Real World Math: Exploratory Lessons - Crop Circles 2

This lesson takes an inquiry-based approach to central-pivot irrigation systems with the use of spreadsheets. 

It isn’t necessary for students to have done the original Crop Circles’ lesson.  More than anything, the 2 in the title indicates an advancement in the skills necessary to solve these problems.  It also shows how these crop circles lend themselves to a diverse range of problems.

I’ve tried to take a new direction in this activity.  Many of the questions are open-ended or absent altogether; that’s because I’d like the students to come up with the avenue of exploration themselves.  Honestly, I am trying to lead to particular problems, but I encourage you to let them determine what direction to take.  For instance, the first placemark is meant for the students to find the area of each ring in a circle, but they may see a circumference/rate problem instead.

What absolutely is intended in these problems is that the students learn how spreadsheets can make groups of calculations easier.  When done by hand, the labor and repetitive nature of the first problem should elicit sufficient anguish.  You may even strike gold and hear someone bemoan, “Isn’t there an easier way to do this?”  This is when you earn their Teacher of the Year nomination and introduce them to spreadsheets.  Even if they have some experience with this tool, they may not have used formulas in them.  This is what makes this activity perfect for this; they’ve already been using the needed formulas on paper and so they just need to understand how to express them on a spreadsheet.

The majority of Real World Math lessons can be done by students independently at home; however, Crop Circles 2 would best be accomplished in the classroom under the teacher’s supervision.  The teacher should act as a facilitator encouraging questions, prodding students in certain directions, and assisting with their spreadsheet skills.  You can break it up and assign specific tasks for homework, or make this a larger part of a project-based learning experience.

As usual, be sure to download and preview the materials provided below before introducing this to students.  The Kmz file includes five crop circle fields to investigate and a 3D model of the system I made with SketchUp.  There’s also a Word doc of teacher notes detailing how I would approach the activity and its problems.  Additionally, a teacher’s guide to Crop Circles 2 is provided as a spreadsheet in Excel and Google Doc formats.  Finally, you are free to adapt or tweak the material as you see fit.  If you’d rather provide students with a spreadsheet template to follow, that should be easy for you to figure out.

Note:  I’ve composed these problems to the best of my ability from my home in Guam.     I scoured the Internet for information, and consulted central-pivot irrigation manufacturers for help.  Big thanks to Reinke Manufacturing, Roberts Irrigation, and T-L Irrigation for their help.  Please email me at to offer suggestions, corrections, or submit a problem of your own.


  • Pursue mathematical questions provided by students
  • Use a spreadsheet to perform functions
  • Solve subject-related problems with spreadsheets




Teacher Notes
Teachers’ Guide

Data, Graphing & Statistics, Geometry, Measurement & Time
Middle School, High School, Educator

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