Day 138: Appropriate Measures

6th Grade Math Standards: 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 3.

6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

The Learning Objective: Determine what measure of central tendency is most useful to use under different constraints in a data set; Identify three types of variation; Find the interquartile range of a data set

Agenda:

1. Jumpstart finding a range and interquartile range
2. Kinesthetic description of vocabulary terms range, interquartile range, and outlier
3. Kinesthetic description of how range, interquartile range, and outlier are measures of variation
4. Review homework problems number three and four from the previous class
5. Appropriate measures of variation notes 
6. Exit ticket of describing what central tendency in a data set of 11, 13, 13, 13, 12, 13, 13 is appropriate and also finding the interquartile range of a different data set
7. Students worked on their homework which included Lawyer's Salaries and NFL Penalties

The Assessment: I warned the students that appropriate measures was not the most exciting topic in the world and for some reason focus seemed to increase. What I initially did was activate their minds by asking what central tendency was used to describe their grades. Students struggled with this more than I would have imagined but eventually stumbled on the mean. I then used the example of a student who had nine 100s on a weekly quiz and one 0, and asked the students to describe this person as a math student. I told them honestly that if I were to describe this student I would use the median or even more likely the mode. That kicked started us to get a more narrowly defined idea of when to use which central tendencies.

I summarized it as the book did by stating that:

• Mean gets used with data sets that have no outliers
• Median gets used when there is an outlier and if the data does not have any gaps in the center (I used the analogy of grandparents day at a nursery school and took the ages of people to explain the gaps). 
• Mode gets used when there are many repeated numbers (there is a little more of a broad definition than the other two).

When the students were left to their own devices after we did two examples together they did not have much trouble. They actually convinced me that a data set of 1, 2, 3, 4, 5 can be appropriate and inappropriate for the median at the same time since there are no outliers but also no gaps.

In addition to the appropriate measures exit ticket, students also had to determine an interquartile range as part of the exit ticket. In one class, this took students twenty minutes. Students are struggling as expected because of the number of steps involved. That said, eventually all students finished this task. It was a clear day where having double-blocked math classes was a benefit because I would never allow twenty minutes for an exit ticket a year ago. Given that there was so much time to play with, I only gave subtle hints and suggestions to help students along in getting the interquartile range. The jumpstart, homework and previous day's work made this process a little less painful than it could have been otherwise.

A third assessment came in checking student homework. Students really struggled to find the error that someone else made in calculating the interquartile range of a data set with nine values. They had no trouble with the median but getting the quartile values proved difficult.

Glass Half-Full Take: In my last class it was wonderful having the Title I teacher in the room. He took half the students and I took half. Students struggled but didn't lose focus as they tend to do when they struggle and cannot get teacher assistance. I was in close proximity to all students and only guided them to the next step at most when they needed my assistance. Sometimes I simply referred students back to their notes. As quick as my feedback was, it would not have been possible without a second teacher in the room.

I was encouraged by the ease of which students recognized what an appropriate measure of central tendency was.

Finally I was also satisfied with how I presented to students what a measure of variation was kinesthetically. Our definition as the textbook provides is that a measure of variation "describes the distribution or spread of the data." This definition means nothing to the students as the term "distribution" does not help anything. I have found that changing the definition to "how spread out the numbers are" is helpful. I will likely modify to this definition going forward as students get a clear picture of what this means.

One Regret: As students worked on the Lawyer's Salary data, I kind of regret not giving them a calculator to find the mean. They get lost in adding and dividing six digit and seven digit number respectively. Consequently when it comes time to analyze an appropriate measure without an outlier (today's objective) they are not in class and could bypass the importance of this question.

Link of the Day: The gender pay gap is still relevant today.