Does 0.3333... = 1/3?
With little grief, fuss, or disbelief most of my students at all levels (from elementary education majors to math majors) will believe that 0.3333... = 1/3.
But, when we change the question and ask:
Does 0.9999... = 1?
The class becomes a room that is suddenly less comfortable. Students look at me and each other with expressions like, "Why is my professor even asking me this?" They chat with each other with strong confidence, "I know that 0.9999... definitely does not equal 1!"
In fact, last semester, in a College Algebra class that I was teaching, we took at vote. The post-it's from the votes are below.
Interestingly, the only student who thought that 0.9999... = 1 was a seventh-grader visiting our class. The post-it in the middle was a "maybe." That is, "maybe 0.9999... equals 1, but only if you round." The rest of the post-it's had written statements with strong acclaimations that 0.9999... definitely did not equal one.
The voting results were similar this semester with a different group of students - adults taking a math for elementary education class.
I began wondering: Why is this so challenging of a topic?