What's Normal

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SWBAT use data from population samples to approximate the mean of the population

Big Idea

What's normal, anyway? How does being normal have anything to do with mathematics?

Set the Stage

10 minutes

You will want copies of the Height Weight data.xls sheet for your students unless you've already handed them out during a previous lesson.  I begin this lesson by asking my students to think about what they know about a normal distribution or normal curve.  I give them a few minutes to reflect, then ask for volunteers to share their thoughts.  It is rare that anyone remembers the simple description of a normal distribution; that it is symmetric around the mean and the mean also equals the median and the mode.  I help refresh their memories by playing an edmodo clip that discusses using a normal curve.  After the video, I ask for volunteers again, this time to post the characteristics of a normal distribution on the board. (MP6)  

Once we're clear on what a normal distribution looks like, I explain that today we will be trying to determine if the height and weight data sets are normailly distributed so we can predict heights and weights more easily.  I ask my students to discuss with their front partner whether or not they think the data will be "normal" and walk around listening for possible misconceptions and/or misunderstandings.(MP2) Examples I've heard in the past include the belief that height and weight must be normal because they are continuous, or that they can't be normal because people can only grow so tall or so heavy and no more. I try to address these issues as I hear them by asking leading questions and gently challenging student assumptions.  I also refer them to the description posted on the board and ask if they are meeting those criteria for "normal".  I answer any other general questions that arise, then tell my students that they will be working with their partner for the next section of the lesson and will need their height-weight data sheet.  I just give another copy to students who can't find theirs to reduce "down" time spent looking or going to lockers.

Put it into Action

35 minutes

Team Graphing: We have been working with this data for two previous lessons (In the Middle and How's Your Spread) so my students should already have frequency tables for at least one of the data sets, however there are always a few who have lost those papers. I briefly review how to make a frequency table, using appropriate class intervals, then tell my students that their challenge for today is to create two frequency histograms, one for height and one for weight.(MP4) I provide graph paper and rulers for those teams who choose to use them (MP5), and walk around as they work offering encouragement and redirecting as necessary. The most common problems are teams that still don't understand how to choose class intervals or who struggle with the specifics of graphing such as setting up the axes. This section should take 10 to 15 minutes, so I adjust the time as needed to allow all teams to complete the histograms.

Individual work: When all the teams are done I tell them that each student should be able to complete the next piece since each team created two histograms. I have a student distribute colored pencils (I keep a tub of assorted colored pencils for student use) and explain that they will be using the pencils to sketch the estimated curve of their histograms. I walk my students through the following steps giving them ample time to complete each step: 1) mark the midpoint at the top of each bar of your histogram 2) connect the points using gentle curves 3) extend your line past both ends of the histogram continuing the gentle curve 4) mark the approximate mean of your histogram (you may use the mean you calculated previously)

Class Discussion:  When all students have completed marking their graphs I ask each team to compare their two graphs and be prepared to share their observations with the class. (MP3)  I randomly select teams to present their graphs and observations, encouraging appropriate questions of the presenters, until all teams have shared.  This allows students to see a variety of histograms for the same data without having to create each histogram themselves.

Wrap it Up

10 minutes

After all the teams have finished making their comparisons, I tell each team to discuss whether or not they think either the height data, weight data or both fit a normal distribution.  I give them a few minutes to talk and then tell them that the final pieces of today's lesson will be completed independently.  I give each student a notecard and ask them to write their opinion about the normalcy of the height and weight data and support their decision mathematically on one side of the card.  I emphasize that I will be looking for appropriate language as well as sound mathematics in their responses to reinforce their understanding and ability to attend to precision. (MP6) I tell them the other side of the card is for them to estimate how many standard deviations above or below the mean their own height/weight would fall, including both their measurement and estimate with explanation of how they made the estimate.