What is the biggest box you can make?
You are given a 279 x 432 mm sheet of paper (11" x 17") and asked to create an open box from it by cutting out four identical squares from each corner (see Figure 1) and folding along the dotted lines. See full lesson plan HERE |
Reflection - Knowledge of the Learner
Portfolio question: Knowledge of the Learner
The problem described above is typical of the kinds of problems encountered by high school students in the Ontario math curriculum - they require abstract thinking skills. Yet according to research (Meece & Daniels, 2008), only about 30 - 40% of high school students are cognitively equipped to do tasks of this nature. What this means for the classroom is that the majority of students still need the help of concrete aids to do conceptualize such problems.
To meets these needs, I incorporated a hands-on “Open Box” activity (Miller & Shaw, 2007) into a Grade 12 Calculus lesson on optimization. In this activity, each student was given a sheet of paper and assigned a value ‘x’ from 5 to 100 (in multiples of 5). They were then asked to cut out squares from each corner using x as the side length in mm, fold up the sides and tape the edges together to form an open box (Figure 2). Stacking the boxes starting from the smallest value of x to the largest gave the class a concrete visual representation of the problem. |
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Students then completed a worksheet in which they calculated and shared the volume for their own box on an excel file that was projected onto the screen. The class results were plotted on a graph so that they could visualize how the volume changed in relation to the value for x (Figure 3). By finding the highest point on the curve of this graph, the students were able to solve the Open Box Problem.
Once they had completed this activity and had a solid grasp of the problem, I then showed the students how calculus could be used to find an answer with greater precision and speed. With the input of the students, an algebraic expression was created to describe the volume of the box for any given value of x (Figure. 4). Next, we revisited the connection between the local maximum of a function and the zeros of the first derivative, something we reviewed at the start of the class with a function and derivative card-matching game (Figure 5). Finally I guided them through a mathematical solution of the problem using calculus. Learnings and GoalsWhat I did not anticipate was how long it would take the grade 12 students to make the boxes. It seemed to be an easy activity that could be completed in 5 minutes, but some took as long as 25 minutes, even with all the supplies ready at hand and in sufficient supply, suggesting that many had not had too many experiences of that nature. One of the high points of the lesson was that one student, who I had often seen sleeping during class, was actively engaged. Not only was he the first to complete his box but he even volunteered answers during the follow-up discussion that demonstrated his understanding of some of the concepts.
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Figure 6. Open boxes displayed on classroom bulletin board
As part of an assessment FOR learning, students were given an exit card to complete. In it, they were asked to communicate a set of general instructions for doing a volume optimization problem using what they had learned and to rate how well they felt they understood the day's lesson.
A better idea would have been to ask students to use what they learned during the lesson about volume optimization for an open box and write out a series of general steps that could be used to optimize a the volume of a cylinder with a restraint on the amount of material used. That way, it would tie in nicely with the handout from the next lesson (a pop can optimization problem) as well as providing them an opportunity to write down what they understood of the process they learned in this lesson.
A better idea would have been to ask students to use what they learned during the lesson about volume optimization for an open box and write out a series of general steps that could be used to optimize a the volume of a cylinder with a restraint on the amount of material used. That way, it would tie in nicely with the handout from the next lesson (a pop can optimization problem) as well as providing them an opportunity to write down what they understood of the process they learned in this lesson.
Resources
Meece, J. L., & Daniels, D. H. (2008). Child and adolescent development for educators (3rd ed.). New York, NY:McGraw-Hill.
Miller, C. M., & Shaw, D. (2007). What else can you do with an open box? Mathematics Teacher 10: 470-474
Miller, C. M., & Shaw, D. (2007). What else can you do with an open box? Mathematics Teacher 10: 470-474