I start this lesson with a Warm up to access the students' prior knowledge on mean, median, mode, and range. I expect the Warm Up to take about 10 minutes for the students to complete and for me to review with the students.
The purpose of this lesson is for students to look at patterns in the data to recognize the best representation for the Center of the Data. Students should be able to explain their observations both quantitatively and qualitatively(MP2).
I do not present students with any formulas to identify outliers in this lesson. Students are to rely on the quantitative values that they find for the Central Tendencies and the patterns that they notice that will effect any of the Central Tendencies.
The problem that I provide students in the Warm Up is about the mileage of cars per gallon of gasoline. The students are given a list of data, and asked to find the range and the best representation for the Center of the Data.
The given set of data does not vary much between numbers, and it has the following statistics:
1. The mean is 22.3
2. The median is 23
3. The mode is 23
4. The range is 19 (The minimum is 13 and the maximum is 32)
The median and the mean are close together, so both could be a good representation of this data set. The mean provides the average of the data set, so the mean is probably the best representation of the data.
In the final question of the Warm Up, I ask students if the best representation of the Center of the Data would change if 66 was added to the data set. Students should recognize that 66 is different from the pattern of the numbers in the data set, it is an outlier.
Most of the students recalculate all of the statistics above for the new data set. They found that the median, and mode remained the same. The range was now 53. Range is not a Central Tendency, but we discuss in class how it effects the mean when the data is more spread out with an outlier. The mean for the new data set is 24.5, and is not a good representation for the Center of the Data. Students should also use the units of miles per gallon when explaining the differences to refer to the context of the problem.
After reviewing the Warm Up with the students, I provide students with a Yard Sale Activity that they are to work on with their table partner. I already have the table partners seats in homogeneous pairs, so that students are working on the problem with a student with similar skills. This allows all of the students time to think about the problem, and solve it at their own pace.
The Yard Sale Activity is open-ended questions that requires the students to think backwards from the given information. This requires students to create lists of data, and understand the concepts of how the Central Tendencies and Range are formed and affected. I demonstrate two of the three problems in the video below.
After we discuss the Yard Sale activity as a whole class, I provide each student an Exit Slip to complete individually. I use the Exit Slip as a formative assessment to check each students ability to identify and explain the Best Representation of Center for a set of data. I access the Exit Slip from the following website:
http://www.cpalms.org/Public/PreviewResource/Preview/71159 (last accessed 7-12-15)
In this Exit Slip, students have to synthesize what they have learned and use it to compare two given data sets. The data sets are comparing the number of text messages sent per day. The students should already recognize by glancing at the data, that the outlier of 168 is going to affect the mean in Group B.
Students then should calculate the mean and median to support their intuition. When calculating the mean and the median for each group, the students get the following values:
Looking at the results above for Group A, students should be able to identify the mean as the best Center for the data. Students should explain that the mean is the average number of text messages sent each day. The numbers do not vary much and the median is only one away from the mean.
When looking at the results for Group B, students should be able to identify that the outlier of 168 has affected the mean. The best representation for Center of the data for Group B is the median. One way that students may explain that the average amount of text messages was actually higher for Group A if 168 is removed from Group B. Without the outlier, Group B has a mean of approximately 19.25 text messages per day. Since part of a text message cannot be sent, it would refer to about 19 text messages per day compared to about 25 text messages a day for Group A.