The Box Investigation

Shown below is the picture of a word problem that is often found in homework problems when discussing relative maximums and minimums. This year, I decided to give it as an investigation to my Algebra 2 Honors class with very little help. I broke them up into groups and then gave them the document below. What followed was a 40 minute class of strong discussion, aha! moments, and deep learning of how what we do in class can be applied to the real world. Students recently learned how to find the maximum and zeros on their calculator, but were not adept at setting their window before this activity. We had 40-minute classes due to a special assembly, but 50 minutes would have been perfect for students to have created the boxes in class. it is interesting how, even after solving everything, some students were unsure how to make the boxes. Seeing the whole problem through was important. I assigned the creating of the boxes for homework.


(also found here.)

Graphing calculator pictures:

A picture of the maximum volume

A picture of one non-extraneous value of x where the volume = 400.

I think this problem tied together many things:

• Why domain of a function is important in the real world
• How to adjust windows on the graphing calculator
• How to find the relative maximum on a graphing calculator
• Why the relative minimum of the graph didn't matter (out of the domain.)
• What an extraneous root looks like (the graphing calculator gave a third answer outside of the domain for a volume of 400 cubic cm.)
• How to work in groups as a team (though each student did have to create their own boxes)
• How letter e) could have been solved two ways: by using the so far unknown intersect feature of the graphing calculator or by setting the equation equal to 400, subtracting 400 and finding zeros...this was cool for kids to see!
• The eventual tie-in to polynomial graphs
• The actual making of the boxes

Perhaps my favorite comment (several mentioned that they loved doing this) was when a student said that when he heard we were doing an investigation today, he was nervous, but then during and afterward, he was excited and really enjoyed himself.

I was observed during this class, and in the write-up, my department chair noted that every student was focused, on-task, and working together for the entire period. He wrote that students exclaimed: WE GOT IT!!!! and that "the teacher smiles and laughs with glee." Ha. Yep. Every damn day.