Mean Absolute Deviation Through Dice.

Here’s the setup: my 6th grade general classes were working with Mean Absolute Deviation the past couple days.  I wrote about my approach last year using line plots as an anchor for understanding, but I talked back and forth with Nathan Kraft this school year about his dice activity to have students develop a need (and process) for MAD.  I was looking forward to teaching this topic from the outset of our statistics unit.  Check out his post before reading further or my words will just seem like rambling.

When I used Nathan’s activity with my first period, it flopped pretty badly.  In hindsight, most of the struggle was due to how I had implemented Nathan’s lesson.  I gave 12 pairs of students dice to roll and I directed them to record their results on a dot plot.  I ended up choosing 6 of the dot plots at random to have students rank, then I used questions to guide students toward the idea of deviations and MAD.  Because of the choice to use only half of the data, some students got disenchanted as soon as we began analyzing the chosen graphs.  Beyond this misstep, I noticed my language wasn’t specific enough when I asked students to rank the data and brainstorm ways to analyze the graphs.

Armed with the knowledge I gained from my first period, I tightened everything up for second period and I got great results.  I had groups of 4 students take turns rolling dice and I directed every group member to make a dot plot of the results.  With only 6 unique graphs in the class, no one was left out during the subsequent activities.  I quickly snapped pictures of the dot plots with my document camera, then I arranged them a slide.  From this point in the lesson, I had a lot more traction already because I kept using the phrase “distance from 7” from the start of class.  When it came time for students to rank the graphs, a student asked, “Does best mean the rolls are closest to 7?” Bingo.  This class also generated more ideas for ranking the graphs mathematically, including suggestions for using mode, range, and median, and mean.  One by one, I met these suggestions with enthusiasm, then I drew dot plots that poked a hole in the idea.  As I thought about it later, I wonder if this hypothesis wrecking could be given to students in some way?

Finally, I suggested the idea of distance from 7 again.  It took a little scaffolding, but eventually students realized we should look at the distances (deviations) for the data.  To speed things up, I had students work with the data for their group.  This moment was where the dot plots students made really came in handy.  Some students chose subtraction, but most students just looked at their dot plot to see the distance from 7.  Once groups compared results, I had asked students why we couldn’t just use the distances as is to make a comparison.  Students noticed that we have a lot of distances written and some groups rolled more than others.  Eventually, we got to the point where an average (mean) of the distances (absolute deviations) seemed like the best option.  Conveniently, all students had to do at this point to calculate a mean for the distances they already found for their dot plot.

In both classes, the benefit of the activity was fully realized the next day.  As we took some notes and completed some examples for MAD, I was able to refer back to the work students derived the previous day for the concept.  Beyond this benefit, I noticed that students beginning to anticipate whether a MAD would be large or small based on the data or a dot plot for data.  All in all, I would recommend this lesson (with careful attention to planning, student activity, and language of questioning) as a way to develop student understanding of Mean Absolute Deviation

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