1. A Story of Functions Module 2: Modeling with Descriptive Statistics
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July 2013
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A Story of Functions
Module 2: Modeling with Descriptive Statistics
Introduce the module as the second module of the 9th grade A Story of Functions. The module title, Modeling with Descriptive Statistics, continues students’ work with data distributions and data representations.
This module extends the introduction to data distributions started in grade 6, and continued in grades 7 and 8.

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Modeling

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Modeling Cycle

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July 2013
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Session Objectives
Discuss the key ideas of what students learn in the module Modeling with Descriptive Statistics.
Examine sample problems to learn how students progress through the Common Core Standards in Statistics and Probability.
Participate in the process of how students learn the big ideas of this module.
Introduce the objectives of this presentation. Discuss with participants that the primary objective of this session is to learn more about the lessons in this module and what students will investigate. These objectives will be developed by working through a few problems, and connecting these problems to the standards.

5. Module Topics Topic A: Shapes and Centers of Distributions
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Module Topics
Topic A: Shapes and Centers of Distributions
3 Lessons
S-ID.1 / S-ID.2 / S.IS.3
Topic B: Describing Variability and Comparing Distributions
5 lessons
Mid-Module Assessment
Summarize the next two slides as the organization of the grade 9 module, and the connections to the standards.

6. Grade 9 – Module 2 Module Focus Session
November 2013
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Module Topics
Topic C: Categorical Data on Two Variables
3 Lessons
S-ID.5 / S-ID.9
Topic D: Numerical Data on Two Variables
9 Lessons
S-ID.6 / S-ID.7 / S-ID.8 / S-ID.9
End-of-Module Assessment

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Update
1. Where are you, or the teachers you work with, in this module? 2. How is it going? 3. What specific challenges are you and your colleagues facing?
Explore with the participants the general questions posted on the slide. Chart their responses. There may be several responses, and then there may be very few. After several questions or comments have been stated (and posted), provide a general answer if possible. Move to a specific topic as outlined in this powerpoint (Topic A or Topic B or Topic C or Topic D) that would best address the questions. If participants have just started the module, and question the overall goals of the Module, start with Topic A. If several questions indicate that participants have completed the first 4 lessons and are asking about the other topics, begin with Topic B or C or D based on participants’ comments.

8. Activity 1: Distributions and Their Shapes Topic A Examples
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Activity 1: Distributions and Their Shapes Topic A Examples
Organize in small groups. Discuss and complete selected problems from Lesson 1 of this module. After you have completed the problems, we will discuss how students learn to uncover the story behind the data. What do students specifically address as they summarize data distributions?
Point out to participants that the first activity is to complete selected problems from Lesson 1 of the module. Briefly go over the format of the first lesson, namely, it is series of questions connected to data distributions represented by either a dot plot, a histogram, or a box plot. Summarize the primary goal of the lesson is to provide students a bridge from what they learned in their previous grades to where they are going with this work. The data distributions provided in this lesson include distributions from their work in the statistics and probability modules of grades 6, 7, and 8. It also opens up new questions that will be addressed in this module.

9. Key ideas of this set of problems
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Key ideas of this set of problems
• Students use informal language to describe the shape, center and variability of a distribution based on a dot plot, histogram, or box plot. • Students recognize that a first step in interpreting data is making sense of the context. • Students make meaningful conjectures to connect data distributions to their context and the questions that could be answered by studying the distribution.
After participants complete lesson 1, summarize the points made in this slide as the primary goals of the problems in lesson 1.

10. What can you summarize about each of the following data distributions?
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What can you summarize about each of the following data distributions?
Use the question on this slide as a way for participants to connect the problems they completed in Lesson 1 to the goals of this module.

11. A data distribution represented by a dot plot:
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A data distribution represented by a dot plot:
Ask participants to summarize what they think is the story behind this data distribution. Essentially summarize that this is a skewed distribution, with a tail to the right. The distribution indicates that most of the delays of an airline were around 20 minutes. 20 minutes would be approximately the median delay time. If necessary, review with participants how to find the median.

