The lesson focuses on the concepts of dot plot representations and the mean absolute deviation for data sets. Students will: 
- create and explore dot plots. 
- analyze data using mean absolute deviation. 

- How can we use probability and data analysis to make predictions? 
- How may data be arranged and represented to reveal the relationship between quantities? 
- How can mathematics help to quantify, compare, depict, and model numbers? 
- How are mathematical properties of objects or processes measured, calculated, and/or interpreted? 

- Dot Plot: A graph showing the frequency of data values plotted on a number line. 
- Mean Absolute Deviation: A measure of the spread from the mean of a data set.

- Exit Ticket (M-6-3-3_Exit Ticket and KEY)

- Use a comprehensive approach to evaluate the construction and computation of the Brothers and Sisters dot plot and mean absolute deviation, as well as the discussion questions concerning hypothetical changes to the data. 
- Use a comprehensive approach to evaluate the production of groups' dot plots and the class's ranking of questions based on mean absolute deviation. 
- The Exit Ticket can be used to assess students' knowledge. 

Scaffolding, Active Engagement, Modeling and Explicit Instruction 
W: The lesson teaches students how to generate dot plots using their own or provided data, allowing them to investigate the spread and variability of the data. Students will study how a numerical measure (such as mean absolute deviation) can reflect the overall nature of a dataset, as well as how outliers affect such measures. 
H: Students will analyze data generated from the world around them, including data generated from their own personal experiences. They will be able to notice how some categories of data have data points that are widely distributed, showing significant differences in personal experience, whereas other categories contain data points that are closely clustered together, representing similarities. 
E: Students will develop dot plots based on their own and their classmates' data. They will next use the data that reflects information about their world to calculate the mean absolute deviation and consider what that means about that data. They will also look at multiple data sets holistically and make educated assumptions regarding the mean absolute deviation. This exploration is detailed enough to educate students how to calculate the mean absolute deviation but being wide enough to teach them how to estimate the same. Students may quickly and efficiently study the concept of mean absolute deviation by eliminating the need for repeated calculations. 
R: Students can learn about mean absolute deviation through dot plots and predictions using a range of real and theoretical data samples. Students will be able to compare their predictions to the actual values generated from data sets. 
E: Students will assess their grasp of the lesson by comparing their predictions to the actual, calculated outcomes. They will be able to compare their calculated outcomes to those of their classmates to ensure accuracy. 
T: Students who struggle with manipulating data sets and performing a complex series of computations will benefit from focusing on the overall concepts and predictions in the lessons. In contrast, students who struggle with holistic concepts will be able to concentrate on manipulating data sets and using the data to perform a complex series of calculations. Students will be encouraged to create their own data sets customized to their specific interests. 
O: This lesson takes an investigative, hands-on approach. Students will begin working in large groups, directed by the instructor, to learn fundamental concepts such as dot plots. Students will continue to work in large group, but with guidance from classmates, to further develop these concepts. The transition from instructor-to-classmate-guided activities will help students recognize their knowledge of the content. Students will next demonstrate their mastery on an individual level, allowing them to see how the concepts can be applied broadly. 

Remind students of the various ways they can express data in a single variable, such as bar graphs, circle graphs, line graphs, etc. (You can ask them to go to the Representations handout from Lesson 2.) Explain how a bar graph can be used to display data in non-numerical categories. For example, to demonstrate how many people have dogs, cats, fish, and so on, you could make a bar graph with each bar representing a different type of pet.

Activity 1: Creating a Dot Plot

Draw a number line on the board. Tell students that in order to illustrate a dot plot, they will help you in creating a dot plot using data about themselves. A dot plot can only represent numerical data sets. For example, a dot plot could represent the number of victories in a baseball league, plant heights, or temperature. Tell students that their first dot plot will be on how many brothers and sisters they each have. 

Under the number line, write "Brothers and Sisters."

Say, "Now we will make a dot plot. We understand that the data represents the number of brothers and sisters we have. What number would be appropriate to place at the left-most edge of the number line? In other words, what is the smallest possible response to the question, 'How many brothers and sisters do you have?'" 

Encourage student responses to use zero as the smallest number, and designate the leftmost vertical mark on the number line with a zero. 

Say, "Now think about the largest reasonable answer someone could give when asked how many brothers and sisters s/he has."

Work with the class to determine what a reasonable answer would be for the class. For instance, if someone suggests ten, ask whether anyone in the class has ten or more siblings. Once you've determined the largest reasonable answer, label it and place a vertical mark at the right end of the number line. Next, fill in and label the vertical marks between zero and the largest reasonable answer. 

Ask students if the numbers on the number line would be the same in another class. Discuss how the specific values on the number line are determined by both the question and the population being asked.

Instruct students to approach the board and insert an X (indicating the number of brothers and sisters the student has) over the appropriate on the dot plot. 

When the plot is finished, show students the similarities and differences between dot plots and bar graphs (the "height" above each value along the x-axis represents how many times that value appears in the data set) and differences (a dot plot would not show how many people have different pets, whereas a bar graph would). Remind students that their number line is really just an x-axis and should have consistent spacing between values.

