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.

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.]

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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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 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.

Session Title

Games And Puzzles That Develop Sequential Reasoning

Better Title

OMG MICHAEL SERRA!!1!

Presenter

MICHAEL SERRA!!1!

Narrative

A structure not dissimilar to Megan Taylor’s yesterday, where Serra debuted games and puzzles and gave us time to tease them out.

I sat with two former colleagues in the back – all of us now at different schools. One teacher enthused over Sudoku puzzles. They challenge kids. Kids like them. It gets them comfortable with numbers. The other enjoys Serra’s games and puzzles, like Lunar Lockout. Both cite improved student disposition toward math and improved deductive reasoning.

I disagreed with them. In general, I find it dangerous to put too much distance between “fun time” and “math time” preferring, instead, to have that cake and eat it too, creating as many challenges as I can that are both fun and mathematically rigorous. (Which Sudoko, to put it plainly, isn’t.) My task is harder, I think, and I know I fail at it more, but I’m more satisfied on balance.

It was a good conversation. Feel free to interrupt us.

Serra’s best offering for my money was Racetrack Math:

It’s like this:

1. Draw a racetrack on graph paper, however crude.
2. You and your opponent start anywhere on the starting line.
3. You travel along vectors. You may increase or decrease either the x-value, the y-value, or both, but only by one unit per turn.
4. First person to the finish line wins.
5. (P.S. No crashing.)

This gets very interesting very quickly. You start out with tiny vectors which lengthen by one unit every turn. If you fail to notice the side of the track off in the distance, though, and fail to slow down in time, you crash. (Which I did in the example above.)

I hereby toss all of my battleship exercises in the recycling bin. This is a much more straightforward introduction to positive/negative coordinates since each new turn is relative to the last turn rather than relative to this strange coordinate axis thing.

Plus, your students can create racetracks of their own, of infinite complexity, within seconds. Serra cited some kids who created a pit lane, which you had to enter on your second lap, and oil slicks, on which you could not adjust your vector at all. I’m impressed.

Visuals

PowerPoint. Which is tough when you’re asking people to solve a puzzle. If someone suggests an alternative route to the one you have programmed into your slide, you have to dodge their answer a bit.

Handouts

Blank puzzles and games to draw on. Again, paper is not dead. How do you do this digitally? Load each picture one at a time into Skitch and pass a stylus back and forth? Moderation, please.

Homeless

• “There is no research that demonstrates these games improve outcomes in other mathematical procedures like two-column proofs,” Serra admitted reluctantly. “It has to be there. I know it is.

At the start of winter semester, maybe a month ago, I told them they’d have homework every night, even weekends.

I called it The Feltron Project. I showed ’em mine and asked them to identify the mathematical forms. I told them we were going to take their lives and make math out of them.

Track Your Life In Four Ways

I told them they had to track four variables this semester. I shared with them my ownAnyone crazy enough to try this with me: it’s essential you play along with your students.:

• where I’ve been [cities per day]
• text messages sent / received [quantity per person per day]
• movies I’ve watched [title per medium (dvd, theater, ipod) per day]
• coffee drinks i’ve purchased [accessory per drink per location per day]

The Feltron Notebook

While they thought on it, we made Feltron notebooks: graph paper, folded, cut into quarters, and bound with repurposed file folders the last teacher left behind.

I showed them how I designed my own Feltron notebook (Coudal’s Field Notes, natch) to maximize page use.

How Do We Grade Your Life?

We discussed grading. What would an A look like? An F? A C? I steered the conversation towards three criteria:

• the interesting-ness of the variables chosen
• their consistent tracking
• their clear & pretty design

We discussed interesting and un-interesting variables. Some students are rocking this thing all semester long, counting calories, tracking everyone they text over a semester, tallying every ounce of everything they drink.

Other students are skating, tracking the number of days they’re late to school, tracking the number of times they sneeze, etc.

We conferenced, each student and I, and I suggested changes, both to add value to their final project and to make the assignment easier for themFor instance, 100 kids decided to track “TV Watched.” “What does that mean?” I’d ask. “Uh.” they’d reply. “So make it min/channel/day or min/show/day, whichever you prefer.”.

Checkpoints

This thing runs on bi-weekly checkpoints [pdf] where I move around the class and verify that everyone’s keeping up.

One Indication This Assignment Wasn’t Stupidly-Conceived

Not one student has taken exception to the workload. Several students, without my prompting, have integrated a notebook update into their daily classroom routine.

The Moment I Fell In Love With The Thing

One freshman decided to track the cigarettes she smoked each day. Not because she wanted to scandalize me or her classmates. She just “always kinda wondered.”

One Month Later

I surveyed 99 students last week: “how much time do you spend updating your Feltron notebook each day?”

The average response was 5.5 minutes with a maximum of 31 minutes and a minimum of 0 minutesNo idea what the minimum’s about..

Next Steps

• I ordered a hard copy of Nicholas Felton’s annual report (to which my assignment pays seeerious homage). We’ll pass pages around and develop a written narrative of his year.
• Then I’ll fabricate entire data sets. eg. some girl’s caffeine intake over the course of a semester. We’ll run through several infodesigns and discuss which ones tell the most effective, truthfulAll better? story. We’ll use other data sets (eg. hours spent studying) to introduce some superficial correlation.
• Uh. That’s all I have.

The Big Questions

• Do we make the graphs in Excel or work out the math by hand? One option gets ’em dirty with the math. One is more useful to their post-grad experience.
• What do I do when a student comes to class a month into the project and claims her dog ate her Feltron notebook? The question, as of first period today, ain’t hypothetical.

