The Absolutes of Mean Absolute Deviation

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Objective

SWBAT calculate the mean absolute deviation of a given data set.

Big Idea

Mean Absolute Deviation: The Absolute Truth

Curriculum Reinforcer

10 minutes

The curriculum reinforcer, is a daily practice piece that is incorporated into almost every lesson to help my students to retain skills and conceptual understanding from earlier lessons. My strategy is to use Spiraled Review to help my students retain what they learned during the earlier part of the year. This will help me to keep mathematical concepts fresh in the students mind so that the knowledge of these concepts become a part of students' long term memories.

Engagement

10 minutes

To foster understanding of the meaning of the word deviate, I will show my students a map of a trail. I will tell them that we are taking a virtual nature walk and that this is the path that we are to take. I will provide the students with three different scenarios. One person who stayed on the path as directed and two who deviated from the path… one person more than the other. This will be used to demonstrate the meaning of the word deviation. I will then show them on average how much the two deviated from the path.

Please view the attached video of the Nature Walk Scenario to see how I use it in my classroom to help my students to understand the concept of Mean Absolute Deviation.

Also attached to this section of this lesson is a picture of the nature walk that you can download and place in a worksheet or present on a digital projector... ultimately it is there for you to use as you please.

Instructon & Teacher Modeling

10 minutes

To begin instruction, I will first ask the students to provide the definition of mean absolute deviation. We will then discuss as a group what we think that means based upon what we just did during the opening activity. During this discussion, I will make sure that I emphasize that the mean absolute deviation is a measurement of variation and that the larger the number the more the data varies. The smaller the number the less the data varies.

 

To begin the instructional portion of this lesson, the students will participate in an activity where they will be able to see what mean absolute deviation is and discuss the significance of this measurement as it pertains to data. In this activity, the students will be given a sheet of grid paper. However, the squares created by the grid lines will be 1 inch by 1 inch. The students will graph a given set of data in the format of a bar graph using the grid paper (DATA SET: 5, 6, 2, 2, 4, 1, 7, 4). They will color all of the bars one color. Then, the students will find the mean of the data and mark the mean as a horizontal line that crosses the y-axis at the number that represents the mean. Next, the students will take a look at each of the bars and if the bars are below the line that crosses the y-axis, then they will place an X in the boxes that it would take to reach that line. If the bars are over the line then the students will place an X on the boxes that they would need to take away to be at the line that represents the mean. When the students do this, I will then demonstrate to them that this is the same as subtracting the mean from each of the data values. Once they understand this, then I will have them to find the mean of the differences. When they have done this, I will let them know that they have found the mean absolute deviation. I will then have my students turn their paper to see how the mean absolute deviation is a measurement that gives the average distance that each data point is away from the mean. The students can compare this to what was presented to them during the engagement portion of this lesson.

 

After completing this activity where students have the opportunity to explore the concept of mean absolute deviation, I will then, ensure that my students can relate the activity to the standard algorithm for finding mean absolute deviation. It is very important to relate the two so that students can visualize the activity when using the standard algorithm.

Try It Out

5 minutes

In this next section of this lesson, I will guide the students through one example of finding the mean absolute deviation. Then, I will give them one data set to try on their own. The students will have 4 minutes to complete the example and then, I will choose one student to come up to the front of the classroom. They will place their work underneath the document camera so that they can present to the class how they found the mean absolute deviation of the data set they were given.

 

Example #1:

6, 4, 2, 6, 7, 4, 4, 3, 7, 4

 

Example #2 - Try it on your own:

2, 4, 7, 2, 3, 5, 3, 5, 2, 4

 

 

Independent Exploration

20 minutes

Once we have worked out all the kinks in the examples, I will then provide my students with an opportunity to work on some problems on their own.

 

Task #1:

First, I will put my students in partnerships. Where uneven numbers occur, I will have a group of three students. My students will find the mean absolute deviation of the following two sets of data using the "fair share" method on chart paper with grid lines.

  • Set #1 - 8, 13, 7, 9 ,2, 11, 5, 3, 10
  • Set #2 - 23, 15, 31, 27, 19, 20, 18, 29, 13

 

Attached to this section are some videos of students using the "fair share" method to find the mean absolute deviation of a given data set. They did an excellent job, asked excellent questions, and they were able to articulate their understanding once they were finished. 

 

Task #2:

I will also have my students complete a worksheet individually, so that they can demonstrate that they understand the concept of the mean absolute deviation and have the skills necessary to calculate the mean absolute deviation without the help of a peer.

Closing Summary

15 minutes

To close out this lesson, I will allow students to collaborate for about 10 minutes, with their partner on the independent exploration worksheet. While collaborating, the students should do the following:

  • Discuss their method of solving the problems
  • Discuss any problems they may have run into
  • Compare and contrast their solutions and solution pathways
  • If any of their solutions do not match, they are to determine, together, the correct solution to each problem.
  • Discuss what they they liked and did not like about this lesson, what they learned, and how what they learned applies to real life. They are to create a bulleted list of their answers to this particular part of their collaboration

Select partner sets will be chosen to present their work from the independent practice portion of the work period. They will present their chart paper as well as share the key points discussed during their collaboration time.

Observing students will comment, critique, and ask questions and show that they are actively engaged in the discussion that is taking place during the presentations.