Category Archives: Geometry

Class Opener – Day 19 – A Blast From Geometry Past

During yesterday’s group work, which included a discussion of Pascal’s Triangle, I overheard some groups mention Sierpinski’s Triangle, which they had seen some in Geometry last year. That led to today’s opener, an applet from the awesomely mathy site Cut The Knot:


In the “Chaos Game” a point be-bops about a triangle under specific rules:

  1. The red point starts one of 3 randomly selected vertices of the triangle.
  2. Next, one of the 3 vertices is randomly selected, and the red point moves half-towards this new point.
  3. The process is repeated over and over, and all landing points are marked.

At first, I have the applet run slowly, and students don’t quite absorb what is happening. But as we speed up the animation, something interesting develops….


Our old friend, Sierpinski’s Triangle! Later in the period we saw this famous structure again when discussing Pascal’s Triangle and factors. Check out this cool coloring remainders applet and have fun!


A Math Teacher Ventures Into a Western Civ Class

During my prep period last week, I came back to my classroom after a trip to the main office and overheard some familiar language: Euclid, Pythagoras, The Elements.  What’s intriguing here is that my next-door neighbor isn’t a geometry teacher; rather, my colleague Glen is a social studies teacher, with 3 sections of Western Civ each day.  Excited, I popped my head into his classroom.  And after some good-batured ribbing out how he was advancing on my math turf, I went back to my prep.  But Glen and I later talked about our shared interest in the Greeks, which ended with an invitation to come into his class to share a brief math history lesson.  I’m no stranger to the occassional cross-curriculur lesson, so this represented a fun opportunity.

One on my favorite courses from my time at Muhlenberg College was “Landmarks in Greek Mathematics”, where I was fortunate to have William Dunham as a professor.  His enthusiasm for math storytelling has shaped my approach as a teacher, and his book, Journey Through Genius, was not only used in the course, but is a book I often come back to for inspiration and contextual reminders of math concepts.  The book both walks you through the mathematical landmarks (like Euclid’s proof of the infinitude of primes) and provides a backdrop of the places and people (like the fascinating battling Bernoulli brothers) which shaped the surrounding culture.  It’s a great resource for any math teacher.

For Glen’s classes, I chose an example which 11th graders could easily understand and which would provide a glimpse into the genius of the greek mathematicians: Eratosthenes’ approximation of the Earth’s circumference.

Eratosthenes observed that on the longest day of the year, sunlight would shine directly into a well, so that the bottom of the well could be seen.  But that farther from the well, in other towns, this did not occur.  The well was located on the town of Syene, which we now lies directly on the tropic of cancer.

Syene Well

In Alexandria, a known distance away from Syene, Erotosthenes measured the angle produced by the sun’s rays off a post in the ground.


Taking this further, we can use alternate-interior angles to use this same measured angle as one coming from the center of the earth.


This central angle, along with the known distance from Syene to Alexandria, yielded an estimate of about 25,000 miles (or the Greek stadia equlivalent), an estiamate with an error of less than 1% of the actual circumference!  Both classes I visited seemed to enjoy this math diversion in the Western Civ class, with one student wanting to know more about how the Greeks approximated pi.

So find your local Social Studies teacher, and offer to bring in a little math!  There are some fascinating stories to tell.


Excerpt from String, Straightedge and Shadow

From the Mathematical Association of America

From Jochen Albrecht, CUNY

Finally, from Carl Sagan’s landmark series “Cosmos”

Problem Makeover – Carpeting

This week’s problem makeover from Dan Meyer features a room in need of carpeting:


There’s a lot of stuff going on in this problem:

  • Overlaying rectangular carpet over an irregular shape.  We’re going to need to worry about waste at some point, perhaps.
  • The double-sided tape.  We’ll have to look at the perimeters of the individual carpet pieces and add them up later.
  • Costs.  We definitely don’t want to waste carpet or tape, or at least minimize the amount of waste.
  • Meters vs Yards vs Feet.  Yeah, this aspect of the problem is just mean  Maybe if Lesa had her fill of coffee before measuring the room, she would have used feet.

I think the piece of this problem which may be easy for kids to overlook, and least comprehend, are those seams.  Seams are bad, and add to cost.  This problem needs something tactile to get kids thinking about the seams and coming up with their own ways to  think about them and minimize them.


Groups of students will be given a bunch of 4×4 inch sqaures and a bigger shape to tile. Tiles1

The instructions are simple:

Use the squares to cover the triangle in the most efficient manner possible.

As groups complete the task, they will then meet with other groups and defend why they feel their coverage method is the most efficient.

Tiles2I’ve left the phrase “most efficient” open for interpretation here.  I’m hoping that a strong definition of efficiency will emerge from group discussions.  As groups continue discussions and start to reach agreement on the characteristics of efficiency, have members of teams approach the front board, and start a class-wide list of these characteristics.  Here’s what I am hoping will emerge:

  • Using the least number of squares as possible is important.
  • Having squares overlap, if it can be avoided, is preferable.
  • Seams are ugly, but necessary.  When we are able, we want to minimize these seams.

This last bullet may be the toughest to coax from kids.  But this is a good time to bring in carpeting as a nice analogy.  Does the carpet at your house have seams?  Where are they?  If you were laying carpet, where would you try to put them?

Once we have a stronger definition of efficiency, it’s time to try a second tiling:


This time have students write up their solution and defend their efficiency.  There’s a nice opportunity for differentiation here.  Students with a strong geometry background could use rules regarding secants and chords to generalize the optimal solution, or even use software to model the scenario.  Middle school students, or those with less exposure to geometry, could simply measure the seams.

Now for the main event.  Students will be asked to tile one last shape:


This time, we add a new wrinkle to the problem.  To get students thinking harder about efficiency, we add some costs to the mix.

  • Each square used costs $10.
  • Every inch of seam in the tiling will cost $2 per inch.

It’s Saturday morning during the summer, so I’m not sure if these are “good” costs for the task, but we can adjust them.  Teams will calculate their costs, defend their solutions, and be assessed on how well they minimized the costs.

Geometry isn’t usually my “thing”, but this challenge appealed to me, because the original task was so filled with stuff that I know many students would shut down.  And I feel there is a natural opporunity here for students to assess their solutions and communicate.  In the end, I want my students to:

  1. Communicate their problem-solving process
  2. Defend their process, and evaluate the process of other teams
  3. Utilize their findings in different situations.