Category Archives: Geometry

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.

Happy Summer Pi Day!

Math teachers love March 14, the day where we have a built-in excuse to strong-arm students into bringing cookies, cakes, and pies to class, all under the clever guise of celebrating our irrational friend.  But while we celebrate and embrace 3.14, its fractional buddy 22/7 often trudges on without fanfare.  So, on this July 22, consider this challenge:

  • Which approximation of pi,  3.14 or 22/7, is better?

What spirited debates which can take place by assigning students a side to defend?  A quick visual inspection of the protagonists, labeled on a number line, provides some initial evidence:

Number Line 1

Additionally, this is a great time to discuss and compute error.  Just how far away are we from what we would like to estimate?  And how good of a job have we done?

Number Line 2

Evelyn Lamb provides some pi anecdotes in this month’s Scientific American.  How many digits of Pi does NASA utilize in calculations?  Why do people seek to memorize the digits of pi?

So raise a glass to the “real” pi day!  Off to seek some fresh blueberry pie……

NCTM – Thursday

Very exciting day….quite a buzz as I arrived today.  Looking forward to some interesting sessions in the next few days.  The hard part is deciding which sessions to choose.

Teaching Proof: Lessons From an Action Research Study.  Pete Johnson, Eastern Connecticut State University

Yesterday, I attended a research session on proof,where’s a 5-person panel each explained research they had done on encouraging justification and proof in both middle and high school math courses.  From that session, there were two takeaways for me:

  • When we provide a proof, for whom is the justification intended?  Who is the audience for the communication? Let’s focus on the end user.
  • An interesting activity is to provide a number of givens, then having students work to  support the “strongest claim”.  We often tailor arguments to fit our pre-determined conclusions.  But what are the possibilities, given provided information?

My first session today continues my focus on proof.

“writing proofs” is not a topic or a bit of content, it is a a process, a way of thinking that evolves over time

Much of the discussion in this session focused on the following challenge:

Prove that if n is an odd positive integer, then n squared is an odd positive integer.

A few approaches emerged.  Let n = 2k + 1, then simplify n-squared.  Are we guaranteed that the result is odd?

Another attendee suggested letting n = m + 1, but then how do we know that m-squared is even?  What is the assumed toolkit of knowns and agreed upon principle in this problem?  What does it mean for a number to be odd?

Also, the group tended to focus on the oddness, but have we proven that the result is positive, or  an integer?

Findings: Teaching proof as a “separate topic” does not work.  Also, instruction in formal logic does not seem to transfer well to mathematical proof.

Engaging Activites for Your Classroom: technology in Middle School Mathematics

The main event of this session featured activities utilizing the TI Nspire CX Navigator system.  A few years ago, I acquired the Navigator system for the TI84 calcs, and had used it in some of my classes.  But, over time, I found the system cumbersome, and that the classroom payoff was not often worth the set up required.  I was eager to try this updated system, as it is now wireless, and integrates with the new color CX calculators.

My first impression is that sending files has become more intuitive, and the entire interface is cleaner and less clunky than the 84 software.  The examples demonstrated today were pulled from the TI Activity Exchange, and could easily be edited for use with TI Publish View, which mentioned in an earlier blog post.

Philadelphia-20120426-00191, Uploaded by Photobucket Mobile for BlackBerry

Looking forward to more great math discussions tomorrow!