**ERIC Number:**EJ1051645

**Record Type:**Journal

**Publication Date:**2015-Feb

**Pages:**6

**Abstractor:**ERIC

**ISBN:**N/A

**ISSN:**ISSN-0025-5769

**EISSN:**N/A

Math in a Can

Cupillari, Antonella

Mathematics Teacher, v108 n6 p434-439 Feb 2015

Practical problems that use mathematical concepts are among the highlights of any mathematics class, for better and for worse. Teachers are thrilled to show applications of new theoretical ideas, whereas most students dread "word problems." This article presents a sequence of three activities designed to get students to think about something they see every day--the humble tuna can. Why did manufacturers pick the usual squat cylindrical shape for it? As calculus students can quickly show, this squat shape is not the one that minimizes the surface area (i.e., the amount of material needed to build the can) for the given volume (Larson and Edwards 2011), and they can easily calculate the optimal answer for the best cylindrical can. But is the cylindrical shape a requirement? If so, why? How could students who have not yet studied calculus investigate the problem using only arithmetic and geometry? Questioning the design of the tuna can generated three activities, designed to bring together several ideas and skills--for example, geometry, writing, and calculus--and reinforce them by their use in an everyday context. The sequence of activities presented here emphasizes the logical ties among several mathematical topics, increasing the length of time that they are discussed in class. Thus, students have a better chance to gain ownership of the material as they have time to mull over it, making the new ideas harder to forget. The activities are organized in this sequence: (1) Students complete a geometry exercise on solids (volumes and surface areas) to get ready for further investigation and discussion of the results obtained; (2) Students work (with calculus if appropriate) to minimize the amount of material used to make a can and discuss the results obtained, brainstorm to put the cans on the shelves, and present the results in writing; and (3) Students consider questions such as, Why do manufacturers stick with the current design? Did we work on the wrong optimization question? What else could we think about? Presenting the context does not take up much class time, depending on the amount of discussion and details the teacher wants to introduce, and can be modified to fit the students' mathematical level and interests.

Descriptors: Mathematical Concepts, Word Problems (Mathematics), Mathematics Activities, Geometric Concepts, Geometry, Numbers, Teaching Methods

National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail: orders@nctm.org; Web site: http://www.nctm.org/publications/

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**Teachers

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

**Grant or Contract Numbers:**N/A