Math Mondays: When my error became an opportunity for student problem solving

A funny thing happened that actually flummoxed me for awhile.  I had a fellow teacher working with the trigonometry rope adventure project.  She kept getting different areas than I had.  I used Heron's Formula; she used the law of sines for oblique triangles.  The sides of the triangles were the same regardless of the method.  So, why were we getting different answers?  I tried running the numbers by hand and through online calculators for the formulas.  Still, the answers were different.  I admit it.  I was stuck.  

I walked away (literally) from the problem.  I went outside, took a walk around the block, and thought about the problem.  What if...?  

I went back to our calculations.  We were working with sides that had been rounded to the nearest hundredth.  The first part of the project requires finding the lengths of the sides of the triangles using the law of sines and law of cosines.  So, I went back and re-calculated the sides, this time not rounding to the nearest hundredth.  I also didn't round sine when using the law of sines for oblique triangles when finding area.  The rounding made a difference.  If the numbers were rounded to the nearest hundredth, the areas varied by as much as 0.05.  In fields such as engineering, astronomy, and chemistry, numerical accuracy is important.  This was definitely something to share with my students.

When my students got to this project, I asked half the class to use Heron's Formula for triangles A - D and Law of Sines for Oblique Triangles for triangles E - H.  Half of the class did the opposite.  I paired students up to "check" their answers.  It didn't take long before the students discovered that their answers were different.  They compared each other's methodology and still found different answers.  Students brainstormed why the areas were different.   Eventually, the topic of rounding came up.  Students tested their theories and discovered as I had that when you round it matters.  We further tested the theories by rounding to the tenth, hundredth, and thousandth and comparing the accuracy of our answers.  

Beyond the simple assessment of students' trigonometric skills, this was one of the best activities we did because:

  • Students discovered errors
  • Students problem solved
  • Students developed theories and tested them
  • Students assessed for themselves the value of numerical accuracy
After we finished, I shared this video with my students.  Although the focus of the video is measurement, the development of precision in measurement is discussed.  How do you make the most of "errors" in your classroom?  Share your ideas in the comments below.

Math Mondays is a bi-weekly blog post (2nd and 4th Monday of each month) sharing tips, ideas, resources, and products for teaching math.  If you have questions or think there is something I should include, you can leave me a message in the comments section below or at the store in the question and answer section.