© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 6 – Module 6 Statistics (Lessons

© 2012 Common Core, Inc. All rights reserved. commoncore.org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

A Story of RatiosGrade 6 – Module 6Statistics (Lessons 1 to 18)



Session Objectives

1. Brief overview (big ideas) of module 6: Statistics

2. Overviews of Topic A (Lessons 1 to 5), Topic B (Lessons 6 to 11), Topic C (Lessons 12 to 16), and Topic D (Lessons 18 to 21)

3. Examine sample problems in select lessons to become familiar with the progression of the statistics concepts in Grade 6.



Agenda

• Introduction to the Module• Overview of Topic A (Lessons 1 to 5)• Overview of Topic B (Lessons 6 to 11)• Mid-Module Assessment• Overview of Topic C (Lessons 12 to 16) • Overview of Topic D (Lessons 18 to 21)• End-of-Module Assessment• Final comments and questions



Curriculum Overview of A Story of Ratios



Module Overview and Standards

Topic A: Understanding Distributions (6.SP.A.1, 6.SP.A.2, 6.SP.B.4, 6.SP.B.5b)• Lesson 1: Posing Statistical Questions• Lesson 2: Displaying a Data Distribution• Lesson 3: Creating a Dot Plot • Lesson 4: Creating a Histogram• Lesson 5: Describing a Distribution

Displayed in a Histogram




Topic B: Summarizing a Distribution that is Approximately Symmetric Using the Mean and Mean AbsoluteDeviation (6.SP.A.2, 6.SP.A.3, 6.SP.B.4, 6.SP.B.5)• Lesson 6: Describing the Center of a Distribution Using

the Mean• Lesson 7: The Mean as a Balance Point• Lesson 8: Variability in a Data Distribution• Lesson 9: The Mean Absolute Deviation (MAD)• Lessons 10–11: Describing Distributions Using the

Mean and MAD




Topic C: Summarizing a Distribution that is Skewed Using the Median and the Interquartile Range (6.SP.A.2, 6.SP.A.3, 6.SP.B.4, 6.SP.B.5)• Lesson 12: Describing the Center of a Distribution Using

the Median• Lesson 13: Describing Variability Using the Interquartile

Range (IQR)• Lesson 14: Summarizing a Distribution Using a Box Plot• Lesson 15: More Practice with Box Plots• Lesson 16: Understanding Box Plots




Topic D: Summarizing and Describing Distributions (6.SP.B.4, 6.SP.B.5) • Lesson 17: Developing a Statistical Project• Lesson 18: Connecting Graphical Representations and

Numerical Summaries• Lesson 19: Comparing Data Distributions• Lesson 20: Describing Center, Variability, and Shape of a Data

Distribution from a Graphical Representation• Lesson 21: Summarizing a Data Distribution by Describing

Center, Variability, and Shape• Lesson 22: Presenting a Summary of a Statistical Project



Terminology

Statistical QuestionMedianMeanDot plotHistogramBox PlotVariabilityDeviations from the meanMean Absolute Deviation (MAD)Interquartile Range (IQR)



Representations (Dot Plot)



Representation (Histogram)



Representation (Box plot)



Agenda




Topic A Understanding Distributions

• Lesson 1: Posing Statistical Questions • Lesson 2: Displaying a Data Distribution • Lesson 3: Creating a Dot Plot • Lesson 4: Creating a Histogram • Lesson 5:Describing a Distribution Displayed in

a Histogram 5 Instructional Days



Lesson 1: Posing Statistical Questions

In Lesson 1, statistical questions are introduced in the context of a four-step process for posing and answering questions based on data. As students begin to explore data, they see the need to organize and summarize data.



Lesson 2: Displaying a Data Distribution

In Lesson 2, students are introduced to the idea that a data distribution can be represented graphically and that there are several different types of graphs, including dot plots and histograms, commonly used to represent a distribution of numerical data. This lesson then builds on students’ previous work with line plots, introducing them to dot plots (line plots, but in a data context where students are to think about the distribution of data rather than to think of individual points plotted on a number line).



Lesson 3: Creating a Dot Plot

In Lesson 3, students construct dot plots and begin to describe data distributions.



Lesson 4: Creating a Histogram

In Lesson 4, students are introduced to histograms as another way of representing a data distribution graphically and the advantages and disadvantages of histograms relative to dot plots are discussed. * shape of a data distribution (symmetric versus skewed)* introduced to the idea that different numerical summary measures of center and variability



Lesson 5: Describing a Distribution Displayed in a Histogram

Lesson 5 gives students additional practice in constructing and describing histograms and introduces relative frequency histograms (histograms where relative frequency rather than frequency is used for the vertical scale).



Lesson 1: Posing Statistical Questions

Student Outcomes• Students distinguish between statistical

questions and those that are not statistical.• Students formulate a statistical question and

explain what data could be collected to answer the question.

• Students distinguish between categorical data and numerical data.



