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Day 167: Hyperbola Take Over

One of the things I really like to have happen in my classroom is having students take over the discussion.  It’s occurrence is more rare than I would like it to be, so when it happens, I celebrate with little “yippees” in my mind. [Note to self: good summer rumination-how to foster more student-led discussions]

Hyperbola Opener

We are studying conics in Precalculus and we were looking at this Opener problem dealing with two hyperbolas.  Even though I suggested that we only work for 5 minutes, my students just couldn’t let it go.

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They worked diligently, and eventually realized that they needed to graph the situation using their Nspires….so they began to put the hyperbolas into function for to enter.  Of course, they asked, “Isn’t there an easier way?!”  I suggested that they explore that more thoroughly on their own.  And one student, excitedly (and somewhat smugly) jumped up and said, “I can show you all!”  And he did.

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How do you get your students to take over your classroom?  I’d love to hear some new ideas to try!

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Day 166: Real-World Conics

This week-end, after reading through my students’ exit slip answers to this question: What questions do you still have about the three conics we have studied this week, I realized from their comments, that we needed to actually see where conic sections are useful in real life, not just in my “geeky math world.”

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So today we did some applications:

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It was so satisfying to hear students exclaim, “Is that why satellite dishes have that funny thing sticking out beyond the dish?! and “I see how the ellipses foci actually show up in real life!”  The most precious comment was “oh, that makes so much sense now!”

Day 165: Hyperbola Dreamin’

Its the day before a four day Memorial week-end!  We were so lucky this year because we didn’t need to use all of the weather-related make-up days in the district calendar.  Thus, we have this glorious 4-day, so needed, break!

So what do I do with my Precalculus classes during the last periods of the week?!  Derive the hyperbola equation from the geometric definition, of course.  And I was so So SO proud of my kiddos because they hung in there during the entire derivation contributing next steps they whole way.

derive hyperbola equation

And we were able to see where the Pythagorean relationship “falls out” from the derivation.  All in all, I am really lucky to have such engaged students!

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.

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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!

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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.

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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.

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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!

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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.

eccentricity comic

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!

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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.

Day 155: Coneheads

 

There are so many cool ways to help students get the ideas behind the geometry of conic sections, that it’s hard to choose.   At one of my TI User Group sessions, we explored the conic sections using patty paper, led by Art Mabbott, to develop the locus definition of various conics.  We then followed it up with a really neat Nspire activity called Introduction to Conics and Patty Paper Ellipses.

This year I decided the tactical approach would be fun, so out came the patty paper. The students followed a guided exploration and they were amazed when the parabolic shape emerged out of their folding.  We compared what happened when the focus moves closer to the directrix and further away.  Using geometry, they determined the actual geometric definition of a parabola on their own.  Very cool!

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As a side note, I was observing my colleague as he had his students explore the ellipse construction using string, tape, and the white boards.  I thought it was quite ingenious using the white boards rather than pushpins.  Watching the students draw the ellipses was fun, but the development of the relationship between the foci and any point on the ellipse through questioning was really well done.  When the students determined the relationship, it was also so interesting to watch the students talk through their formal definition, revising and cleaning up the language.

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Day 153: Any Way You Slice It.

My percalc kids took their test on Matrices today and then we begin our last area of study: conics. Interestingly, when I polled my classes, very few had worked with them before. So, I’m excited. Conics are so much more fun the first time around.

A great resource for classroom-ready Nspire activities is Math Inspired.

MathInspired

Using the Nspire document, Introduction to Conics, they were introduced to some vocabulary and images to help build a basic foundation.  The second part of this Action-Consequence document had students explore the relationship between the focus, directrix and points on a parabola.

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Since we don’t have class tomorrow, and we’re in the midst of AP testing, I also had them do an exploration using a Shodor (a national resource for computational science) called Conic Flyer for homework.  The goal is to have the students look at the basic equation for each type (how they are alike and different) as well as apply their understanding of transformation of functions to conic sections.

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