Download Lesson Plan - Howard County Public School System

Lesson Title: Mean Absolute Deviation Course: Mathematics 6
Date: _____________ Teacher(s): ____________________
Start/end times: _1 – 50 minute class_
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
6.SP.B.5 Summarize numerical data sets in relation to their context, such as by:
5c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or
mean absolute deviation), as well as describing any overall pattern and any striking deviations from the
overall pattern with reference to the context in which the data were gathered.
MP2: Reason abstractly and quantitatively.
MP4: Model with mathematics.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
The amount of snowfall for the 2015 winter in cities
along the East Coast was recorded in the table below:
City
Boston
Snowfall
Total (inches)
99
NYC
37
Atlantic City
13
Washington, DC
15
Richmond
Philadelphia
7.5
27
Baltimore
20
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Circle- UP!
Since this is a lesson that covers many aspects of
statistical data, have the students stand up as a class and
pass a ball to each other. When they catch the ball, the
student has to say one thing about statistical data from the
lesson today. Once a student has said something they
pass the ball to another person, that person has to say
something different. If a student repeats, or cannot think
of something within 10 seconds, they have to sit down.
The last student standing wins!
1. Create a box plot for the data above.
2. Use the data below to determine the mean amount
of snowfall for these cities along the East Coast.
3. Which meaning of the mean are we looking for in
this data? (leveling)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Review the lesson launch with students. (The goal of this lesson launch is to engage in mathematical discourse
among students to discuss how the average or mean is not a representative statistic for this data set. This is
because Boston’s statistic of snowfall is considered an outlier.) Lead a discussion with students about the
distribution in the box and whisker plot and where we can see most of the data is clustered below the mean.
2. Review student understanding of outliers. (Note: If students have not yet calculated outliers in data, show them
how to use the formula Q1 -1.5 ´ IQR or Q3 +1.5 ´ IQR .) In this case, 99 is a clear outlier for this data. Show
students how to then create a modified box plot to show the outlier in the data. Discuss with students why it is
important to identify outliers in the data set and how the outliers affect measures of center, specifically the mean.
3. Assign students to groups of 2-3 and distribute sheet protectors, dry erase markers, student resource and this
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Mean Absolute Deviation Course: Mathematics 6
Date: _____________ Teacher(s): ____________________
Start/end times: _1 – 50 minute class_
problem.
Mr. Statdad, was shocked to learn about the amount of battery life his 6th grade son’s cell phone had when he
returned home from a school day. He decided to contact a sample of 6th grade students in his neighborhood
and asked them to record their hours of battery life remaining after a school day.
4. Allow students about 5-7 minutes to work together to answer questions a, b, and create graphs for c. Discuss the
answers students have for a and b, then have students share out which graphical representation they chose to use
and why. Ask, “What does each display reveal about the data?” What (if anything) does it conceal about the
data?”
5. Discuss types of displays (dot plots & stem & leaf- reveal individual data and distribution, histograms-reveal
distribution and begin to group data, box plots-see a snapshot of distribution based on quartiles but do not reveal
individual data, mean, or mode).
6. As a class, discuss which graphical representation would be the best one to use to get a good representation of
the data (shape and spread). Say, “Let’s begin by focusing on the dot plot.” If students did not create a dot plot,
have them do this at this point in the lesson or have a group place one of theirs under the document camera.
Battery Life Dot Plot:
7. Say, “What does the display reveal about the data?” and “What is the measure of center?” Allow time for
students to determine the possible measures of center. Students will use all the numbers because they are not
outliers and the mean and the measure of center. Students may fine the mean =8, median=7, mode-12, or midrange=6 as a measure of center. These are all possible measures of center. Ask, “Does the measure appear to
be a reasonable representation of the various hours of battery life remaining?” Say, “Let’s focus on the mean.
What is a ‘mean’? The question is what does the mean represent in terms of this data?” Remind students that
there are two meanings of the mean. In this context, the mean may be seen as a balancing point for the
distribution. The data point at 2 represents 6 units from the mean (below the mean). The data point at 12
represents 4 units from the mean (above the mean).
