Day 158: The Eccentric Ellipse

Today was a red-letter, AWESOME day as a teacher…in fact, I was pinching myself continually because I thought I was in a dream.  As I mentioned last week in post Day 155: Coneheads, my colleague used taped string on the whiteboards to get his students to develop the geometric definition of an ellipse…and I wanted to do that too!!  But I had already committed to the patty paper activity.  So what to do?  I just wanted to have fun too!

Since we had not done any of the formal notes I had planned to do yesterday, I saw a wonderful opportunity!  I put up 6 stations of taped string around the room; I had the foci at different distances along an imaginary horizontal major axis, one with a vertical major axis and one with a slanted major axis.  All of the strings measured 100cm (and were they surprised when they all measured their major axis!).  I also put out 5 different color markers, a yardstick and an extra piece of tape. At this point, I wasn’t sure what I would actually do.

 

I asked my students to divvy themselves up between the 6 stations.  Of course, they immediately picked up a marker and drew in the ellipses.

Then, with a time restriction of 5 minutes, I asked them to identify these 5 things on their ellipses and color-code them.  They were not to use their iPad to go to the internet at this time.  They had to discuss what they thought these words meant and label their ellipse.  A little whining, but very little.  They talked and looked around at what the other groups were doing and saying, and revised their thinking and/or stood firm to their original thoughts.  Love seeing the tilted heads and moving arms!

After this initial time, I had them go back to their seats so we could generate some generalizations.  Some things they said were:

1. Distance from one focus to a point on the ellipse added to the distance from that same point to the other focus is always the same.
2. The closer together the foci are, the more the ellipse looks more like a circle.
3. String length = major axis length
4. eccentricity is the ratio between the major axis and the minor axis.

From yesterday’s exploration, they knew the first conjecture was true, although it was nice to have them check this around their personal ellipse.  We looked at the various ellipses around the room and determined that the second conjecture seemed to make sense (and a nice lead in to the idea of eccentricity).  They weren’t so sure about the third conjecture.  So I sent them back to a different ellipse (not their own) with the tasks to verify the three conjectures AND to see if they could find a relationship between the semi-major axis, the semi-minor axis, and the focal radius.

 

More great discussions and most did the typical “string forming an isosceles triangle” approach to find the relationship.  Again, the deep discussion with probing questions and sense-making almost brought tears to my eyes!

We came back together as a class and debriefed once again (we had a block class today which means a 90 minute period).  Last of all, I had originally put this cartoon up for the exit slip today, but actually used it to explore eccentricity.

I simply asked if they could determine the eccentricity of their conic using the cartoon as their guide.  I reminded them that they could only use measurements we had already discussed: major-axis, semi-major axis, minor axis, semi-minor axis, and the focal radius. Absolutely wonderful discussions.  Most groups came up with the ratio of the focal radius to either the semi-major axis or the semi-minor axis.  They argued/reasoned through why they thought the denominator was the semi-major axis.  YA-HOO!

   

Back in March, I attended and blogged about a one-day workshop with some of my school colleagues  given by Dan Meyer on Intellectual Need in the Classroom.  Looking back, I realize now that I used Dan Meyer‘s three essential components of an engaging and relevant lesson: Start a fight, turn up the math dial slowly, and create a headache – to provide the “aspirin.” And maybe that’s why I got the comment from one student, “That was the best lesson I’ve ever had!”

What is one spectacular lesson of your career?  What made it so?  Have you been able to replicate it in subsequent years?  I’d love to hear about your Red-Letter day in teaching.