“First Past The Post” And The Dreaded Disease “Z”

This is a lesson I adapted from a model I first learned about at a math teaching conference that I attended at the Dana Hall School.  This conference was without a doubt, the best professional development workshop I’ve attended.  The faculty was amazing, the workshops were specifically geared by grade level, and the resources were practical and rich.  My teacher for the grade 6-8 workshops was Tom Harding, who is the current math department head at Shady Hill.  He was a wonderful and caring teacher.  I can’t say enough good things about this workshop.

This problem has a fun, if highly unbelievable back-story.  As the kids picked apart the story – and mine spent some time doing this – there was no pretense of the specific situation actually being useful, except perhaps for creative writing, but the ideas are highly transferable.  It is not a real world problem.  Perhaps even better, it falls under the Realistic Math Education (REM) umbrella described by MathEd.net HERE.

I used this as an integrated lesson about forms of government in addition to the math lesson.  I’ll set-up the problem and then explain what I did with it.

THE SET-UP

A group of 5 people are boarding a spaceship to Mars. Their health is one of the following:

• Normal
• Mono
• Allergies
• Common cold

And one has…

Students must identify who has The Dreaded Disease Z, as it is highly contagious – and fatal!  If the infected person boards the ship with everyone else, they are all doomed.

THE PROCEDURE

The only way to identify each person’s health is through blood testing; a bag for each person has representative chips for their blood levels (per chart included below).  I used color tiles in brown paper bags for this, and let students take 40 “blood samples” (a sample is taken by blindly taking one chip out of the bag, noting its color, and putting it back in the bag).  With 40 chances, students had to be strategic about which bags to sample from.  Should we take 5 samples from one bag or 1 sample from each bag?  Should we decide on 20 samples and then re-evaluate, or re-evaluate our plan every 5 samples?

Of course, students need some time to just play with the tiles and make patterns too.

You’ll need to set up at least one set of bags with tiles to match the histogram.  If you want separate groups to choose their own ways of sampling, you’ll need multiple set-ups.  40 samples were just about enough to make students pretty confident in their guesses, but not enough for them to be sure.  For me, the big math idea here is sample size, and given a situation, how many samples do you need.  I had students make individual predictions at 0, 10, 20, and 30 samples, and then come to a group consensus about who to kick off the spaceship.

INTEGRATION

I was working on forms of government as part of an integrated Social Studies unit, and saw a nice opportunity here.  When I did this at the conference with other teachers, there was a certain amount of strife about how we should decide to take our samples.  We each had ideas about how to go about finding the bag representing the Dreaded “Z,” and it was sometimes hard to let someone else decide.  And we were adults in a situation where we shouldn’t have cared about solving the problem, but instead should have been focused on examining the merits of different strategies, and why a student might choose one or another.  I imagined that for kids, this would be much harder than it had been for us – and I couldn’t resist leveraging these potential conflicts.

I did this in a whole group setting.  Each group of 4-5 students got one bag, and I asked them to give it a name.  In naming their bag, they grew attached to it, and were more invested in having their bag not be “infected.”  We later talked about how statisticians remain impartial (…or not).

I divided the 40 allowed samples into eight groups of five, and gave specific instructions for who got to decide which samples were taken.

1. Samples 1-5: By Consensus:  Students all had to give a thumbs-up (I’m totally behind this method), thumbs-sideways (I can live with this method), or thumbs-down (I can’t live with this method).  They had to keep talking it out until everyone had at least thumbs-sideways, and then we took the samples.  As you might imagine, there were some strong ideas and this took a while.
2. Samples 6-10: “First Past The Post” Popular Vote:  Anyone could suggest a method, and we voted on which method to use.  The method that received the most votes was the one we used.
3. Samples 11-15: Alternative Vote: Students suggested methods, and voted for their top three choices in ranked order.  We tallied votes, and eliminated the lowest ranks until we had a vote of over 50%.
4. Samples 16-20: Students voted for three classmates by secret ballot.  The three classmates got together and decided how to take the samples.
5. Samples 21-25: I chose one student to make the decisions for the whole class, and didn’t allow any consultation.
6. Samples 26-30: I kept the single decision maker, but added two advisors to help with the decision.  The final decision was still up to the one student with the power.
7. Samples 31-35: I chose four students to make the decisions for everyone.
8. Samples 36-40: I chose names out of a hat, and each student whose name was picked could choose one sample.

After each set of 5, I asked students to name the decision-making method, and reflect on how it felt in writing (The organizer that I used for this is included below).  At the end, I had each group choose who to kick off the spaceship, dumped the tiles for the big reveal, and debriefed the math.  We talked over statistical analysis, probability, and sample size.  This was an excellent big picture statistics discussion, and worked well as a hook for probability exercises to come.

After the math discussion, we debriefed the decision making process.  As you likely figured out, the decision-making methods roughly correspond to different forms of government: Variations of Democracy, Republic, Dictatorship, Socialism, Oligarchy, and Anarchy (OK, Anarchy analogy sort of falls apart).  The analogies only went so far, but the written reflections gave us a great opportunity to talk over the benefits and liabilities of different systems, and the different perspectives of those in charge vs. those not in charge.  “It was awesome! I got to decide for everyone.”  Or… “It was really hard.  There was a lot of pressure and I had to decide for everyone!”  I referred back to both the mathematical and the philosophical content of this discussion several times over the course of the news few weeks’ studies, and the kids are still talking about this and asking to do it again.

THE MATERIALS

Organizer for Recording the Decision Making Process