1792 Penny

1792 U.S. Copper Penny

Recently a rare copper penny from 1792 was auctioned off in Florida for  $1,150,000. In addition to lamenting the fact that my ancestors didn’t squirrel one of these away for me, my next thought was GEOMETRIC SEQUENCES! Isn’t 2012 simply another year on this penny’s valuable march of perpetuity? The penny fetched such a steep winning bid because it is both rare and in uncirculated condition.

I brought it up to my class and jump-started the wonderment: “How much did it go up in value each year?”

I was blown away by the thoughtful remarks:

1) “Well, it probably didn’t go up by the same amount each year.”

2) “It depends on if it went up by the same amount or the same percent.”

3) “It probably didn’t increase beyond its face value for years. In 1793 the penny was probably still worth a penny. And in 1801 it was probably still worth a penny.”

4) “Why would someone spend so much?”

5) “It probably depends on how many were minted.”

6) “What makes things valuable?”

I got more questions than we could answer. I especially liked the last question.  I kept our focus trained on “If the penny was worth $.01 in 1792, and in 2012 it was worth $1,150,000, what percent did it increase per year?” full-well knowing that we were assuming constant growth, which would not be the case. Fresh out of a unit on sequences and series, we explored.

I had no idea what kind of answer we’d get, and I let them know this. Let’s put our math to work.

Solving for the common ratio, it turns out the pretty penny increased in value by about 8.8% every year. It’s as though the penny were tied to the stock market’s historical average rate of return, before there was a stock market! We verified our algebra, recursively charting the penny’s 220 year existence.  I stored the exact value in the calculator, cleared my screen, initialized the sequence to 0.01, multiplied the answer by the stored ratio, which I named R.  Every time I pressed ENTER the value of the penny revealed itself with each passing year.

After a minute or two of thrilling tedium, the value of the coin reached exactly $1,115,000. Priceless.

Extensions? If the coin continues at this rate of appreciation, when will it hit $2 million?

To Kill A Mockingbird, Apple, and a US Postage Stamp

Given a known starting value and a current value, what other interesting objects can we examine to determine their average rate of growth? I threw together a Growth Rate Comparison Worksheet to compare Apple’s stock value from its IPO to roughly today, a first edition of To Kill a Mockingbird, and the Inverted Jenny U.S. Postage Stamp. The worksheet is bare bones but should provide a place for everyone to give their “two cents!”

Storytelling and Teaching

On the heels of an off-colorful  joke that would make a nun blush, Andrew Stanton states in his recent TED talk: ”We all love stories. We’re born for them.” As I got deeper into his talk I thought, wow, this has Dan Meyer and his 3 Acts brain-child all over it. Not the joke part. But Dan’s funny too. Think “lesson” when Stanton says “story.”

Exceptional Lessons Make Promises

Stanton shows a scene from his new movie, John Carter, and then states:

What this scene is doing, and it did in the book, is it’s fundamentally making a promise. It’s making a promise to you that this story will lead somewhere that’s worth your time. And that’s what all good stories should do at the beginning, is they should give you a promise. You could do it an infinite amount of ways. Sometimes it’s as simple as “Once upon a time … “

Great teachers make promises! They’re great story tellers.  They’re raconteurs. They hijack their students’ attention without permission in order to take them to a place of new understanding.  They coax out the reluctant and enthrall the engaged. Shouldn’t we all promise to lead our students somewhere that’s worth their time? When the promise is direct and obvious, the buy-in is easy and strong. When the promise is tenuous, so is the buy-in.  Doesn’t this sound like Meyer’s Act 1 hook?

Stanton goes on to describe “the clues to a great story”:

Storytelling without dialogue. It’s the purest form of cinematic storytelling. It’s the most inclusive approach you can take. It confirmed something I really had a hunch on, is that the audience actually wants to work for their meal. They just don’t want to know that they’re doing that. That’s your job as a storyteller, is to hide the fact that you’re making them work for their meal. We’re born problem solvers. We’re compelled to deduce and to deduct, because that’s what we do in real life. It’s this well-organized absence of information that draws us in.

