Outline of a Lesson Plan for Max-Min Problems

(Aimed at Algebra Students)

by Margaret Morgan

This lesson is influenced by the Mathematics in Context curriculum for middle school students, but could be used for College Algebra students as well. You would need to allow different amounts of time for different age groups.

Divide the students into small groups (approximately four people). Begin by handing out a rectangular sheet of papers (all the same size), tape, and scissors. Ask them to work in their groups to figure out how to make a lidless box and then for each student in the group to make a box. Hopefully, students will realize that they must cut squares out of each corner and fold up the sides to get a box.

Once multiple boxes (hopefully not all with the same dimensions) have been created, we can begin to talk about the size of the box. The discussion could be initiated by asking the students who has the "biggest" box and once they have made a choice to explain why that box is the biggest. Hopefully, different students would give different answers and explanations. Some possible choices would be the box with the largest height, the box with the largest width, and the box with the largest volume (perhaps described as the box that could hold the most stuff).

After a discussion of the different ways of looking at the size of the box, I would focus the class on the box's volume. After looking at maximizing volume we could look surface area and address issues related to height and width. I would now give each group tools to measure the volume of the boxes (rulers and cubes come to mind) and ask the groups to determine the volume of each box. After this task was completed we could discuss the different methods used to determine the volume of the box. Hopefully, this discussion will lead to the possibility of using the formula V = lwh.

At this point we will have some data to explore. I would make a table displaying the size of each student's box. A sample is below based on initial paper size of 5 x 8.

 Student Name Volume of Box (in cubic inches) Emily 8 Vance 15 Keesha 8.4375 Griffin 14 Will 18 Maddie 8 Jack 8.5

Now, we would need to address why we are getting different size boxes if that has not already been determined. The goal of this discussion would be to see that the length of the side of the square we cut off becomes the height of the box and determines the length and the width. We could create a table with the dimensions of each students' box. A sample is below.

 Student Name height (in inches) length (cut from 8 in side) width (cut from 5 in side) Emily 2 4 1 Vance 1.5 5 2 Keesha .25 7.5 4.5 Griffin .5 7 4 Will 1 6 3 Maddie 2 4 1 Jack .25 7.5 4.5

From this data, we could determine that l = 8 - 2h and w = 5 - 2h.

Now, I would ask if they think we have created all possible boxes. Hopefully, through this discussion the students would determine that we had not and since we had not created all possible boxes, we could not know for sure that Will's box has the largest possible volume. And now, finally, we have motivation to use technology to help us. Technology can allow us to simulate making many more boxes.

The type of discussion used to introduce excel would depend on the amount of experience the students had with excel. Our first discussion with regards to excel would be how to choose the values of h--the length of the side of our box. If possible I would like for each group to work in excel and come up with a list of possible values for h, l, w, and V. Then come back together as a large group and discuss the problems we encountered and how we determined the possible values for each variable. This discussion would hopefully lead to an understanding of the constraints of the problem.

Hopefully at least one group would include 0 as a possible value for h and we could discuss why 0 gives us a volume of 0. Or, if no group includes 0, maybe at least one group will have consciously excluded it because it does not result in a box being made.

It is more likely that groups will, at least initially, choose values of h that equal and exceed 2.5 and we can discuss why h must be less than 2.5. We could even allow them to go back to their paper and scissors and see what happens when we cut off a square with side lenght 2.5 inches.

Below are some possible tables the students might come up with and some possible points of discussion:

Choose only integer values for h: discuss why this might not be sufficient--discuss meaning of negative values

Choose h = 0 as start and increase by .5: Is this enough values to try? Where should our table stop?

Choose h = 1 as start and increase by .1: Is it okay to start with 1? Why is 0 not on the list? What other values could we choose to start with? Do you have enough values?

My ideal chart for this question:

What we have done so far could be done with many levels of students. Depending on the level of the students, I would next go into defining the volume of the box as a function of the height V = h(8 - 2h)(5-2h) and graph it:

We can now discuss that h is our dependent variable and V is our independent variable and that we often use x and f(x) or y to play these roles. We can also discuss how the graph relates to the data we collected. In small groups, I would ask the students to come up with an interpretation of the graph to share with the larger group. In particular, I would try to focus them in on which part of the graph represents our data and why the rest of the graph is not of interest to us in this case. This could lead to a discussion of domain and why we sometimes restrict the domain of a graph and how restricting the domain may result in restricting the range.

Finally, I would ask them if they are convinced that the box with the maximum volume is created when h = 1? I would expect them to be convinced at this point. Then I would ask them if we have proven that this is the case. This question would hopefully lead to some discussion and would create a desire to know how to prove what seems so obvious at this point.

Depending on the level of the students, I might at this point introduce the concept of the tangent line and its slope--not at the level of determining the derivative, but more as a way to motivate interest in calculus. Sometimes students seem to see Calculus as this separate subject that lies ahead of them. I think it is nice to introduce Calculus concepts early on so it does not seem as intimidating of a subject.

Alternately or in addition, we could go back to quadratic equations and identifying the max or min of those functions algebraically.