Sue Van Hattum’s Optimization Problem

Last night John Golden included  one of our Family Math projects in a list of some neat math-related posts he’d seen recently.  Thanks, John, for including us in the Math Carnival:

Reading through some of the other projects in the list, I found a really neat activity by Sue Van Hattum and decided to try it out with the boys this morning.  The idea in this activity is to find the box with maximum volume that you can make using a sheet of paper:

Although this is really a calculus activity, it still seemed like there was plenty of interesting math for kids to talk through and I figured that I’d just show them the final optimization step on a computer.

The first step was an introduction to the problem.  We’ve talked a little bit about geometry, so the basic ideas behind how to find the volume of a box were at least something that they’ve heard before:

The next step was to go to the kitchen table and look at some examples.  Before we turned the camera on I had each kid prepare one of the sheets of paper we’d be using in this section.  It was really interesting to me to hear them talk about the patterns that they thought would be in these numbers, as well as where they thought the maximum would be.

The last step was going to the computer to see what the graph of our volume function would look like.   I’ve recently been talking about graphs of functions with my older son as part of his normal school work.  Some of the lessons I’ve learned from those discussions have made me want to talk a bit more about graphs with both of the kids, that was part of what attracted me to this lesson.  I though it was neat to see them connect the specific calculations we’d done in the prior step to the points on the graph and then see that the maximum volume actually occurred at a slightly different value of x than they were expecting.

Thanks again to John for including us in his post, and thanks to Sue for this great exercise.