### Summary

Students review how to determine the surface area and volume of a rectangular prism, that all dimensions are equal in cubes so the volume of cubes are the length of any side raised to the third power, or cubed. This prepares them for two associated activities. First, students find the volumes and surface areas of rectangular boxes such as cereal boxes and then figure out how to convert their boxes into a new, cubical boxes having the same volume as the original. As they construct the new, cube-shaped boxes from the original box material, students discover that the cubical box has less surface area than the original, and thus, a cube is a more efficient way to package items. Students consider why consumer goods are generally not packaged in cube-shaped boxes, even though this would require fewer materials and ultimately, less waste. Then, to display their findings, each student designs and constructs a mobile that contains a duplicate of his or her original box, the new cube-shaped box of the same volume, the scraps that are left over from the original box, and pertinent calculations of the volumes and surface areas involved. The activities involved provide valuable experience in problem solving with spatial-visual relationships.

### Engineering Connection

### Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the *Achievement Standard Network (ASN)*, a project of *JES & Co. *(www.jesandco.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics; within type by subtype, then by grade, *etc*.

Click on the standard groupings to explore this hierarchy as it applies to this document.

### Pre-Req Knowledge

### Learning Objectives

- Determine the dimensions of a cube when given its volume.
- Assert that a cube has less surface area than a rectangular prism of the same volume, and then prove this assertion with examples.

### Introduction/Motivation

### Lesson Background and Concepts for Teachers

### Associated Activities

- New Boxes from Old - Students find the volumes and surface areas of rectangular boxes such as cereal boxes and then, using the same box material, construct new cube-shaped boxes having the same volumes as the original boxes.
- The Boxes Go Mobile - To display the results of the New Boxes from Old activity, students design and construct mobiles that contains duplicates of the original boxes, the new cube-shaped boxes of the same volume, the leftover scraps from the original boxes, and pertinent calculations of the volumes and surface areas involved.

### Lesson Closure

**After students have conducted both associated activities, lead a class discussion.**Begin by asking students to share with the class their answers to the last question on the New Boxes from Old student pages. While the total areas of their scraps should equal the differences in surface areas of their two boxes, it is unlikely that they will actually be very close. Measurement inaccuracies, rounding, and the difficulties of cutting straight lines and right angles all combine to make their answers not as closely matched as they ought to be.

### Attachments

### Assessment

- Identify the equations used to calculate the volumes and surfaces of cubes and rectangular prisms.
- Determine the difference in surface areas of a rectangular box of given dimensions, and a cube-shaped box having the same volume as the rectangular box.
- Sketch a cube-shaped box, including its dimensions, that has the same volume as a provided rectangular box.
- Identify realistic criteria and constraints that should be considered in packaging design.
- Write a paragraph explaining why consumer goods packaged in cube-shaped boxes would use less packaging material than rectangular boxes containing the same product volumes. Require each student to provide an example, including sketches of the boxes and their dimensions, to substantiate his or her explanation and identify why more cube-shaped boxes aren't used.

### Lesson Extension Activities

### Contributors

Mary R. Hebrank, project writer and consultant

### Copyright

© 2013 by Regents of the University of Colorado; original © 2004 Duke University

### Supporting Program

Engineering K-PhD Program, Pratt School of Engineering, Duke University

### Acknowledgements

Last modified: March 4, 2015