, and invited students to now look at the outside of the box, instead of just the inside.  Most groups figured out that if they used the formula (2 x L) + (2 x W) + (2 x H) to determine how much cardboard they’d need.  The cheapest boxes to make would be ones that are closest to the shape of a cube, as opposed to a long, skinny box.

3.  If the factory created three sizes of cube-shaped boxes–2 x 2 x 2, 3 x 3 x 3, and 4 x 4 x4, how many units could each hold?  If it costs 12 cents per unit, how much would each box cost?  This one looked at the inside again, and added another layer of multiplication (with money) to figure out the final answers.

All throughout this investigation (which goes for about 10 days), the focus is on kids discovering strategies for volume, rather than just giving it to them.  Through the posters they create and the Math Congress conversations we share, they are also working on sharing and representing their thinking.  They are learning how to make their representations clear and concise so that other people can understand exactly what they did.

This poster-sharing part is not new to me in Math Workshop.  But Fosnot’s unit added a layer I’ve never thought of before in math–revision.  Much like when mathematicians publish proofs (and like we’d just spent time on in Writer’s Workshop!), students were able to get feedback from others on what worked, what was confusing, what they should add or take away.  They they had the opportunity to revise and edit their posters before they shared.  With each new poster they created, they added new ways of showing their thinking clearly.   They did this by discussing with their group, and then leaving suggestions on post-its.  We used the “Plus-delta” model to share something we liked and something we’d change:

So by the third time around, we were pretty great at showing thinking on our posters.  Even though you didn’t see all the steps, you can still appreciate the clarity and organization of these:

Have you ever done a Cathy Fosnot unit before?  How have you used revision and feedback in math to clarify thinking? What strategies do you use for teaching volume? We’d love to hear about it!