Polya’s Problem Solving Process

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back



Statistical Problem Solving

1. Pose a question that can be answered by data.

2. Determine a plan to collect the data.3. Summarize the data with graphs and

numerical summaries.4. Answer the questions posed in Step 1 using

the data and summaries.



What is a statistical question?

A statistical question is one that can be answered with data and for which it is anticipated that the data (information) collected to answer the question will vary.



Statistical Question?

What are the favorite colors of 6th graders in my school?

A statistical question should have:Stated populationMeasureable variableAnticipate variability



Statistical Question?

How old is the principal at our school?

Why isn’t this a statistical question?



Types of Data

Quantitative variables are numerical. They represent a measurable quantity. The height of a person, the weight of German Shepherd puppies, the circumference of a ball are examples of quantitative variables.



Types of Data

Categorical variables take on values that are names or labels. The color of a ball (e.g., red, green, blue), the breed of a dog (e.g., collie, shepherd, terrier), or favorite pizza topping would be examples of categorical variables.



Lesson 1: Problem Set

• Organize in small groups.

•Discuss and complete all the parts of problems 1 and 2 from Lesson 1 of this module. (S4)

•After you have completed the problems, we will discuss the answers.



Lesson 2: Displaying a Data Distribution

Student Outcomes• Given a dot plot, students begin describing the

distribution of the points on the dot plot in terms of center and variability.



Lesson 3: Creating a Dot Plot

Student Outcomes• Students create a dot plot of a given data set.• Students summarize a given data set using

equal length intervals and construct a frequency table.

• Based on a frequency table, students describe the distribution.



Lesson 3: Example 2

A group of 6th graders investigated the statistical question: “How many hours per week do 6th graders spend playing a sport or outdoor game?”• Here are the data the students collected from

a sample of 26 6th graders showing the number of hours per week spent playing a sport or a game outdoors: 3 2 0 6 3 3 3 1 1 2 2 8 12 4 4 4 3 3 1 1 0 0 6 2 3 2



Frequency Table



Dot Plot



Lesson 4: Creating a Histogram• The boys and girls basketball teams at Roosevelt Middle School

wanted to raise money to help buy new uniforms. They decided to sell hats with the school logo on the front to family members and other interested fans. To obtain the correct hat size, the students had to measure the head circumference (distance around the head) of the adults who wanted to order a hat. The following data represents the head circumferences, in millimeters (mm), of the adults:

• 513, 525, 531, 533, 535, 535, 542, 543, 546, 549, 551, 552, 552, 553, 554, 555, 560, 561, 563, 563, 563, 565,

• 565, 568, 568, 571,571, 574, 577, 580, 583, 583, 584, 585, 591, 595, 598, 603, 612, 618





Building a Histogram



Frequency Histogram of Head Circumferences



Shape of the Distribution: Mound Shaped or Symmetric



Skewed Distribution



• Organize in small groups.•Discuss and complete all the parts of problem 2 from Lesson 4 of this module. (S25)•After you have completed the problems, we will discuss the answers and some areas of student concern.



Lesson 5: Describing a Distribution Displayed in a Histogram

Student Outcomes• Students construct a relative frequency

histogram.• Students recognize that the shape of a

histogram does not change when relative frequency is used compared to when frequency is used to construct the histogram.



Frequency Table of Head Circumferences



Relative frequency table



Relative Frequency Histogram



Frequency Histogram of Head Circumferences



Agenda




Topic B: Summarizing a Distribution that is Approximately Symmetric Using the Mean and Mean Absolute DeviationLesson 6:• Describing the Center of a Distribution Using the Mean Lesson 7: • The Mean as a Balance Point Lesson 8:• Variability in a Data Distribution Lesson 9:• The Mean Absolute Deviation (MAD) Lessons 10–11:• Describing Distributions Using the Mean and MAD6 Instructional days



Lesson 6: Describing the Center Using the Mean

In Lesson 6, students learn to calculate the mean and to understand the “fair share” interpretation of the mean.



Lesson 7: The Mean as a Balance Point

In Lesson 7, students develop an understanding of the mean as a balance point of a data distribution—the point where the sum of distances of points to the right of the mean and the sum of distances of points to the left of the mean are equal.



Lesson 8: Variability in a Data DistributionLesson 9: Mean Absolute Deviation

Lessons 8 and 9 introduce the MAD as a measure of variability, and students calculate and interpret the value of the MAD.



Lesson 10 and 11: Describing the Distribution using the Mean and MADLessons 10 and 11 give students the opportunity to use both graphical and numerical summaries to describe data distributions, to compare distributions, and to answer questions in context using information provided by a data distribution.



What do we mean by “average”?

What students say:“add em up and divide”“it’s the average”“it’s the middle number”“it’s in the center”



Lesson 6: Describing the Center using the MeanQuestion: How many pets do five randomly sampled sixth graders have?

