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Archive for the 'geometry' Category

Shoulda Woulda Coulda

Two things I'd do if I were still doing the job instead of just talking about it:

Set Up The Expected Value Spinner

I don't think people who understand expected value understand how hard it is for other people to understand expected value.

Let's say I roll a die. I ask if you want to bet on an even number coming up or a five. You're bright. You pick the even number. It has a 3/6 shot versus a 1/6 shot for the five. But what if I said I'd pay you $150 if the even number comes up and $600 for the five. What if I said I'd keep on giving you that same bet every day for the rest of your life? This is where expected value steps in and puts a number on the value of each bet, not its probability. The expected value of the even number bet is (3/6) * $150 or $75. The expected value of the five bet is (1/6) * $600 or $100. The five bet will score you more money over time.

This is tricky to fathom in gambling where superstition rules the day. ("Tails never fails," betting your anniversary on the pick six, blowing on the dice, etc.) So one month before our formal discussion of expected value, I'd print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I'd seal my own bet in an envelope.

I'd ask a new student to spin it every day for a month. We'd tally up the cash at the end of the month as the introduction to our discussion of expected value.

So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?

BTW: You could make an argument that a computer simulation of the spinner would be better since you could run it millions of times and all on the same day. My guess is that your simulation would be less convincing and less fun for your students than the daily spin, but you could definitely make that argument.

Host A Steepest / Shallowest Stairs Competition

Tonight's homework: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.

Your students should then determine whose stairs were the steepest and the shallowest and you'll post those photos at the front of the classroom. You'll make a big fuss over them. Then you'll post a bounty for stairs that will knock them off their perch.

One interesting thing about slope is that it doesn't have a unit, so you don't need a measuring tape or a ruler to calculate it. Anything your students have on hand will work, including their hands.

Be prepared for a contentious discussion about the difference between the tallest steps and the steepest steps. It's possible to design steps that are extremely shallow but too tall for anyone to climb up. Wrap your students' heads around that one.

Be prepared also for students who can't shake the sense that math is here every time they climb up a new set of stairs.

What a cool job the rest of y'all have.

[photo credits: moyogo, vulcho]

2012 Jan 17: Useful description and modifications from James Cleveland.

Nick Hershman is running laps with this one. Check the blog post or the screencast, in which he explains how he built a Python script around an algorithm from the comments.

Summer school right now involves six hours of Geometry instruction followed by three hours of planning for the next day's Geometry instruction, which basically leaves me fully tapped for tweeting, blogging, smiling, anything but sleeping. I'd say something laced with regret here but the fact is I enrolled some truly incredible students who challenge me and crack me up for the better part of those six hours. These kids make for light work.

Their proficiency does cause its own kind of trouble, though, because my strongest and weakest students space themselves out dramatically over six hours, requiring all kinds of differentiation. My favorite recent method, particularly with today's investigation of reflections, is to say, "okay, now do that with just a compass and straightedge."

I had a method in mind but several students each did me one better.

One student made kind of stunning use of SSS congruency. Another dripped sweat all over the page constructing perpendicular bisectors, copying angles, copying sides in an incredible (but functional) mess. Another used the method I chose but did it in three fewer arcs.

I have five more days to enjoy this.

[BTW: I have determined that at least 20% of this is garbage.]

Five uninterrupted hours of Geometry differentiated between credit recovery students and enrichment students turns out to be exactly as easy as everyone predicted it would be. After misjudging time-on-task about a dozen times and grossly overestimating our ability to construct an orthocenter by Just Playing With It, I did something at the end of class that I didn't hate.

I put up this slide and asked Mika to pick a point out. I asked her to tell Jason across the room which point she was thinking of. She stumbled and stammered a bit. "It's sort of to the left of the one that's near the center," etc.

And then I added labels.

And it became a little clearer why we label points. Mika relaxed. Everything looked easier.

In 2007, I told my students that we name lines using two letters and I gave several examples. Today, I asked Mike how he would tell Kelsie across the room which of these lines he was looking at. First, it was easy.

Then it was difficult.

The same went for how we name angles.

This math thing is easier to approach if I ask myself, what about this concept is useful, interesting, essential, or satisfying, and then work backward along that vector, rather than working toward it from a disjoint set of scattered skills. There is probably a book I should read somewhere in all of this.

Postscript

Also: I didn't hate our opening exercise in which I gave each student a) a compass, b) a straightedge, and c) a map of the Meyer family's South Pacific archipelago, Meyeronia, and d) five questions. [pdf]

  1. How many miles is it from Kenneth to Christy?
  2. Which island is farther from David? Barbara or Christy?
  3. List all the islands that are three miles from Kenneth.
  4. Find a location in the water that is the same distance from Tom & Bob. How many are there?
  5. Find a location in the water that is the same distance from Tom & Bob & Kirsten. How many are there?

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Flavors:

2012 Nov 24. Of course you could just take the concept straight on — defining the terms and defining the notation. No one would have any idea what purpose that notation served or why you'd need two letters to define a line. The concept would be just something else to memorize. But you could do that.

I like this. The iPhone application RulerPhone will measure anything, in any photo, so long as the photo includes a credit card. It's a great use of proportional reasoning, which, if pressed to name one, would be The Mathematical Skill I'd Most Like My Students To Retain After High School.

I added it to the What Can You Do With This? segment featuring The Bone Collector, which seemed like an obvious pair to me. In trying to find the best classroom entry point for this program, I can only think of the question, "How can we break this thing — trick it into giving an incorrect measurement?" I imagine someone can do better.

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