On a Desert Island with Unit Sticks, Continued Fractions and Lagrange

Victor Ricchezza
University of South Florida, School of Geosciences
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Initial Publication Date: June 6, 2019

Summary

This lab style activity has students measure the length of an object (generally a tabletop) using an unmarked stick (a disposable chopstick being the most common choice). Students use the single stick as a fundamental unit of measure, then use remainders of the stick (and remainders of remainders) when measuring the total length to solve for the length in "sticks" using continued fractions, after Lagrange (1795). Students estimate error in their own measurements, but must grapple with what that really means, as they have no easy decimalized marks to choose as halfway points for error judgments. Excellent source of class discussion about fundamental concepts of units, error, and measurement. Applicable beyond the geosciences.

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Description and Teaching Materials

On a Desert Island with Unit Sticks, Continued Fractions and Langrange (Acrobat (PDF) 391kB Apr30 19)
Stick Lab Student Handout (Microsoft Word 2007 (.docx) 235kB Apr30 19)
Stick Lab Pre-lab Quiz (Microsoft Word 2007 (.docx) 50kB Apr30 19)

Teaching Notes and Tips

Recommended that you read the paper, give the pre-test, then have students perform the lab. Students working in groups of 2-3 is optimal. Fewer is problematic and more results in idle behavior. A bag of uniform disposable chopsticks can be obtained from many Asian food markets for $2-3 for a bag of about 100 sets; these are perfect for this purpose. I recommend having students report results a few minutes before the end of class in a spreadsheet.

Assessment

Students measure an object with a chopstick and write entries in a spreadsheet. It will be fairly obvious if one or more groups doesn't have values matching their classmates. Estimating error values is tougher but is a great teachable moment.

References and Resources

https://serc.carleton.edu/earth_rendezvous/2016/program/demos/thursdayA/136743.html'

This material was included as part of a teaching demo at the 2016 Earth Educators Rendezvous in Madison, Wisconsin.