# David Wiggins

## Mathematics Education University of Minnesota

Introduction: This lesson can be adapted to suit the level of the student. Algebra students can work on skills such as collecting data, plotting points on a graph, and using formulas for volume. Higher algebra or calculus students can develop functional models and use more sophisticated techniques to interpret the data from the models.

1. Instructional Objectives: Vary by level of student

• collect and organize data into a table
• transfer data to a graph by plotting points
• analyze and interpret tabular or graphical data

• design algebraic/functional models
• use technology to graph functional models
• analyze/interpret model with graphing technology
• use calculus to solve optimization problem

2. Opening Activities/Purpose:

• Review: Computing geometric measures (lengths, area, volume)
• Purpose: Many of the algebraic models will require students to use information about perimeter, or area, or volume. This exercise will determine their readiness for making these calculations. It will also refresh the memories of many students about formulas used to make the calculations.

3. Developmental Activity: (Steps 6-8 are for the more advanced students)

1. Introduce the problem: (See transparency #1)

2. Demonstrate: Using an actual 20 by 25 cm piece of paper, the teacher demonstrates:

• the construction of the box
• the calculation of the volume

3. Group Work: Student should work in groups of two or three, depending on class size
• Hand out prepared 20cm by 25cm sheets of paper, each with a different size corner to cut out
• Students construct the box by cutting, folding, and taping up the sides
• Students compute the volume of the generated box (Volume = L*W )
• Students record data: "size of cut" and the resulting "volume"

4. Collecting the data:
• Gather the boxes in front of the class: which looks the biggest
• Students report their findings: data is collected in tabular form
• Tabular data transferred to a graph: plotting points

5. Analysis of the data: Which box was largest? Is this the best box? What next?

6. Developing an algebraic model: Express volume as a function of the size of the cut.

7. Using a graphing aid: Enter and graph this function on a graphing calculator

Graph: Volume vs Corner Width
x-scale = 0.5 cm
y-scale = 50 cubic cm

8. Interpret the graph: How do you find the largest box from the graph? Using calculus?

4. Closing Activity: Once discussion as to whether or not the class found the "best" solution to this problem, students can work together in groups to solve the following problem:
Problem: "Find the dimensions of the largest (in area) rectangle with perimeter 600 cm."

### Reflections

I have given this lesson to students ranging in ability from high school pre-algebra to college calculus. I can honestly say that I have never been disappointed that I did! Students of all levels seem to like the constructions, and then like to see the display of varying sizes of boxes at the front of the class. I often let them "vote" to see which they think is the biggest by placing scraps of paper in their chosen box. I am always surprised that the biggest box is never the unanimous choice! I guess it is a fine balance between being a deep box, and being a wide box!

The strength of this lesson lies in the multiple representations of the data. Namely we have the data from our physical model (corner size and volume ) being collected and recorded as:

1. numerical/tabular data (ordered pairs)
2. graphical data (plotted points)
3. symbolic data (functional expression)

Students seem to somehow gain an understanding of the many approaches to solving this simple problem. Those with higher levels of abilities can finally see in a very concrete way what it is that they have been studying about - yes, derivatives can be useful!

I always have a great time with this lesson, and really would hate to have to let a substitute teacher use it on a rainy day for a time killer. It deserves much more attention than that!