The Biggest Box

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SWBAT use a cubic polynomial to model the volume of a box. SWBAT estimate the maximum value of the polynomial on a given interval.

Big Idea

How can you maximize the volume of this box? With a cubical parabola, that's how!

Introducing the Problem

10 minutes

At the beginning of class, I present the problem by drawing three pictures on the board to model the stages of taking a flat rectangular sheet to make an open-top box.  Initially, the diagrams are completely unmarked aside from the original dimensions of the rectangle, the image includes green and red markings we added during the class. 

To demonstate in 3D, I'll be sure to have a piece of cardboard ready with the squares already cut out.  I can fold this one up into a box for the class to see.

Once everyone understands the problem they're being asked to solve, I give them a good five minutes of quiet, individual work time.  This is when students begin marking up their diagram, building models (if they like), and making a first attempt at creating an equation.  The most difficult step is often the first, labelling the side of the square cut-out, x. (MP 1MP 4)


Creating an Equation

5 minutes

Now, I will call a student to the board to explain the equation that he or she has arrived at. With my prompting as needed, the student explanation should begin with the choice of the variable (x = side of the square), articulating the expressions for the dimensions of the box ((12 - 2x) by (10 - 2x) by x), and explaining how these can be used to create an algebraic model for the volume.

Other students are encouraged to ask questions, and I ask them to share alternative forms of the equation if they have them.  In the end, I will make sure that we have both a factored form and the expanded form on the board for all to see. (I make sure to use both forms because one of the main goals of this entire unit is to understand the relationships between these forms and the behavior of the function.)


Creating the Algebraic Model

20 minutes

"So, how does this equation help us," I ask.

Well, now we can quickly figure out the volume of the box for any sized square we like.  For instance, if the square has 1-inch sides, then the volume is V(1) = 4 - 44 + 120 = 80 cubic inches.

"Ok, what are the largest and smallest squares we could use?"

This question is not so easy, but it leads to the domain of our function, which is 0 < x < 5.  Once we've established the domain, I tell the students that I want three things from them:

  1. A table of V(x) values for x = 0, 0.5, 1, 1.5, ..., 5.
  2. A graph of the function from x = 0 to x = 5.
  3. An estimate of the maximum volume, and how to get it (MP 5).

I give the class about 10-15 minutes to complete this either individually or in small groups.  For the sake of efficiency, I encourage them to collaborate to complete the data table more quickly.  Please see this sample of student work for an idea of what I'm looking for here.

Seeing the Big Picture

10 minutes

At this point, I will find a student whose work is exemplary, and I will ask him or her to share their work with the class using a document camera.  This gives everyone a way to quickly check their work, and also give us a single table/graph to discuss together.

I ask students what they have noticed about the graph, and I expect to hear the following:

  1. It's like a parabola, but it's not symmetric.
  2. The maximum occurs around x = 1.8.
  3. It seems like it's going to go back up after x = 5.

The first and last points are the most important to me, and they lead to the final part of this assignment.  I now ask the class to quickly evaluate a few more x-values outside of our restricted domain to figure out what this function is like overall.  The students should share the values with me, and I'll add to the graph on the whiteboard.  Please see this student's work for an example.

Within a few minutes, we should have a full picture of the cubic parabola, and students will begin to comment.  What a weird graph!  Why does it turn around like that?  Will it just keep going up?

[Class can end with students simply marveling at the strangeness of the graph, but if there's some time, I like to ask why the graph has these three x-intercepts.  The answer I'm looking for comes from the factored form of the equation, but it isn't easy for students to see this right away.  When they do, they're often surprised at the similarity to quadratic equations. (MP 7)]