8. Using the sheet protectors and dry erase markers, have the students show the mean on the dot plot and the
balancing point of the distribution of the data (see below). Discuss the meaning of the data points each have a
balanced value on each side of the mean.
9. Next, have students use the visual data to complete the table provided on the back of the student resource. If
necessary, students can use a calculator, as they may not have the skills to subtract to get negative values. In
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Mean Absolute Deviation Course: Mathematics 6
Date: _____________ Teacher(s): ____________________
Start/end times: _1 – 50 minute class_
this case, the above picture is displaying distances and not actual values. The data point of 7 is -1 away from
the mean or 1 away below the mean.
10. Once students have completed the table, and determined the sum of the distances from the mean, they will then
be ready to determine the Mean Absolution Deviation. (Note: Students can also calculate the sum of the
deviation of the mean and notice that it will sum to 0, which reinforces a balancing point for the mean.) Have
students calculate the Mean Absolution Deviation (MAD):
Calculating Mean Absolute Deviation:
sum of the distances from the mean
= MAD
total # of data
11. Emphasize the following: The mean absolute deviation of a set of data is the average distance between each
data value and the mean. The MAD is a statistic (because it is describing a sample) and is a useful measure of
the amount of variability within a distribution. If we divide this sum of the absolute distances on either direction
of the mean (14 +14=28) by the total number of data points (9) we get 3.1 hours. Thus, for these 9 battery life
hour times, the individual battery life remaining varies from 3.1 hours from the mean. On average by about 3.1
hours in either direction above or below the mean. The smaller the MAD, the less variation, the larger the
MAD the greater the variation.
12. Have students mark the MAD on each side of the mean on their dot plots. Then consider how many data points
are in between—this will not be 50% because it is not quartiles! This is showing that all of the data points are
closer to the mean than the average distance. If a lot of points are clustered here, then it may indicate that the
data within is “typical”. If a 6th grader says that his/her cell phone battery life has 11 hours remaining, this value
is inside of the MAD, so it would be within a typical range of battery life. This is NOT to say that if a value is
outside of the MAD that it is atypical, since you also have to account for the distribution and whether or not the
data is an outlier.
13. Have students revisit Mr. Statdad’s problem with his son’s batter life hours on his phone. Based on his findings,
what conclusions should he make? (Possible student response: If his son’s battery life falls within the mean
absolute deviation of the data, 3.1 hours in each direction of the mean from the data collected, then he would
have no reason to replace the phone or get a new battery.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Students should be able to identify outliers in a data set and the impact outliers have on the measures of center,
specifically the mean.
Students should be able to easily explain the two definitions of the mean (leveling of data and balancing point of
data).
Students should be able to understand and explain the mean absolution deviation of a set of data as the average
distance between each data value and the mean. It useful in measuring the amount of variability within a data
distribution.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
For the batter life data, the sums of the distances below and above total to 14 (if we neglect direction). If we account
for direction and take the sum of all of the distances, we get 0, which reinforces this idea of balance.
The mean absolute deviation of a set of data is the average distance between each data value and the mean. The
MAD is a statistic (because it is describing a sample) and is a useful measure of the amount of variability within a
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Mean Absolute Deviation Course: Mathematics 6
Date: _____________ Teacher(s): ____________________
Start/end times: _1 – 50 minute class_
distribution. The smaller the MAD, the less variation, the larger the MAD the greater the variation. The MAD
simply gives a statistic about the variation of the data from the mean. It is precursor to students understanding
Standard Deviation in Algebra 1.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Sheet protectors (sleeves)
Dry Erase Markers
Mean_absolute_deviation_resouce
Calculators
Use the data below to:
a. Create a dot plot
b. Determine the mean
c. Show the deviations from the mean on the dot plot
d. Determine the Mean Absolute Deviation for the
data and explain what it means in the context of
the problem.
The following data was collected on the amount of runs
scored by the top 12 Baltimore Orioles players during the
2014 season:
87, 88, 81, 56, 48, 65, 51, 38, 33, 27, 22, 17
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Do students understand what the mean absolute deviation represents for a given data set?
What will I do tomorrow based on my students’ current understanding?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.