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Archive for the 'algebra' Category

Odds / Ends

  1. Bill McCallum runs a "standard of the week" contest through his Illustrative Math Project, the goal of which is to illustrate what the different Common Core standards look like in student tasks. I submitted a task for 8.4.F, linear modeling, which was accepted. It's called "Graduation" [pdf].
  2. Key Curriculum Press posted the Ignite talks from CMC-South. I did five minutes on the question, "When will I ever use this in the real world?"

2012 Feb 18. Patti Smith used this task in class. Fun feedback:

My students finally understood the meaning of y-intercept as something more than "when it all began". They also understood slope to mean rate – or how fast they read the names – rather than rise over run!

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.

Guess The Eggs

So you have here a fairly straightforward carnival estimation game, which I decided to complicate by filling up a smaller container with the same kind of (horrid) malted eggs and making that quantity known.

I surveyed my students, my math-department colleagues, some of their students, my principal, and the central office staff. A little over 100 guesses all told. I tagged each guess with the following metadata:

  • name,
  • guess type (gut check, visual estimate, math computation),
  • job description (student, math teacher, staff member, principal),
  • current math class (eg. Algebra 1, Geometry, AP Calculus, etc.),
  • grade level (freshman, sophomore, junior, senior),

I showed my students the raw data and asked them what they wanted to know. I wrote their questions on the board.

  1. who won?
  2. who guessed worst?
  3. what was the ranking of everyone in between?
  4. what type of people used math computation for their guesses?
  5. were there any tied guesses?
  6. what was the highest/lowest guess?
  7. which grade level guessed the most?
  8. which grade level guessed the best?

Define "Bounty"

I said I was offering a "bounty" for answers to those questions and asked them to define the term. Some kids had seen Dog the Bounty Hunter and explained it from that angle. I assigned each question a point value that corresponded roughly to a) the difficulty of the question and b) its relevance to my objective — how are absolute value and percent error useful for calculating accuracy? I offered 20 points for a picture of an interesting fact. (See "Interesting Pictures" below.)

They had to scrape together 100 points for the day and I offered extra credit for initiative, divergent thinking, etc.

What Happened

Students worked in pairs on laptops. They downloaded an Excel sheet with all this data, including the real name of every guesser. Naturally, they were into that.

The great part about a sample size of one hundred guesses is how easy it was to determine which groups were taking a tedious, manual approach to these questions and which were using Excel's built-in capability for sorting and calculating. I circulated the classroom and could tell that a group was ready to learn more about Excel because they were using hash marks to count up every freshman, sophomore, junior, and senior. Those students were wandering the desert on foot, ready for the water, compass, and camels I could offer them.

Likewise, I saw another group of students subtracting all one hundred guesses from the actual answer (1831) one at a time on cell phones. It didn't take much to convince them to experiment with another approach.

The Constructivism Multiplier

My favorite conversations with students centered around a definition of "accuracy," as in, "who were the top ten most accurate guessers?" Our earlier trick of just subtracting the guesses from the answer messed with Excel's sort mechanism, unhelpfully stacking positives on top of negatives, when, really, we didn't care if you guessed 100 eggs too high or 100 eggs too low. For our purposes, those two people tied1.

Two students were so close to constructing that operation themselves I had to bite off my tongue to keep from spelling the whole thing out ("ABSOLUTE VALUE! SUBTRACT AND THEN TAKE THE ABSOLUTE VALUE!!") and then the bell rang. We didn't graph anything. We didn't get to percent error. Half the groups got to absolute value.

Off moments like this, I have determined my constructivism multiplier to be four, which is to say it takes me four times longer to bring a student to conceptual understanding through conversation and questioning in a social situation the student helped create than it does to get up in front of the class and simply give it to them straight, no chaser, through direct instruction and a handout of questions I wrote.

What I find maddening about conversations with committed constructivists (cf. the conversation here) is the reflexive assumption that educators choose direct instruction because they're either power-drunk or self-obsessed or because they lack faith, courage, or high expectations. I can't, personally, wave so dismissively at the massive institutional impediments to student-constructed learning.

Interesting Pictures

Percent Error By Grade Level

Percent Error By Guess Type

It's worth pointing out here that "Math Computation" isn't the same thing as "Correct Math Computation." The most accurate guessers verified their correct math computation with a visual estimate.

Percent Error By Math Class

Percent Error By Job Description

That last graph is what I meant at TEDx when I said that math gives your intuition a certain vocabulary. The math teachers have a more descriptive vocabulary for expressing their own intuitions than the students do. This is also a fair answer to the question, "when will I ever use math?" You might not. You can live without it. But it makes a lot of intuitive tasks a lot easier. And you should also understand the risk that you'll one day be fleeced by or passed over for those who know how to speak with that vocabulary.

The Creative Feedback Loop Of Teaching

Where else can you get this? In all of the creative fields that have ever tempted me professionally — I'm talking about graphic design, screenwriting, and filmmaking — ideas often take months to generate and refine, years to produce, and, in many cases, you can't do anything with the feedback except hope it's good enough to get you your next job.

With teaching, you can get any old harebrained idea on Friday, challenge your students with it Monday morning, then adapt it for your afternoon class based on feedback from the morning. The feedback loop is fast enough to give you whiplash. It's so much fun, this job, it seems impossible sometimes that anyone could ever walk away from classroom teaching.

The Grand Prize

Not those horrid malted eggs, that's for sure.


  1. This is a good companion exercise, in that sense, to How Old Is Tiger Woods?

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.

In these exponential times, I admit that even I find it easy to nod my head credulously at a passage like this:

When it comes to [Facebook's] online chat function, 1.6 billion messages are sent every single day and 1.4 million photos are uploaded a second.

Not so Nat Friedman who crunched some numbers in an utterly classroom-appropriate exercise in unit conversion and calculated that this means everyone on Earth is uploading approximately 20 photos per day.

Which means I had better hurry up and get a Facebook account.

[via daring fireball]

[N.B. The Internet has been pretty generous today, right?]

[1] http://www.census.gov/ipc/www/popclockworld.html

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