Jigsaw: Histograms With Differently Sized Bins

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SWBAT create histograms and compare those with differently sized bins.

Big Idea

Exploring the idea that the same data can tell different stories, depending on the decisions we make in its representation.

Opener: Choose Your Level

3 minutes

I very rarely tell students where to sit in my classroom, but flexible grouping is a powerful tool in keeping a class engaged.  So throughout the year I employ a variety of strategies for getting students - at least temporarily - to sit in other parts of the room with different people than usual.  

Today's class opens with one such strategy.  Students can choose their seats based on the level of challenge in which they'd like to engage today.  I write this on the front board, and then as students enter I direct them accordingly.

Group Work: Fill in Frequencies and Create Histograms

20 minutes

Today's task is for each group to take a set of data and create a histogram.  For data, we use the greed scores for all students who played the game a week ago.  The different levels of challenge are based on bin width.  At the easy level are wide bins like 100's and 90's, while at the challenging end are narrower bins with widths of 25 and 30.  In between are 50, 60, and 75.  In the Greed Histograms Jigsaw, you'll find the data on the first page, followed by different frequency tables set up for each group.

As students form their groups for the day, I circulate and ask if everyone is ready to get started.  Each group gets one copy of the data and one frequency table.  Every group receives a frequency table with different intervals (see Greed Histograms Jigsaw). The first part of the assignment is to work together to fill in the frequency table.  About half of my students need to revisit the word frequency, and I tell them that it just means "how many".  Once they get the idea, they're off and running pretty quickly.  It's fun to see the strategies students employ here.  When I see something like this, I applaud students for embracing the opportunity to invent new mathematics.

I tell groups to tell me when they're done.  When they do, I ask them to make sure that the sum of their frequencies is 96, which is the total number of students who played Greed last Monday.  They engage in a round of error checking.  If a group's number don't add up to 96, I tell them to find and fix the errors and to let me know when that's done.

When a group is ready with bona fide frequency tables, each individual student gets a two-sided graph paper (with this on the front and this on the back).  This is the first individual task of the day.  Before class, I wrote these instructions on the side board.  Now, I also post an example histogram (see the first page of today's notes) from the projector.   Students make a histogram from their shared frequency table, and because they are working in groups, they are able to help each other.  The pace of the class is such that I can move around and help one group at a time.  The different-sized bins mean that students are ready at different times for next steps, and the work spaces out nicely.  They make their graphs.  Our work on the number line from Unit 1 really pays off now.

As students finish, it's time for the jigsaw, so there's a gradual transition to the next part of class.  If anyone finishes particularly early (it's common for this to happen when highly skilled students choose the easy side of the room at the start of class) I'll have them create another histogram, with smaller intervals, on the back.  But if at least the first two groups finish around the same time, I can set them to the next task.

Jigsaw: Find Two More Histograms

15 minutes

In their initial groups, students collaborated and everyone produced one histogram of a given bin width.  Each table was assigned a different bin width, so now it's time to jigsaw: students take their completed histograms and find two other classmates who graphed the same data with a different parameter.  Check out this narrative video for details on this activity

It's terrific to see watch students as they help each other out, share information, and start conversations about the decisions we can make when representing data.  Here's an example of what two students might see as they compare their work.

For your perusal, here are seven different Greed Histograms.  What questions do these raise in your teacher-mind?

Announcements and Homework

5 minutes

With a little less than five minutes left, I call everyone to attention to announce homework, my extra help hours, and anything else that might be necessary.

Then I try to start an informal conversation about the different histograms.  "You all graphed the same data today, but do all of your histograms look the same?" I ask.  They don't.  I try to get students to explain why this is the case.  If they need more prodding, I ask students to think about the story that their particular histogram tells.  It's just enough to start getting at the idea that the choices we make on a data representation can affect the message we're trying to deliver.  With Greed Scores, this message is of little consequence, but as the year proceeds, there will be more to see.