12. A distribution represented by a histogram:
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A distribution represented by a histogram:
The above distribution is summarized by a histogram. A summary would indicate a skewed distribution, with a tail to the right. Ask participants what they can summarize about the data from this histogram. They may indicate various descriptions of the population based on different age groups; for example, 17% of the population of Kenya is 0 to 4 years old. Or, they main summarize that a fraction of a percent of the population are 80 or older. Discuss any summary that participants suggest can be derived from this graph. Also ask participants what summaries cannot be determined from this graph (for example how many people are in this data distribution or how many people are a specific age category).

13. A distribution represented by a box plot:
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A distribution represented by a box plot:
Ask participants again to summarize what is indicated by this graph. Look for descriptions that include the median number of pets owned, or the minimum number of pets owned, or a summary that indicates 50% of the people surveyed owned 2 or less pets. Ask participants to summarize what they think the “ *” at the end of the box plot means (an outlier). Ask them to suggestion an interpretation of the outlier.

14. What questions do students answer to summarize a data distribution?
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What questions do students answer to summarize a data distribution?
What is the shape of the distribution? What is the center of the distribution? What is a measure of the variability of the distribution?
Carefully read through this slide as it summarizes the primary questions that are investigated in Topic A. The slides that follow indicate how these questions are investigated.

15. Shape What is the shape of the distribution? Is it nearly symmetrical?
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Shape
Ask the questions in the slide. Summarize this shape as a data distribution that is skewed.
What is the shape of the distribution?
Is it nearly symmetrical?
Is it skewed?

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Center
Describe the center of a distribution as the value that provides a typical description of the distribution. For a skewed distribution, the median is considered a typical description. In this example, an estimate of the median age would be located within the interval of 15 to 19 years old (approximately 17 years old). This age would describe the typical age of the people in this sample. Indicate that if the distribution was nearly symmetrical, the mean would be a description of the typical age.
What center is the best indicator of the typical value of the data distribution?
Mean?
Median?

17. Spread or Variability of the Data Distribution
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Spread or Variability of the Data Distribution
Variability is linked to the shape of the distribution. For a distribution that is skewed, the variability is described by the interquartile range (IQR), or the difference of Q3 (upper quartile) and Q1 (lower quartile). The IQR is approximated by making a box plot of the distribution.
What is the best measure of spread or variability for a data distribution?
Mean Absolute Deviation (or MAD)?
Interquartile Range (or IQR)?

18. Activity 2: Comparing Distributions Topic B Example (Lesson 8)
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Activity 2: Comparing Distributions Topic B Example (Lesson 8)
Country Data A science museum has a “Traveling Around the World” exhibit. Using 3D technology, participants can make a virtual tour of cities and towns around the world. Students at Waldo High School registered with the museum to participate in a virtual tour of Kenya, visiting the capital city of Nairobi and several small towns. Before they take the tour, however, their mathematics class decided to study Kenya using demographic data from 2010 provided by the United States Census Bureau. They also obtained data for the United States from 2010 to compare to Kenya.
The next example is a comparison of two data distributions. In this example, students compare the populations of Kenya and the United States.

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The representation of a data distribution tells us about the story behind the data. Each representation tells us various summaries of the story. Examine the representations of two distributions. The first representations are histograms of two samples. The first sample is 200 people from Kenya and the second sample is 200 people from the United States. The second presentations are box plots of the same samples. Think of 2 questions students could answer from either the box plot or the histogram or both. Also, think of a question that cannot be answered by either representation.
Read the slide. Challenge participants to provide 3 questions that they might ask of students as they compare two data distributions represented by histograms. Similarly, ask them to compare the data distributions represented by box plots. Indicate that they should consider questions that can be answered by a histogram and not a box plot, or questions answered by a box plot and not a histogram.

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July 2013
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Allow participants to examine the two distributions from the histogram. These two histograms are also part of Lesson 8. Encourage participants to discuss in small groups the exercises related to histograms in Lesson 8, or exercises 1 – 8..

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Discuss the two box plots. Ask participants, “What can you summarize about the data from a box plots that you could not from the histograms?” Encourage participants to work through the exercises involving box plots in Lesson 8, or exercises

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Key Points of Topic B
• Data distributions are represented by dot plots, histograms, and box plots. • Data distributions are defined by their shape, center, and spread. • Distributions that are skewed use the median and interquartile range (IQR) for measures of center and variability • Distributions that are nearly symmetrical use the mean and the mean absolute deviation (or MAD) for measures of center and variability
Read and summarize this slide.

