What is Three?

A middle-school teacher’s family emergency pulled me into the classroom last week to teach an honors Geometry class to 8th graders.  Geometry…sigh….the course I always put on my “please do not ask me teach this” list during my time as a high school teacher.  And since it is the start of the year, the class is learning basic terms and definitions, all the great stuff I dreaded as a teacher.  Oh, and I have 10 minutes to plan before the kids walk in.  Ready?  And scene…..

This is the 3rd day of school.  Students have been exposed to the class rules, some algebra review, and textbooks look clean with their grocery-store paper-bag-covered  exteriors.  This is the first real geometry lesson for these kids.  I am their first impression of geometry, and the precision and argument they will experience.  No pressure.  Today’s lesson: basic vocabulary and terms.  Let’s look at the terms we need to understand by the end of today.

  • Undefined Terms: point, line, plane
  • Ray
  • Segment
  • Collinear / Coplanar points

How nice.  I drew the short straw.  Essentials of geometry vocabulary, and I get to be the boring guy.  Not the role I was born to play.

And just what are “undefined terms” anyway?  According to the textbook, these are terms which we understand, but don’t need to define.  Seems a bit hinky to me…

So how to build some discussion, based off previous knowledge, and ease our way into a structure for geometry?  As students entered, I had the following warm-up ready as they prepared to take notes:


  • Three
  • Line
  • Odd number

After some initial snickering about my strange challenge, the students took to their definitions.  So, how do 8th grade geometry students on the first day of class define “three”.  A similar response was given by a number of students:

It’s the number between 2 and 4

Thankfully, a few students identified the flaws in this definition: that, first, there are an infinite number of “numbers” between 2 and 4, and that in order to understand this definition, you need to understand what 2 and 4 mean, which seems unreasonable if you don’t know what three means.

So, should we consider “three” to be an undefined term?  Are we OK with NOT having a formal definition of “three”?  What do we need to consider?

Do we all understand what three means?

Yes, when asked to represent 3, everyone in the class demonstrated the same understanding of its quantity.

Would we expect any alternate understandings of the term, if we asked others?


Would having a definition increase our precise understanding of three?

Nah, I think we all get it.  Three is three, and that’s that.


This discussion turned out to be a nice opener to the traditional undefined terms in geometry: point, line and plane.  And hopefully a good start as these students begin to experience the logical structure of geometry.

Tomorrow, ask your students to define “Three”.  Would like to hear what they say.


By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

One reply on “What is Three?”

“easy words with hard definitions” is one of the most important parts of mathematics in my opinion – I’ve used a similar opener for this lesson in the past but I have them try to come up with definitions for abstract human concepts like “love”

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