Teach 180: A Little Competition Can Be a Good Thing (Day 42)

Today was our first day in Probability and Statistics for analyzing bivariate data.  One of the concepts that students sometime struggle with is estimating the strength of a linear relationship from a scatterplot.  They think there is no association when there is a weak, negative linear association.  Or they think there is a strong, positive linear association when it is more of a moderate, positive linear association.

Today, we used the Rossman-Chance applet called Guess the Correlation to improve our estimation skills with correlation.  Student were arranged in a bracket-style competition.  Each student was shown a randomly produced scatterplot with 25 dots and the student had to guess the correlation coefficient.  The student in the pair that was closest to the correct value moved on to the next round of competition.  An example is shown below.  You can see that I did fairly well. But not as well as the winner of the tournament, who was within 0.005 with his estimate!

Teach 180: Sometimes I Don't Listen (Day 41)

One of the things I love about working with the other math teachers at my school is the fact that we enjoy collaborating.  If we had a common office space, I am guessing it would be a challenge for us to get anything accomplished individually, because we would be sharing ideas all the time.

Even though we share ideas frequently, I don't always listen.  Marilyn told me that this one activity took her longer than she planned and I didn't listen.  In my plans, I had even made a note that I thought the activity would only take about 15 minutes.  Then, 20 minutes elapsed, then 25 minutes and finally the last group completed the activity in about 30 minutes.  Students were to match a function with its graph, domain & range and characteristics. You can see a grouping of four such cards below.

It was a valuable activity and students did really well working together.  Why did it take so long?  Part of it was the fact that there were 40 cards in front of them and I gave them no guidance in what might be easiest to match first.  In addition, I let students debate which cards were to be matched together and did not step in when there was a disagreement.  Usually, the student with the right answer prevailed and convinced the other students at the table as to why the answer was correct.

So, what would I do differently?  I might do eight groups instead of ten and I might model the thinking process for completing one match.  This would allow us to review key ideas, like open and closed intervals, prior to having them work on the activity in their groups.

Teach 180: The German Tank Problem (Day 40)

I spent about two hours on Friday finalizing what I needed to do to prep for a talk that I was giving on Saturday.  The talk was given at the Association of Math Teachers of Philadelphia and Vicinity (ATMoPAV) fall conference.  The title of my talk was "The German Tank Problem: Simulating a Statistic".  For those of you who teach AP Statistics, this is an activity you likely do to introduce sampling distributions.  When I first did this activity, it bombed, because I wasn't comfortable with using Fathom to create a simulation of a sampling distribution.

Then in 2009, I took an online course to learn how to use Fathom.  I still wasn't comfortable with using it entirely and spent about a year screencasting demos to show Fathom in class.  Then in 2010, I started to use Fathom with my students.  It is a great visualization tool and it is great for helping students understand the difference between the distribution of a population, the distribution of a sample and the distribution of a sampling statistic. (Note: Fathom has a free trial version and a 1-year subscription is $5.95.)

For the German Tank problem, students are given a bag with N "tanks" numbered in their bag consecutively from 1 to N.  They randomly select 7 tanks and use the numbers on the tanks to create a statistic to estimate the total number of tanks. The eight participants at my workshop on Saturday worked in pairs and spent a good fifteen minutes to create their statistics.  At the right you can see the common statistics students often create.  The Partition Method is the one that was used by the mathematicians in WW II to estimate the number of tanks.  At the workshop on Saturday, participants created several of these common statistics.

I had been doing this problem for several years and last year I had a pair of students create a statistic that was good as (or perhaps better than) the Partition statistic.  I named the statistic after the two students, Cecily Redfern and Neelam Ferrari.  It's called the Redfern-Ferrari statistic.  Two hundred trials of the simulation can be seen on the screenshot at the left.  Note that 344 is the true number of tanks in the bag and that both Partition and Redfern-Ferrari are centered near 344 with similar variability.

What actually is the Redfern-Ferrari statistic?  The formula for the statistic is Max. + Range/6.  The students found the average difference between two consecutive numbers by calculating Range/6. Then, they added it to the maximum number they drew to get the approximation for the value of N.

If you have tried this activity and abandoned it, I would encourage you try it again. If you are interested in the Fathom files to do this activity and/or the handout I give to my students to do the activity, send an email to me at mathteacher@ptd.net or leighnataro@gmail.com.