## Saturday, October 16, 2010

### Measures of Center in Relation to Shape

I had a lesson that did not go as I thought it would when I planned it out.  I reached out for help and @druinok was able to lend a hand.  Below is our exchange (I didn't have a blog yet).

We have been working on how to display data (stem plot, dot plot, histogram, relative frequency histogram, frequency table).  We know how to describe data (SOCS) with a slight except for outliers.  For those we are just gut feeling it right now.  The official calculation will come shortly.  We have been playing around with characteristics/properties of mean, median, mode such as:

• outliers cause the mean to change more than median and why that makes sense,

• if few values in your data set then that one outlier can have bigger effect on mean then if lots of values.

So my next step was to show how in certain shapes (skewed left/right and symmetric) mean, median and mode generally follow a pattern in where they are located in relation to each other (skewed right the mean tends to be greater than median etc.).

Instead of just telling or showing I thought I would have them do the following:

• broke the class into thirds.  Each third had to create a histogram with their assigned shape (skewed left or right or symmetric).

• calculate the mean, median and mode and mark/label it on your histogram.

• students then go gather in a corner of the room with others who created the same shape to see if they notice anything.

• go back to table group and share what you observed, "hey guys when you have skewed right the mean is tends to be greater than median, cool huh? what did you find?"

• pat ourselves on the back and move on

Things that did not go as planned:

1. Creating the  histogram took longer than anticipated, actually much longer.  They really liked thinking up a pretend scenario and then coming up with data values based on how it might actually happen if they did the survey/measurement.  I was thinking just quick draw a histogram with meaningless values as long as it meets the required shape and then do the calculations.

2. Many of the scenarios they came up with (like how many pets you own) did not have a very big spread so the mean being greater than the median was not as obvious.

3. I think because of #2 when they went to their corners to talk with other people who had a similar shape the "ah ha" factor did not happen.  They were looking at the actual values of mean, median and mode instead of how they were in position to one another.

So basically I ended up having to spoon feed on this one.   I suppose I could supply them with a variety of data which when displayed on a histogram would lend itself to the "ah ha" moment more easily.  On the other hand I did like how some of them were really thinking what the actual survey results would look like if they did it.  Some nice thought processes.  They seem to get into it more when it is their data instead of "fake" data.  Another possibility would be to use technology (TI calc or statcato) to make histogram for them so that would take out the time there.  I'm not sure about that option either, by having them make the histogram by hand I was getting in some more practice time as far as how a histogram is made and what would make a bin be higher or lower. Not many but some kids think tall bin means higher values not more frequent value so it was nice to hear convos like "I need lots of that number to make the bin higher".

Random thoughts...

I love the idea of the activity!!  A couple of ways to revise it for next year...

1)  Have the kids brainstorm some distributions they think would be symmetric vs skewed first, then assign them a dataset to gather?  That way you have a bit more control over graphs that have a wider spread

2)  I like that the kids had to make their own histograms and had those convos - the kids definitely need that additional practice in creating a histogram!

3)  If you have access to a computer lab, statcato might be fun for them to play around with, but you aren't going to have the same discussions I don't think as you did by hand.

4)  I plan to steal your idea for next year :)  LOL  Thanks for sharing!!!  (and btw, this would make an amazing blog post!!!)

Overall, the only thing that I think would be the kicker for change is making sure they understand they are looking for *patterns* in the *relationship* for mean, median, and mode rather than the actual numbers.

Her last comment was definitely a Homer Simpson moment, dohh!  Before the students met with their shape groups all I said was something along the lines of "see what you notice."  Grrr.   The other issue of students mainly coming up with distributions with small spreads may be helped by 1).

Thank you again @druinkok for your thoughts.  If anyone else has more to add please do so.  This process has been very helpful to me.  I look forward to posting more lesson ideas and failures.