Oh, yes, I want my audience to work for their meal. In fact, I’m having to train myself to quit doing the work for them and to be “less helpful,” as Dan Meyer suggest all teachers do.  An intriguing opening (visual, video, story, joke, anecdote), coupled with the correct lack of information, launches a lesson. Exceptional teachers are experts at organizing this absence of information and prey on the fact that students are inherently prone to filling in this void, if we let them. The struggle for piecing it all together is the cornerstone of Act 2.

Stanton goes on to discuss a story’s conclusion, similar to Meyer’s Act 3:

When you’re telling a story, have you constructed anticipation? In the short-term, have you made me want to know what will happen next? But more importantly, have you made me want to know how it will all conclude in the long-term? Have you constructed honest conflicts with truth that creates doubt in what the outcome might be?

Dan Meyer is a Jedi Master at negotiating the terrain of motivating students with perplexing math scenarios, much like Andrew Stanton with movies.

Want another dynamo at setting the bait? Dave Burgess is a colleague who’s genius at hooking kids in the social sciences.  I’ve seen him get students to spend a Saturday walking 20 miles to the beach to understand the increase in societal mobility that accompanied the invention of the automobile.  Check him out, and another TED talk.

My Friday Ritual

When students walk into my class on Friday they’re greeted by a fast paced revolving slide show of my outstanding students for the week. It happens between passing periods and only on Fridays. I decided last school year to start recognizing students more often. They deserve more attention than they get. All of them.

I choose one student, sometimes two, who have shown me something I find notable. It might be one discrete event that punctures the day with the preterite tense or a continuous build up that might elicit integrals and the imperfect.  Obvious ways to get chosen are the typical ways one might expect to earn accolades: showing significant improvement over time, evidencing academic consistency, earning perfect marks on a concept quiz, articulating insightful comments in class, showing work that makes me write “lovely.”  But there are other not-so-obvious ways too. For example: being vulnerable enough to ask for help, being part of the school’s extracurricular offerings, offering to clean desks, volunteering to help someone on crutches get to their next class, standing up for the defenseless, showing the initiative to create something (a club, a blog, a personal habit) that aligns with their passion. If I notice it, it’s on my watch list.

Students talk about this slideshow, wonder who’ll be next, ask to get a copy of their slide (which I provide), inquire about others, see how I write, see how I create, see how I extol a student who may not be popular to others, witness my commitment to making it happen every Friday, and know that what they do matters. Many times I’ve had a winner tell me after class that I made her day. Now we both feel good.

Most weeks deciding who to choose is easy. When it’s not, that’s good. It makes me look deeper and plumb the depths of my own awareness for perhaps what I’m not seeing. Every one of my students is someone’s child. What would his parents find? Why am I not seeing something worth an extra look? What can I do to find that something? What kind of questions can I ask of the student to learn more about him? If our relationship is lacking communication, what can I do to foster that?  Does she dislike math? Does he dislike me? All questions worthy of my own reflection. Making myself do this every week forces a time of reflection.

I have roughly thirty-six Fridays to honor students. With creative planning to sprinkle the all-stars with the developing all-stars, I can honor an entire class of forty throughout the year in a way that I strive to keep not phony.

I keep the music the same. This year it’s “Don’t You Forget About Me” by Simple Minds. Last year it was “If I Had a Million Dollars” by Bare Naked Ladies. Students come running to me all the time with stories of where they were when they heard the music outside of school and think of our class. Let me repeat that, paraphrased:  I’m doing something that makes students have a positive connection to school when they’re not at school. That’s time well spent, in fact, usually about 20 minutes a week. Quite honestly it depends on the shots of espresso that morning.