23. Activity 3: Two-way Tables Topic C Example (Lesson 9)
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Activity 3: Two-way Tables Topic C Example (Lesson 9)
Students in a mathematics class to pose the statistical question, “Do high school males have different preferences for superhero powers than high school females?” Use the following table to think about how that question might be answered.
The following example is from Topic C of the module. It indicates how students also work with categorical data. The categorical data is summarized using a two-way frequency table, and examined by frequencies, relative frequencies, and conditional relative frequencies. If time permits, read and discuss the standards connected to the two-way table, or S-ID.5. If time permits, organize participants in small groups to discuss and work through the exercises of Lesson 9.

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To Fly
Freeze Time
Invisibility
Super Strength
Telepathy
Total
Females 49 60 48 1 70
228
Males 51 71 27 25
220
What summaries can you make of the above data that could be used to answer our statistical question? Think of at least two, and discuss.
Continue discussion of the two-way table. Ask the questions in the slide and discuss with participants. Introduce the terms frequency, relative frequency, and conditional relative frequency. Point out to participants that the terms frequency and relative frequency begin in grade 8. The terms conditional relative frequency and association are introduced in grade 9, and further developed in grade 11 as conditional probability.

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July 2013
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X minutes
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Key Points of Topic C
• Categorical bivariate data are summarized by two-way frequency tables. • Conditional relative frequencies are used to evaluate the possible association between two categorical variables. • Many of the 9th grade standards and learning expectations are started in grade 6, 7, and 8, and are continued in this grade.
Read and summarize this slide. If time permits, move to a discussion of the mid-module assessment. Ask participants to complete the first problem. Discuss with participants the connection to the standards.

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Activity 4: Numerical Data on Two Numerical Variables Topic D Example (Lesson 12)
Examine a scatter plot of elevation and mean number of clear days for 14 cities. Do you see a pattern in the scatter plot?
The example presented on this slide is from Topic D, or Numerical Data on Two Variables. The opening problem from Lesson 12 summarize the focus of this topic. Discuss with participants how this investigation involves S-ID.6, or “Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.”

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The scatter plot does not have a strong pattern. Participants may respond that it looks like the data points are randomly scattered. If participants look carefully, however, there is a pattern that suggests as elevation increase, the number of clear days also appears to increase. Direct participants to work in small groups to complete the exercises of lesson 12, or exercises If time permits, encourage participants to work on exercises 4 – 7, or “Thinking about Linear Relationships.” After discussion of the exercises, direct participants to work on exercises , or “Not Every Relationship is Linear.”

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Key Ideas of Topic D
Topic D continues to present scatter plots. Students particularly focus on deciding whether or not a scatter plot has a linear model and that that model tells us about the data. A linear model is introduced in grade 8, and then expanded in grade 9 as students develop a “best-fitting” line. The decision of whether or not the linear model is a good model is based on a residual plot. The next slides introduce the topic of a residual and a residual plot. The topic involving residuals concludes Topic D and the module.
Read and discuss the points presented in this slide. There will not be enough time to fully develop the topic of residuals, however, this may be a topic unfamiliar with participants. As a result, introduce that a residual “looks like” and a general summary of a residual plot. (An entire session could be developed just on these topics!) If interest is expressed by the participants and time is available, encourage small groups work through the exercises in lessons

29. Residuals TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X
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Residuals

30. Residual Plot TIME ALLOTTED FOR THIS SLIDE: X minutes
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Residual Plot
Discuss with participants that a special scatter plot of the residual is used to decide if a linear model is a good model for the scatter plot. The topic of how to use the residual plot in deciding whether or not to use a linear model is developed in lessons 15, 16 and 17. If time permits. Involve participants in selected problems from those lessons.

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Summary
Grade 9 is a key grade level. Review in the Module Overview the Focus Standards. Key ideas: • Data Distributions • The role of shape, center, and spread (variability) in summarizing a data distribution. • Analyzing categorical data from two-way frequency tables. • Association • Linear models and Best-fitting Lines
Discuss with participants the points that were developed in the overview of this module. Read the points on the slide and ask if there are any questions related tot them. If time permits, move to the End-of-Module Assessment and ask participants to complete the first problem.