Students should generate a list of potential questions for which the resulting data could be represented using a dot plot. Students should write out their suggestions.

Activity 2: Describing the Distribution

Say, "Imagine you're explaining the data in the 'Brothers and Sisters' dot plot to people who haven't seen it before. What would you say to them? How would you describe it?"

Encourage students to discuss how the data is "clumped" around specific numbers (likely 2 and 3) and how spread out (or not spread out) it is. Explain to students that showing someone how spread out data is can provide a rapid understanding of the overall "shape" of the data set. In mathematics, we try to quantify things; rather than simply saying something is "kind of spread out" or "clumped together," we like to give particular numbers to these kinds of general statements. In this case, we're going to discuss the mean absolute deviation. (Use the term "mean absolute deviation."

Underline the word "mean."

Ask students, "What does this word mean in mathematics? Can you explain it?" (The mathematical mean represents the average value of a set of data.) 

Follow this up with a quick review of how to calculate the mean. 

Underline the word "absolute." 

Ask students, "What does this word mean in mathematics? How do you explain it?" (Absolute is used to refer to absolute value; absolute value represents a number's distance from 0 on a number line.) 

Underline the word "deviation."

Ask students, "What does this word mean in mathematics? How do you explain it?" (Students may not have a mathematical background for what deviation means, so focus the discussion on the fact that deviation deals with differences.)

Draw a circle around the words "absolute deviation" and instruct students to think of those words as a group—the positive difference between two values. "When we discuss a difference, we must consider the difference between two values. In each case, one of these values will be the mean of the data set." 

Assign students to work in pairs to calculate the mean of the data set represented by the "Brothers and Sisters" dot plot. Write the calculated mean on the board.

Now, return to the word "absolute deviation" and circle the first X on the dot plot, which most likely represents 0. Determine the "absolute deviation" between the value represented by the circled X and the mean. Introduce students to the concept that the difference is just the absolute value of the difference between the mean and value. Write down the difference, then proceed to the next X on the dot plot (which could be in the same or next column). Record the absolute deviation for this value. Note that the absolute deviation will be the same for all values in the same column.

Find various absolute deviations so that students have a general understanding. Working in pairs, have students record all of the data's absolute deviations. 

Return to the term "mean absolute deviation" and place a checkmark above "absolute deviation," since all absolute deviations have been found. 

Say, "Now that we've found all of the absolute deviations, we'll look for the mean absolute deviation of the data. What do you think the mean absolute deviation will tell us? How do you think we can determine the mean absolute deviation?" (Students should guess that they need to calculate the mean of those values.)

Students should continue working in pairs to determine the mean absolute deviation of the data set. Write this value on the board and tell students it represents how "spread out" the data is. 

Ask the following questions: 

1) "What would happen to the mean absolute deviation if a new student entered the class and that student had ten brothers and sisters?" (It would increase.) 

2) "What would happen to the mean absolute deviation if every student in the class had the same number of brothers and sisters?" (It would be zero.) 

The second question should prompt a discussion on the mean being equal to the number of brothers and sisters and the mean absolute deviation being zero, since the absolute deviations are all zero.

3) "What would happen to the mean absolute deviation if every student in the class had two siblings except one had three siblings?" (It would be quite close to zero, just slightly higher.) 

4) "What if every student in the class had two siblings, but one only had one? How does that compare to the case where you have three siblings?" (The mean absolute deviation would be very close to zero, just slightly lower.) 

5) "What if every student in the class had one additional sibling?" (The mean absolute deviation would remain the same.)

Ask students to generate questions that will result in data sets with a mean absolute deviation greater than the Brothers and Sisters data set. Repeat the brainstorming questions to generate data sets with a mean absolute deviation less than the Brothers and Sisters data set.

Activity 3: Making Your Own Data Set

Students should collaborate in small groups (4 to 5 total) to come up with a question to ask their classmates that will provide results suitable for display in a dot plot. Encourage students to survey their classmates and create their own dot plots (one per group).

After each group has created their own dot plot, write the four or five survey questions on the board and ask them to rank the results based on what they think the mean absolute deviation will be.

Before asking each group to determine the mean absolute deviation for their data, investigate the predictions and the reasons behind them. 

After all of the mean absolute deviations have been calculated, have each group report their deviation and determine the order in which they occurred. Did any groups get the order exactly correct? Did all of the groups agree on the question that would produce data with the greatest (or least) deviation?

Investigate some reasons why the actual results did or did not match the class's predictions.

Extension: 

Routine: Instruct students to design and carry out surveys with mean absolute deviations greater (or less) than the mean absolute deviation of the Brothers and Sisters data set, or ask students to generate a dot plot for ten hypothetical data values with a mean absolute deviation less than a given value.

Small Group: Have small groups prepare five survey questions that can be used to generate data for a dot plot. Have groups exchange question lists and rank hypothetical results based on predictions of the mean absolute deviation. Have several groups rank the same set of five questions and compare the results.

Technology: This lesson makes use of scientific calculators to calculate the mean absolute deviation.