Here are a few cheesy lines I extracted from my slides:

“He’s the lead dog in this class! If this were the Iditarod, you’d wanna put a leash on him!”

“Rolling in Pure 5s! Knows the power of reassessment!”

“Observant contribution to Broken Squares Activity!”

“Scorched the final and rocked the Charles Lindbergh glasses!”

This has become a tradition that looks like it’s here for a while.

If you want to see a poor man’s version of it with names abbreviated and no music, click here.

I Made an Error

Today I made my eighth error.

One of my honors Algebra 2 classes now watches me like a hawk. It’s game on when class starts. I made a bet with them.

If you can get past my Wiggle-looking shirt, this video explains it all.

For what it’s worth, the bet was made after one of my very perspicacious students had already tallied six errors and logged them in a notebook.  I showboated into the seventh error just to drum up excitement. Probably not smart. As of today, I have roughly five months to go and three mulligans. Gulp.

I told them to keep this a secret from the other periods. So far so good. If I lose the bet, out to the track we go.  Many of my students already volunteered to run the mile with me.

Processional effect: This bet is providing a conversation piece in continuing our year long quest towards realizing that “Failure Leads to Success.” How I operate and how many risks I am willing to take in front of the class changes when I know I might be penalized.

The student who had the wherewithal to film this bet and whose hand you see in the video signs his papers “Future M.D.” after his name. Apropos. He also is keeping a magical notebook of all the analogies I’ve employed this year. I cannot wait to see this in June.

Etch-A-Sketch and Lite Brite

Image via Wikipedia

This has worked wonders in my algebra 2 classes when we’re discussing ways to write domain and range. Talking about a continuous graph, I’ll describe it as an Etch-A-Sketch graph. When I reference a discrete graph, or as they know it – a collection of ordered pairs-I call it a Lite Brite graph. Kids get it.

When we’re looking at an Etch-A-Sketch graph, we know to usually describe the domain with inequality symbols, whereas with a Lite Brite graph every element (or brite peg) must be listed.

This is even useful for discussing whether or not the graphs (relations) are functions or not. Most cool pictures on either display will obviously not be functions however.  This year I asked for a student volunteer to draw a function on the Etch-A-Sketch while it sits under the document camera. It’s hilarious because if the student turns only the right knob, game over. That’s the vertical knob. It takes a mixture of x and y components, or turning both knobs, to avoid the buzzer that accompanies an undefined slope.

Image by J.G. Park via Flickr

Keeping a Lite Brite and an Etch-A-Sketch in my classroom makes my teaching better and usually brings back positive childhood memories for me and my students. It’s a win-win.

Ineffable Loss

Good night, sweet sir. You will never know the extent to which your life has influenced mine.

You Can Count on this Game

This week I made a few changes to a class game I’ve done in the past. It went over like gangbusters and brought home the concept of relations and functions. Toward the end of my lesson on functions as special relations, I asked my algebra 2 classes if they wanted to play a game. Captive audiences do.

Here’s how it works. Tell the class you want them to count to ten as a class by taking turns shouting out the next number.  Getting to ten on the first try is near impossible. It takes several rounds for them to catch on. These are the rules:

No conspiring.  Otherwise leaders will emerge to organize a winning strategy.

All students must take a turn before anyone can go twice.

You shout “Go.” One student shouts “one,” another “two” and so on. If two (or more) shout out a number at the same time, the counting starts over because there’s overlap. Thus, the counting turns the function into a relation.

Several students mentioned they did something like this in their drama class except they were blindfolded. This twist could make it more interesting. I’m not sure. I like to see the anticipation on their faces.

The game works on many levels. It’s fun. It gives a non-math platform to talk about math (bazinga!), it hones the “failure leads to success” behavior I’m constantly talking about for math success, and it allows me to learn more about the character of the class.

Here’s an example of a round in which the class counted successfully to the number four, but then two students shouted out “five” and derailed the game.