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.

Building a Better Snowman

In a recent tour of my midde school, I observed a 6th grade teacher working with a class to use compasses.  Their goal: to make a snowman with segments of different, but given radii, 3cm, 2cm and 1cm.  Eventually, this will lead to students having their first experience with circumference, and that sneaky number pi.  But why tie students to fixed radii values?  And just what are the “perfect” measurements for a snowman?  Here are some lesson ideas for letting students explore their own snowmen, using technology, then taking it a step further by considering how much snow our snowmen will need.

Snowmen may come in all different sizes, with different accessories.  But it is pretty well established that the traditional snowman is made of 3 body segments.   Snowmen with anything other than 3 segments are blasphemous.



Clicking the image to the right will take you to an interactive graph I made using the Desmos calculator.  You can manipulate the diameter of each snowman segment by pulling the sliders provided, but the height of the snowman is fixed at 6 feet.  Like your snowman with equal segments?  Knock yourself out.  Are you more of a bottom-heavy snowman connoisseur?  There’s room on the bus for you , too.

Once you are convinved that you have built the world’s best snowman, please share your slider settings here.  I’d like to feature them in a later blog post.


For a classroom discussion, have students print out their snowman images (Desmos has a snazzy print feature), and compare their snowmen in groups.  Whose snowman would need the most snow to build?  Whose would need the least?  Or, if all of the snowmen are 6 feet tall, then will they use the same amount of snow?

This is a great time to talk about volume, and introduce the formula for the volume of a sphere.  And, since each student has their own product, have them find the volume of each snowman segment, then add them to get their required snow total.


If it snows 5″ overnight, will we have enough snow on the ground to build our snowman?  For this next stage, we can have students compare the volume of snow needed for their snowman with the volume of snow on the ground.  For example, if your yard is 10 feet long and 10 feet wide, will 5 inches of ground snow be enough to build our snowman?

BUT, we pack snow while we build snowmen.  How much less is the snow volume in a snowman vs its volume when it is on the ground?  2 times less volume?  5 times less?  10 times less?  I really have no idea.  Next time I have a few inches of snow on the ground, it will be experiment time.  With a class, perhaps debate the correct number and use it for calculations.


To help with calculations, and checking student work, I have created this handy Snowman Calculator on Excel.  You can input your a and b values from the Desmos document, along with the dimensions of a yard or rectangular area.  The volume of your snowman, along with needed snowfalls, are then given.

Think warm!  And then we will start working on sandcastles.

Doing the Translation Dance

Last month, I wrote about my talk on Encouraging Perseverance in Math Class, given at the Fall, 2012 ATMOPAV conference.  But earlier that same day, I had the opportunity to hear Scott Steketee‘s thoughts on functions: “Function Dances: Using Transformations to Make Variables Vary and Functions Behave”.


I have found that the approach many teachers take to functions is one of notation only.  That by simply introducing the f(x) and g(x) symbols, and “covering” domain and range, algebraic functions will be understood.  Scott’s presentation provided ideas for introducing the concept of  function, without all of the scary symbols, through dynamic Sketchpad files.  The group worked through a number of progressively intricate functional relationships on iPads.  In this first example, students can grab points and look for relationships.  Some points will not move when dragged, as they are “dependent” upon other points’ movements.    Also, the dependencies vary, from simple linear relationships, to a few which require dilations or reflections.


Later, we were introduced to the Sketchpad “Translations Dances”.  As one point (below, the point on the green outline) travels about its “domain”, we are challenged to trace the “range” of the translated point p.  These start off innocently enough, but become more diabolical as the translations begin to include reflections and rotations.


These were addictive and appropriate uses for the iPad, and I was able to easily load the files into iPad’s Sketch Explorer through my DropBox account.

The second half of Scott’s talk was more kinesthetic, social, and potentially embarrassing, as the group split into partnerships to choreograph dances based on transformations.  My partner acted as the independent variable, and I (the dependent variable) followed her actions, using lines in the floor to act as  axis of reflection.  This would be a fun way to expose kids to functional ideas, but I made sure that no photographic evidence of my dancing ability exists!

What I appreciated most about Scott’s sketches and dances is that they allow teachers to develop functional ideas without having to wade through all of the complex language.  Through play and exploration, students can summarize their observations, and begin to characterize the relationships.  As students begin to understand the relationships between variables, we then can discuss the need to have special notation to express them.  Finally, dilations and reflections, which are often over-looked in our curriculum, become the stars of the show through fun (and addicting) Sketchpad games.  My screen grabs here certainly don’t do Scott’s files justice, so download them, play around, and enjoy the dances!

What is on Your High School Math “Mount Rushmore”?

Today was math club day.

Math club is a strange creature, where the content is always dependent on the interests and audience of kids who show up.  Some days, we will work through interesting past contest problems.  Other days, we will discuss “not taught in class topics” like Mersenne Primes or Cryptography.  One day last year, we watched and enjoyed the “Pi vs e Debate“, which is fun viewing, but will produce odd looks from those who aren’t in on the joke.

Today, I challenged students to think of the “3 most important math equations” they had encountered in their high school career.  This opener caused some discussion of what should be considered “important” as opposed to memorable.  A few students shared their responses on the board:

Top 3

This led to my presenting a  video I came across recently, “10 Mathematical Equations that Changed the World”.  The ranking is by no means scientific, and a few of the equations are well-above high school math.

This also produced discussion of the blurred line between physics and math, especially as students begin to take more challenging courses.  Ask your students tomorrow “What are the 3 most important math ideas you have learned in your life”?  Can our students summarize and prioritize their math experiences and reflect upon their learning?

As a teacher, which concepts belong on your high school math “Mount Rushmore”?  Shrink down the high school math experience to the 4 most central ideas.  Here’s mine:

  • Quadratic Formula.  Is this one a gimme?  One of the first times our students consider a general case and the gatekeeper to algebra 2.
  • A proof of the Pythagorean Theorem.  I’m going back and forth on this one, thinking perhaps law of  sines / cosines would be better.  Pythagorean Theorem alone seems too middle-school, but being able to develop and defend a proof then moves it to a summary of geometry.
  • The Central Limit Theorem.  The backbone of confidence intervals and hypothesis tests.  If you don’t teach stats, this might be out of your wheelhouse.  If you do, I think you are with me.
  • The fundamental theorem of calculus.  Ties together all of our algebraic and abstract-thinking skills in a nice tight package before we send the kids off to college.

Would love to hear your “Mount Rushmore”!

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.