Shape, Center, and Spread

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Objective

SWBAT summarize data sets in relation to their context, giving quantitative measures of center and variability, and relate choices of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Big Idea

A graph is a visual representation that can be used to analyze and interpret data based on the center, spread and overall shape of the data.

Think About It

7 minutes

My students will work on today's Think About It problem in partners. After 4 minutes of work time, I have students share out what data they think should be used.  We discuss the idea that depending on what you think is most important, you would display different data. So if you think being able to score points is the most important this is the data you would want to collect and display.

It's also important to discuss the analyses that could be performed with the data.  Students should say that Ms. Nichols would want to the find the mean score because this is the precise measure of center (if we assume that being able to score is the data that's important). However, Ms. Nichols wouldn't want a player who is inconsistent, so she should collect measures of variability like the range and MAD to make sure that the player is always great.

Intro to New Material

15 minutes

At this point of the unit my students know several ways to represent data visually with dot plots and box and whisker plots.  They have several different measures they can use to talk about and analyze data and know how to find the mean, median, mode, range, interquartile range, and mean absolute deviation.  In this lesson, students will represent data sets visually, calculate quantitative measures of center and variability, and then use those representations and calculations to formulate an interpretation of the data.

To start the Intro to New Material section, I have students read and annotate the first problem.  I ask everyone to make a prediction about who they think is right, which they'll show me by raising their hands for either teacher.  Students construct a box plot independently.  

I then ask students what would be a better measure of center for this data set - mean or median. I'm looking for students to say that the median would be a better statistic here, because there are outliers in the data.  I then ask students how we should determine variability within the data set. After we discuss this question, I will ask my students to find the IQR and MAD.

Once we have a visual representation and we have calculated the median and variability, I guide students to make an assertion, supported by numerical evidence.  I write (and students capture):

  • My assertion is that the typical number of community service hours is 20.   
  • The data is skewed to the lower number of hours and clustered around 10-20 hours. 
  • The median of the data is 20. 
  • The variability is large - the IQR is 90 because there are outliers but when looking at the data you will see that most are clustered around 10 and 20.

 

Partner Practice

20 minutes

Students work in pairs on the Partner Practice problems.  Students have access to calculators throughout the lesson.  For today's work students need to accurately create either a dot plot or box and whisker plot, calculate the measure of center they think best represents the data based on the distribution and variability of the data, and then support their interpretation with evidence.

As I walk around, I have with me my own answer key so that I can easily check for the shape and center of the dot plot or box plot.  I'll ask what the five number summary is in the box and whisker plot.  My answer key lets me verify the accuracy of calculations with the mean, median, mode, range, interquartile range, and mean absolute deviation. As I observe, I will ask students to share their interpretations and the evidence that supports those assertions. 

After 15 minutes of partner work time, students complete the Check for Understanding independently.  I'll circulate as they work.  Students will share their representations and assertions with their partners.  I'll ask for students to volunteer their partners to share an assertion aloud, if they feel their partner has a strong writing sample. 

Independent Practice

10 minutes

Once students have explored today's concepts with a partner, they work on the Independent Practice problem set alone.  

For these problems, students who are struggling to interpret the MAD could look at the data on a dot plot. Here they will be able to see more clearly when data is more spread out. Students who struggle with computation could cross out each value once they’ve entered it into their calculator or chunk numbers so that they have less numbers to enter.

For students who are ready for an extension, they can explain what alterations, additions, or subtractions in the data set would make their interpretation even stronger.  For example, with the apartment home problem in the Partner Practice set, students could argue that the quantitative measures would be more reliable if based on a greater number of data.

Closing and Exit Ticket

8 minutes

After Independent Work time, I ask for 2 students to share their responses to the Great_Pizza_Debate.  I ask the class if there is anything they'd add to the shared responses, so that we craft an exemplar together.

Teacher's Note: The great pizza debate is a very real thing here in New Haven.  If you every find yourself here in our city, make time to try Modern, Pepe's, and Sally's.  I'd add in BAR, too, for their mashed potato pizza.  My personal preference is for Modern, even if the data in this problem suggests otherwise. :) )

Students work independently on the Exit Ticket to close the lesson.