Category Archives: Class Openers

How I start my class each day

Class Opener – Day 66 – Surprising Coin Patterns

A short post today, as I am out the door for a meeting with our NCTM local group, ATMOPAV. Please check out our website, where we have information on local awards, and house our award-winning newsletter!

I enjoy giving problems with solutions which go against our instinct. In statistics, there are many opporunities for this, and today’s opener in my AP class seemed innocent enough:

Which will more likely occur first in a string of coin tosses: HTH or HTT?

After a few moments of debate, there was universal agreement that the two patterns are equally likely, and therefore we should have an equal expectation of seeing them occur first in a string.  But the correct answer goes against this intuitive notion.

Peter Donnelly’s TED Talk – “How Stats Fool Juries” is easily digestible for the high school crowd. I show it over 2 days, first to present the coin-tossing problem. Then in our next class meeting I will show the second half, where conditional probabilities and the multiplication rule make appearances in courtroom trials.  In the video below, fast-forward to about the 5:30 mark if you want to learn about the coin-tossing problem, or watch from the beginning for some statistics humor.

Class Opener – Day 65 – Kohl’s Coupons

After a weekend away from composite functions, today’s opener was designed to bring functions back into discussion, disguised as an innocent-looking shopping problem:

It’s the day of the big department store sale, and you have two coupons you have clipped from the newspaper.  One coupon takes $10 off the price of any item, while the other takes 30% off the price.  In what order should these discounts be taken for you to realize the maximum savings?

After a few minutes of table talk, just about all groups agreed that taking the 30% off first would seem an optimal strategy.  But when asked to provide justification, groups took much different paths.

Some felt choosing a dollar value would provide adequete justification:

trial trial2

How many values are needed to convince ourselves that this strategy is optimal? Is it possible that one strategy is best for some prices, while the other is best for others?

Another group shared the “I know I am right…just because” method

explain

Not very elegant…nor very convincing. But a ray of sunshine appears from the other side of the room, as a group considers defining functions to represent the discounts….but stops just short of pursuing them as a proof.

functions

The eventual “proof” done via composite functions shows that not only is one method superior – it will always be superior by 3 dollars. Add in a domain restriction that our starting value must be at least 10 dollars, and we have successfully reviewed all of our scary function vocabulary.

Class Opener – Day 64 – Can My Students be Random?

Today begins out probability chapter in AP Statistics, which is often deceptively tricky for students. Until now, probability has meant simple experiments – drawing cards, flipping coins or picking marbles from urns (why are we probability folks always so fixated on urns, anyway?). Thinking about probability as a long-term proportion of success is a foreign concept, and separating short-term “bad luck” from a suspected effect requires much deeper understanding. Here is one of my favorite openers to start conversation about short-term probability, which is adapted from an activity done in a college statistics course.

CAN I DETECT PSEUDO-RANDOMNESS?

pic3Students are separated into teams of 2 (or 3).  5 minutes are on the projected clock, and each team is given a grid with 50 squares, along with instructions, face down. Students are told that I will leave the room for exactly 5 minutes, during which time they are to complete the instructions.  At the end of 5 minutes, I will return to the room (after enjoying my coffee) and class will commence.  All instructions are expected to be followed precisely, and without talking to other teams.

In the room, there are two sets of instuctions, which I have passed out without knowing who received which.  The instructions are mostly similar, but with an important difference:

TEAM 1:  YOUR JOB IS TO USE YOUR GRAPHING CALCULATOR TO SIMULATE A COIN BEING TOSSED 50 TIMES.  USE THE COMMAND “RANDINT (1,2)” TO GENERATE RANDOM DIGITS.  LET 1 BE HEADS AND 2 BE TAILS.  RECORD THE COIN TOSSES IN ORDER, USING H’s AND T’s, IN THE GRID PROVIDED.  DO NOT WRITE ANYTHING ELSE ON YOUR GRID PAPER WHICH WOULD IDENTIFY YOUR GROUP.  WHEN YOUR GROUP IS DONE, BRING YOUR GRID TO THE FRONT BOARD AND ATTACH IT WITH A MAGNET.

TEAM 2: YOUR JOB IS TO GENERATE A SEEMINGLY RANDOM STRING OF 50 COIN TOSSES.  GOING AROUND THE GROUP, HAVE EACH MEMBER SAY “HEADS” OR “TAILS”, IN ORDER TO COMPILE A SEMI-RANDOM SEQUENCE.  RECORD THE RESULTS IN ORDER, USING H’s AND T’s, IN THE GRID PROVIDED.  DO NOT WRITE ANYTHING ELSE ON YOUR GRID PAPER WHICH WOULD IDENTIFY YOUR GROUP.  WHEN YOUR GROUP IS DONE, BRING YOUR GRID TO THE FRONT BOARD AND ATTACH IT WITH A MAGNET.

So, one group tosses “real” coins, while another group tries their best to act randomly. When I came back to the room, 8 sheets were hanging on the board.  Without comment, I write “RandInt” and “Guts” on the board to indicate the two methods, and now the challenge is on: can I successfully separate the “real” from the “fake”? As I examine the papers, and slide them into their groups, students begin to sense what I am up to as I hear groans , cheers and grunts…but I don’t want to know who is who yet.  I am sometimes quite good at separating these, but some days I over-think things or believe that a group or two may have sabotaged the experiment.  So, what I am looking for here?

  • Runs of heads or tails: the probability of a run of 5 heads (or tails) in 50 coins is 55%, verified with Wolfram. Usually, non-random students will not let a run go beyond 3, maybe 4.
  • Alternating starting behavior: my hypothesis is that if a group is developing a pseudo-random string, they will alternate often at the start.

For the class, this becomes a rich discussion about short-term versus long-term behavior.  That while we can expect a group of 50 tosses to settle to about 50/50 heads and tails, the short term can yield surprises. And how long of a string should cause us to begin to suspect something is amiss, versus a natural occurrance?

So, how did I do this time?  Unfortunately, not so hot. But that’s OK.

pic1

Asking kids what they thought I was looking for led to many of the big ideas of the section I was looking for, and we were off and running into our probability chapter!

pic2There was one paper I really struggled over, as it started with 5 consecutive tails. While my guidelines should have clearly placed this into the “randint” group, my suspicion was student sabotage. But it was RandInt all the way….even short-term events can fool the instructor.

One student in my colleague’s class then summarizes the entire activity quite nicely, and also provided a needed Friday dose of comic relief:

The calculator gives you runs

Well, if it’s a TI, maybe…