The Regret

I should’ve collaborated with someone here. I don’t know another teacher, period, who’s out there sweating the connection between language and math like I am here which makes The Feltron Project something of a blind jump off the high dive when it ain’t altogether obvious that the pool is filled with water, thumbtacks, or nothing.

I’m about to give you what I’m convinced are good blueprints even though the house I built off of them today was pretty raggedy.

Here, three days before winter break, I wanted an activity that injected math into something mindless. I thought about snowflakes, you know, how you fold some paper, cut it here and there, and open it up only to discover you’ve recreated The Storming of the Bastille.

So here’s (what I’m convinced is) an awesome exercise in spatial intelligence for you and your students: predict what the snowflake will look like before you open it up.

I’m tempted to leave it there and let you decide how this oughtta shake out, encouraging you to please get back to me and let me know. Because what I did today didn’t have the same loose-limbed energy my best stuff usually does. This was second-rate but maybe we can spin something better out of it – you and me:

• I passed out a sheet of standard letter paper and some scissors to each student.
• I had them square the paper and fold it into fourths – now a smaller square.
  
• I put up a series of slides. Each one asked them to make one cut.
  
  
  
• They made the cuts and I said, before you open up the snowflake, sketch what you think the snowflake will look like.
• They sketched it.
• I walked around, observing, sometimes making comments.
• They opened it up and checked themselves.

Then, without passing out more paper, we went backwardsWorking backwards from a solution to the problem, incidentally, is the most reliable way to carry your kids a few rungs up Bloom’s Taxonomy..

• I gave them the result and asked them what cuts had been made to get it.
  
• I called up five volunteers to the board to show their solutions, most of which differed only slightly from each other, a fact which offered up some good conversations starting with words like “compare” and “contrast.”

Then I passed out this worksheet, which asked for eight visualizations, the second half doubling in complexity by adding one fold to the snowflake.

Typing all that here at the end of the day, it’s kinda obvious to me that this was too much even for my Geometry sophomoresNine of whom apparently read this thing so, hey, team, no disrespect.. The spatial learners had a blast but I didn’t manage to transcend that division and pull the other intelligences over the wall like my better stuff tends to. This thing lacked a certain scaffolding. In other words, buyer beware.

Attachments:

• Handout
• Slidedeck [keynote]
• Slidedeck [ppt]
• Slidedeck [pdf for those poor ppt07 people]

Search Engine Chum

1. triangles, quadrilaterals, isosceles, equilateral, scalene, circles, chords, tangents, secants
2. word problems, math games, basketball
3. free powerpoint geometry lesson plans

Macroscope

1. Moving swiftly along, trying to prevent this from becoming yet another time suck.

Filesharing

1. Keynote
2. PowerPoint (which has no one to blame but itself)
3. PDF
4. Interactive QuickTime

Microscope

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   274 mi. 6h 48 m. Follow up question: can you do it in one tank.
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   If you started driving South from Santa Cruz and kept going the length around Hawaii, how far would you get?
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   I post two photos a week on our little Moodle install and ask my students to submit five sentences on five concepts we discussed during the week.
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    Have them fold their notepaper into sixths.
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    Last year didn’t care about the sequencing. Parallelogram is a trapezoid on crack. A rhombus is a kite on steroids. Rhombus + Rectangle = Square. What do you notice about these three?
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    The point being that it doesn’t matter what it LOOKS like. Only what it’s marked up as.
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    What is this?
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    Draw and define.
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    Discuss different ways to write major and minor arcs. Note: we did a LOT of discussion here. I put up letters over every intersection and asked them to name the arcs, point to them, etc.
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   heart disease
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   Tell ‘em you can assume that A is the center.
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   Might ask here what you call a circle that is both concentric and congruent. Laugh at them after they sweat the answer.
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   Which words here are unfamiliar.
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Search Engine Chum

1. pool table problems, angle measurement, angle definitions
2. polygon definitions, coordinate geometry
3. free powerpoint geometry lesson plans

Macroscope

1. The pool table animations took, like, two hours to build last year. This year they took zero.
2. Went to a welcome-home party for a friend instead.
3. I’m quite okay with that, thank you.

Filesharing

1. Keynote (which four out of five doctors prefer)
2. PowerPoint
3. PDF
4. Interactive QuickTime

Microscope

1. 
   $35. Origin story here.
2. 
   Last week we handled problems like these where we GAVE you the incoming path and asked you for the outgoing path. It changes today.
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   Ask them which diamond do they aim for in order to hit it? Take several guesses. Have them come up and make a mark.
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   Alright, we’re gonna define eight terms. Show how we can write m(angle)B = 90° and AB (perp symbol) BC
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    Talk briefly about parallel lines here and how to write ‘em.
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    Notice it’s LINEar angles. Gotta be a line in there.
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    So outta curiosity if this whole angle is 54°, how big is this one? How do you write stuff here?
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• Pool Table Geometry 1
• Pool Table Geometry 2
• Pool Table Geometry 3

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   $528,783,552,000 (I did a coy little unveiling here, first writing $528 and then, as they freaked out, writing the next three digits and so on.) Backstory here. ¶ Underline must, never, and every in the first three. Ask ‘em how many examples it takes to prove must, never, and every wrong. (hint: one)
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   book definition: “a closed figure in a plane, formed by connecting line segments, where each segment intersects exactly two others.” ¶ Have them build their own but get it close. They really enjoy building their own definitions.
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    corresponding sides AND angles are congruent
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