MMM Project - Designing Packages, teacher pages 3

Saving Trees

At a Glance

Launch

Review the shape of the box from Problem 2.1 that had the least surface area.

Introduce the problem of finding what arrangement of cubes will require the least amount of packaging material.

Have groups of three or four work on the problem and follow-up questions 1 and 2.

Explore

Encourage groups to try their conjectures on other arrangements and other numbers of cubes.
Help groups having trouble getting started to look at the case of 8 cubes.

Summarize

Talk about the arrangements students found.
Help the class think more deeply about the minimal surface area of a rectangular box.
Use follow-up questions 3 and 4 as a quick assessment.

Saving Trees

Were you surprised to discover that 24 blocks can be packaged in ways that use quite different amounts of packaging material? By reducing the amount of material it uses, a company can save money, reduce waste, and conserve natural resources.

Problem 2.2 When packaging a given number of cubes, which rectangular arrangement uses the least amount of packaging material? To help you answer this question, you can investigate some special cases and look for a pattern in the results. Explore the possible arrangements of the following number of cubes. For each number of cubes, try to find the arrangement that would require the least amount of packaging material.
8 cubes	27 cubes	12 cubes
Use your findings to make a conjecture about the rectangular arrangement of cubes that requires the least packaging material.

Assignment Choices

ACE questions 8, 9, 11-13, and unassigned choices from earlier problems.

Answers to Problem 2.2
The rectangular arrangement of cubes that requires the least packaging material is the arrangement that is most like a cube. For 8 cubes, it is 2 by 2; for 27 cubes, it is 3 by 3 by 3; and for 12 cubes, it is 2 by 2 by 3.