Monthly Archives: December 2014

Class Opener – Day 67 – Verifying Inverse Functions

We’re finishing up our unit on function operations.  Yesterday we developed the definition of inverse functions (using only linear functions) and graphed to demonstrate the symmetry. Time to see what we have learned:

functions

Many students’ instincts let them down on the first pair, believing them to be inverses. But after my prodding that they somehow verify the inverse relationship, we soon verified that f(g(x)) did not produce the result we desired.   The second example was then complete easily.

But what about that third problem?  They sure seem like inverses.  One student offered his proof for the pair:

They are inverses just because I know.

Sometimes ideas in math are just that obvious, and maybe we don’t need to prove them specifically.

On the board, we “proved” that both f(g(x)) and g(f(x)) both seem to simplify as x.  And a few numerical examples help show this:

  • f(g(5)) = 5
  • g(f(10)) = 10
  • g(f( -6 )) = 6……. ruh roh……

Students in my class have not been exposed to a formal definition of the square root function, and this led to a nice discussion of absolute value, and the need to restrict the domain in order to consider inverses. Planting seeds for algebra 2, which many of my students will take next semester, is always a bonus.

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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…

Class Opener – Day 63 – Function Addition

Not all of the class openers go off as intended.  I suppose if they did, and I had a magic formula for engagement, then I’d be living on a dessert island by now with the money I’d made off bottling the secret.

After a short quiz today, a lecture/exploration on operations on functions would begin. It’s not the most exciting lesson of the year, but there are some “ooh” and “aah” moments as students experience new functions.  Domain and range also frame the discussion – and we finally move beyond “all real numbers”.

Even though we were starting with a quiz, I wanted a visual to get students thinking about function behavior, and start to make some conjectures about addition and subtraction of functions. The Desmos graph here was animated and rolling as students entered.

Click the graph to play on your own, and the animated gif below gives you a flavor the the motion.

evxel

After the quiz, I hoped to generate discussion regarding the graph.  What did the students notice? Any interesting patterns?  How are the graphs related? Can we gain some insight by looking at a table of values?

functions

I was hoping students would eventually notice that the ornage function was the sum of the green and red, or at least note the “betweenness” of it all. But with the rush to get notes and discussion started, this opener ended up on the back burner.  They’re not all winners….. All is not lost, though, as I’ll come back to this one tomorrow to build some connections between our notes and the homework.

Class Opener – Day 62 – When Good Questions Go Bad

Today is our last day in the experimental design unit in AP Statistics, and students started class today with an actual problem (or at least part of one) from the 2013 AP exam. This is a question I read during the 2013 reading, meaning I saw about a thousand different responses to this question. It’s quite an experience to be able to share the good, bad and ugly of responses I saw.

For this question, which was number 5 of 6, I only gave out part (a) of the question as this is the portion of interest to us in our experimental design unit. Here’s the question, with some symbols changed for online convenience…

  • Psychologists interested in the relationship between meditation and health conducted a study with a random sample of 28 men who live in a large retirement community. Of the men in the sample, 11 reported that they participate in daily meditation and 17 reported that they do not participate in daily meditation.
    The researchers wanted to perform a hypothesis test [compare the proportion of men with high blood pressure among all the men in the retirement community who participate in daily meditation and against the proportion of men with high blood pressure among all the men in the retirement community who do not participate in daily meditation.]
    (a) If the study were to provide significant evidence against the null hypothesis in favor of the alternate , would it be reasonable for the psychologists to conclude that daily meditation causes a reduction in blood pressure for men in the retirement community? Explain why or why not.

In additonal to providing a response, I asked students to circle the most important words in the question – which words or phrases are most important when considering part a) of this question.

There’s was a hidden agenda behind having students circle some words and phrases.  The average score nationally on this question in 2013 was 0.57 points (out of 4). Most questions usually have an average around 1.2-1.6…with some creeping below 1 occasionally, and some venturing above 2.  In my memory, this question was BY FAR the lowest-scoring question in recent AP Stats history. And while part (a) was the best opportunity to score points, many students still missed its intent.

After students completed the question, I asked each group to provide me a “top 3” list, and we compiled responses on the board. Here are some words which made our list:

  • STUDY
  • CAUSES
  • RANDOM SAMPLE
  • MEDITATION
  • SIGNIFICANT
  • HYPOTHESIS

It’s not a bad list. And, in looking back, my instructions arern’t totally helpful, as there is one (and ONLY one) word which is important here – CAUSES!

In Statistics, there are big ideas, and then there are BIG IDEAS:

A well-designed experiment can allow us to infer cause-effect relationships. Observational studies cannot.

In this problem, students who tended to write more probably dug themselves deeper into a hole. It’s not easy to tell kids to “write less” as details often matter, but in this question saying “this was an observational study, and not an experiment” was all that was really needed.  In reading this question in Kansas City, I found many students who appealed to the small sample size or to perceived confounding variables, and many who simply seemed to gloss over the word “cause” as important. Note: you can find more details about student errors in the Chief Reader’s summary.

Sometimes the simple questions which assess big ideas become the toughest, especially when there are lots of scary-sounding words surrounding the concept. Having students identify the meaningful words can help facilitate these discussions.

Class Opener – Day 61 – Slicing the Cake

Today’s class started with a review of laws of exponents, including negative exponents. I like to send students up to boards often in class, and sometimes use a deck of cards to have students determine their fate. There are 3 large boards in my room, and those who drew Aces, twos and threes were each directed to a board – 3 at each board. Before I began to bark out problems, it’s time to claim our space –

Split the board into 3 parts so that each person has an equitable space.

This was no problem for most teams, as there is always someone willing to take control and draw big, vertical lines down the board. But you can easily determine the trios which had more than one “type A” personality, as erasing sabotage, arguing, and even boxing-out occurred to just split up the board….they’re freshmen….

After the first group of students had completed their problems, the fours, fives and sixes then went to the board. Remembering some lessons from a Contemporary Math class I had taught at Rowan University a few years back, this seemed like a great time to expose students to the divider-chooser method for fair division. And while we are splitting up a board, this would be ideal for splitting up a cake, land, or other assets.  Here are the instructions:

  • In your team of 3, assign roles of player 1, player 2 and player 3.
  • Player 1 – approach the board and divide it into 2 “fair” pieces, without help from the others.
  • Player 2 – choose one of the 2 pieces to claim as your own. Player 1 now owns the other piece. Both players should stand in fron of their pieces.
  • Players 1 and 2 – divide your area into 3 “fair” pieces.
  • Player 3 – choose 1 slice from the areas of players 1 and 2 to claim as yours.
  • Each player now has 2 “equitable” pieces.

Sometimes it’s fun to do 5 minutes of a math nugget they may never see again, but it’s worth the discussion it generates. It was interesting to see how some players chose to work horizontally, rather than vertically – and we even had a triangular arrangement (seen below). But these aren’t really practical for doing exponent problems, so we eventually went back to a traditional division.

North Carolina State provides a helpful file which summarizes a number of fair-division methods, including the Lone Chooser method for 3 people, and you can also easily search “Fair Division Methods” to find more interesting ways to divide assets.

boards