standard_description: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
problem_activity: Activity
In the previous activity, students evaluated the performance of three students based on the mean and variability. Here they learn the term
mean absolute deviation (MAD)
as a way to quantify variability and calculate it by finding distances between the mean and each data value. Students compare data sets with the same mean but different MADs and interpret the variations in context.
While this process of calculating MAD involves taking the absolute value of the difference between each data point and the mean, this formal language is downplayed here. Instead, the idea of “finding the distance,” which is always positive, is used. This is done for a couple of reasons. One reason is to focus students' attention on the statistical work rather than on terminology or symbolic work. Another reason is that finding these differences may involve operations with signed numbers, which are not expected in grade 6.
Teacher Notes for IM 6–8 Accelerated
The activity Shooting Hoops (Part 2) and the activity referred to in the launch are not included in IM 6–8 Math Accelerated.
During the launch, ask students what they think will be true about distances between data points and the mean, knowing that the mean can be considered a balance point. (The sum of the distances to the left of the mean is the same as the sum of the distances to the right of the mean.) Then tell students that the distances can also tell us something else about a distribution that will be explored in this activity.
Launch
Remind students in the last lesson they found distance between each data point and the mean, and found that the sum of those distances on the left and the sum on the right were equal, which allows us to think of the mean as the balancing point or the center of the data. Explain that the distance between each point and the mean can tell us something else about a distribution.
Arrange students in groups of 2. Give students 4–5 minutes to complete the first set of questions with their partner, and then 4–5 minutes of quiet time to complete the remaining questions. Follow with a whole-class discussion.
Student Facing
The tables show Elena, Jada, and Lin’s basketball data from an earlier activity. Recall that the mean of Elena’s data, as well as that of Jada and Lin’s data, was 5.
Record the distance between each of Elena’s scores and the mean.
{"table": [{"_attributes": {"border": "1", "style": "width:500px"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "Elena"}]}, {"_value": "4"}, {"_value": "5"}, {"_value": "1"}, {"_value": "6"}, {"_value": "9"}, {"_value": "7"}, {"_value": "2"}, {"_value": "8"}, {"_value": "3"}, {"_value": "3"}, {"_value": "5"}, {"_value": "7"}]}, {"td": [{"_attributes": {"style": "width:150px"}, "strong": [{"_value": "distance from\u00a05"}]}, {"_value": "1"}, {}, {}, {"_value": "1"}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Now find
the average of the distances
in the table. Show your reasoning and round your answer to the nearest tenth.
This value is the
mean absolute deviation (MAD)
of Elena’s data.
Elena’s MAD: _________
Find the mean absolute deviation of Jada’s data. Round it to the nearest tenth.
{"table": [{"_attributes": {"border": "1", "style": "width:500px"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "Jada"}]}, {"_value": "2"}, {"_value": "4"}, {"_value": "5"}, {"_value": "4"}, {"_value": "6"}, {"_value": "6"}, {"_value": "4"}, {"_value": "7"}, {"_value": "3"}, {"_value": "4"}, {"_value": "8"}, {"_value": "7"}]}, {"td": [{"strong": [{"_value": "distance from\u00a05"}]}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Jada’s MAD: _________
Find the mean absolute deviation of Lin’s data. Round it to the nearest tenth.
{"table": [{"_attributes": {"border": "1", "style": "width:500px"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "Lin"}]}, {"_value": "3"}, {"_value": "6"}, {"_value": "6"}, {"_value": "4"}, {"_value": "5"}, {"_value": "5"}, {"_value": "3"}, {"_value": "5"}, {"_value": "4"}, {"_value": "6"}, {"_value": "6"}, {"_value": "7"}]}, {"td": [{"strong": [{"_value": "distance from\u00a05"}]}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Lin’s MAD: _________
Compare the MADs and dot plots of the three students’ data. Do you see a relationship between each student’s MAD and the distribution on her dot plot? Explain your reasoning.
[IMG]
Student Response
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Student Facing
Are you ready for more?
Invent another data set that also has a mean of 5 but has a MAD greater than 2. Remember, the values in the data set must be whole numbers from 0 to 10.
Student Response
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Anticipated Misconceptions
Students may recall the previous lesson about thinking of the mean as a balance point and think that the MAD should always be zero since the left and right distances should be equal. Remind them that distances are always positive, so finding the average of these distances to the mean can only be zero if all the data points are exactly at the mean.
Activity Synthesis
During discussion, highlight that finding how far away, on average, the data points are from the mean is a way to describe the variability of a distribution. Discuss:
“What can we say about a data set whose data points have very small distances from the mean?”
“What about a data set with points that show large distances from the mean?”
“Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?”
“What do MAD values of 2, 1.5, and 1 mean in this context?”
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary.  Include the following term and maintain the display for reference throughout the unit: mean absolute deviation (MAD). Invite students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language
Listening, Speaking, Representing: MLR8 Discussion Supports.
To provide a support as students explain their reasoning, invite students to use the sentence frame: “The largest (or smallest) mean absolute deviation value matches with the dot plot with the _____ spread because . . .”. This will help students produce explanations connecting visual and numeric representations of data distribution.
Design Principle(s): Support sense-making


aligns: no