Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
- Students will interpret the differences in shape, center, and spread in the context of the data sets
- Students will identify which measure of center is affected the most by the existence of extreme data points (outliers)
- Students will identify which measure of spread, the interquartile range or range is affected the most by the existence of extreme data points (outliers)
Prior Knowledge: What prior knowledge should students have for this lesson?
- Students should know how to construct a standard box plot and a histogram
- Students should know how to determine the shape, center, and spread of the data
Guiding Questions: What are the guiding questions for this lesson?
How can the statistics and extreme values be used to persuade MLB negotiations?
Teaching Phase: How will the teacher present the concept or skill to students?
After the students have finished the Activating Activity (Formative Assessment), which should take about 10 minutes, show the students how to find outliers using the rule that points more than 1.5 times the interquartile range from Q1 and Q3 are generally considered outliers (Q1 – (1.5*IQR) and Q3 + (1.5 *IQR). The students will then determine which extreme data points are outliers. (For this data the IQR is $44m, IQR times 1.5 plus Q3 is $208, therefore $219 and $273 are outliers.)
After identifying outliers the students should recalculate the mean and medium without them. (Included are these calculations plus both boxplots in the attachment "Activation Boxplots."
Then have the students discuss the changes in the mean and median. They should discover that when the outliers are eliminated the mean and median are much closer with the mean decreasing a lot more than the median. This should lead the students to conclude that the mean was affected more by the outliers than the median.
The students will then construct a new boxplot without the outliers and compare this boxplot to the original boxplot. Tell them to pay special attention to the changes in their 5 number summary values. The students will then discuss how the outliers affected the range and interquartile range. The students should conclude that the range was affected much more than the interquartile range.
This portion of the lesson should also take about 15 minutes to complete.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Next display 3 boxplots, one for each of the teams with the top 3 payrolls. The students will compare and discuss what the boxplots reveal about the data. Included are these as the attachment "Guided Practice Boxplots." The students should discuss how the range, IQR, mean and median for each of the boxplots differ and what kind of inferences they can make about the data.
What the students should see:
- The minimum salaries for all three teams are very close. As a matter of fact, the league minimum salary is $507,500, which is the minimum for both the New York Yankees and the Los Angeles Dodgers. The Boston Red Sox minimum salary is just $1000 more.
- The Q1 values are less than $600,000 for both the Red Sox and the Yankees, which means at least 7 of the 30 man roster of players are within $50,000 of the league minimum salary, where as the Dodgers Q1 value is almost $200,000 more than the league minimum.
- For both the Red Sox and the Yankees the median values are very close (within $25K), where as for the Dodgers it is substantially higher.
- The distance between the median and Q3 value and the distance between Q3 and max value for New York are almost equal, this means the spread in salaries for the top 50% of players is relatively consistent and much greater than the lower 50%.
- The range for the Dodgers is a lot greater than the ranges for the other two teams and since the IQR is much shorter than the distance between Q3 and the maximum value there seems to be at least one outlier salary for the Dodgers.
- The IQR for the Dodgers plot is smaller than the other two plots and since the median is also twice what the other teams are this means that the Dodgers have a lot more higher salaried players than the other two teams.
This portion of the lesson should take 15 minutes depending on how much scaffolding the students need and how their discussions proceed.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
Students will be given a spreadsheet with individual player's salaries for two MLB teams and asked to find the mean, 5 number summary (min, max, Q1, Q3, and median) for each team's salary. The students should also calculate the mean, range and IQR for each team.
Students will construct a boxplot for each set of data and then asked to compare and explain what the differences between the boxplots tell them about the data.
Students should then determine if any of the salaries are outliers and if so eliminate them and recalculate the mean, median, range and IQR.
Students will then be asked to pick a position at the negotiating table as either the owner's representative or the player's representative and then construct an argument using central tendency and spread to support their position.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students will be asked to write a paragraph describing what they have learned about how the mean and median are affected by outliers. They should include how they can identify extreme data points as outliers as well as some ways they can show visual representations of the data (box plots and histograms).
See attachment labeled assessments for the following below:
- Students will be given the box plots for three MLB teams that represents the salaries for individual players and asked to interpret what the box plot reveals about the data.
- Students will be asked to compare and discuss what the differences between the boxplots tell them about the data.
- Student will be asked to take on the role of a sports agent representing a player and determine which measure of central tendency should be highlighted to negotiate the best salary.
- Students will be given a spreadsheet and asked to identify outliers based on data more then 1.5 x IQR + Q3 and Q1- (1.5 x IQR). After removing the outliers, the students will recalculate the mean, media, range and IQR and explain which central tendency and measure of spread was affected the most by the outliers.
- Used as an activating strategy for this lesson.
- Students will be given a spreadsheet containing the Payrols of all MLB teams.(either displayed on a document camera or printed out and distributed)
- Students are to construct a boxplot and a histogram with data from spreadsheet.
- When students are constructingaboxplot (box-and-whisker plot) and a histogram:
- Teacher probing question: What are the 5 numbers needed to construct a standard boxplot?Response: (minimum value, maximum value, median, first quartile and third quartile)
- Teacher probing question: How many bars will you use for your histogram?
Response: (Answers will vary but most will use either 5 or 10).
- Teacher probing question: Does having more bars reveal any thing about the data that you might not see if you used just a few bars?
Response: (Using 10 bars shows that the largest value is probably an outlier where as with only 5 bars this could easily be missed.)
- When students are in small groups comparing answers and discussing extreme data points:
- Teacher probing question: Are there any data points that seem to be much higher or lower than the rest of the data?
Response: (Yes, $273 million and maybe even $219 million)
- Teacher probing question: What does the range of the data represent?
Response: (the difference between the maximum and minimum payrolls).
- Teacher probing question: How do you calculate the interquartile range and what does it represent?
Response: (subtract Q1 from Q3, it represents the middle 50% of the data and how spread the data is)
Feedback to Students
Teachers will be giving feedback to students verbally, as well, as students will be giving feedback to each other when discussing in their small groups.