Data from five randomly chosen sixth graders:2, 6, 2, 4, and 1



How many pets do students have?2, 6, 2, 4, and 1



What do you mean by “average?”The evening out or fair share value



Fair Share value



Fair Share value



Fair Share value



Fair Share value



Fair Share value



Fair Share value



Fair Share value



The results after “evening” out



Question: How many pets do students have?What does “fair share” mean in words in the context of the question?



Question: How many pets do students have?Mean is 3 – the fair share value, i.e., the number of pets that all the students would have if they all had the same number of pets is 3. (There are 15 pets to be divided evenly among 5 students. Connections to math: 15/5)



Fair Share Value

Aside: What if there had been 16 pets ?There would be one pet “left over” that needs to be divided evenly among the five children. What to do?

Well, the pet would have to be “chopped up” into five equal pieces giving each child one piece, or more exactly 1/5 th of a piece. Hence, each child would wind up with 3 1/5 th pets. The mean would be 3 1/5 pets.



Lesson 7: The Mean as a Balance Point

The number of minutes it takes two students to get to school.



If you represented the dot plot with a ruler and pennies, where would the ruler balance?



What is the sum of the deviations?





Will the ruler balance at 6?



Lesson 8: Variability in a Data DistributionStudent Outcomes• Students interpret the mean of a data set as a “typical” value.• Students compare and contrast two small data sets that have the

same mean but different amounts of variability.• Students see that a data distribution is not characterized only by its

center. Its spread or variability must be considered as well.• Students informally evaluate how precise the mean is as an

indicator of the typical value of a distribution, based on the variability exhibited in the data.

• Students use dot plots to order distributions according to the variability around the mean for each of the data distributions.



Lesson 8: Variability in a Data Distribution



Lesson 9: The Mean Absolute Deviation (MAD)Student Outcomes• Students calculate the mean absolute

deviation (MAD) for a given data set.• Students interpret the MAD as the average

distance of data values from the mean.



Measuring Variability

• Mean temperature is 63 degrees

Temperature (degrees F)90858075706560555045403530

City G



Complete the temperature deviations



Mean Absolute Deviation

• Find the mean of the absolute deviations = 3.67• This is MAD



Measuring Variability

• Mean temperature is 63 degrees

Temperature (degrees F)90858075706560555045403530

City G



Mean Absolute Deviation

What does a MAD of 3.67 represent in context?

If MAD is “large” what would that indicate about the variability of the data?



Lesson 10 and 11: Describing Distributions Using the Mean and MADStudent Outcomes• Students calculate the mean and MAD for a

data distribution.• Students use the mean and MAD to describe a

data distribution in terms of center and variability.





•Discuss and complete all the parts of problems 5 and 6 from Lesson 10 of this module. (S69)




Lesson 10 Problem 5



Agenda




Mid-Module Assessment



Agenda




Topic C: Summarizing a Distribution that is Skewed Using the Median and the Interquartile RangeLesson 12: Describing the Center of a Distribution Using the Median Lesson 13: Describing Variability Using the Interquartile Range (IQR) Lesson 14: Summarizing a Distribution Using a Box PlotLesson 15: More Practice with Box Plots Lesson 16: Understanding Box Plots5 Instructional Days



Lesson 12: Describing the Center of a Distribution Using the Median

In Lesson 12, students learn to calculate and interpret the median.



Lesson 13: Describing Variability Using the Interquartile Range (IQR)

Quartiles are introduced in Lesson 13, and the quartiles are then used to calculate the IQR. Students also learn to interpret the IQR as a measure of variability in a data distribution.



Lesson 14-16 Summarizing a Distribution Using a Box Plot

Lessons 14–15 introduce box plots. Box plot is a graph of five key summary statistics of a data set (the minimum, lower quartile, median, upper quartile, and the maximum).Lesson 16 has students use box plots to compare groups, setting the stage for future work on comparing groups in Grade 7.



Lesson 12: Describing the Center of a Distribution Using the Median

Student Outcomes• Given a data set, students calculate the median

of the data.• Students estimate the percent of values above

and below the median value.



Lesson 13: Describing Variability Using the Interquartile Range (IQR) Student Outcomes• Given a set of data, students describe how the data

might have been collected.• Students describe the unit of measurement for

observations in a data set.• Students calculate the median of the data.• Students describe the variability in the data by

calculating the interquartile range.



Finding the Median and Quartiles

The owner of the chain decided to check the number of french fries at another restaurant in the chain. Here is the data for Restaurant B: 82, 83, 83, 79, 85, 82, 78, 76, 76, 75, 78, 74, 70, 60, 82, 82, 83, 83, 83.




Number of french fries (n=19) in order from smallest to largest:

60 70 74 75 76 76 78 78 79 81 82 82 82 83 83 83 84 85 85

Lower quartile Median Upper quartile




n= 18:

60 70 74 75 76 76 78 78 79 81 82 82 82 83 83 83 84 85





n= 16:

60 70 74 75 76 76 78 78 79 81 82 82 83 83 84 85




Lesson 14: Summarizing a Distribution Using a Box Plot

Student Outcome• Students construct a box plot from a given set

of data.



Lesson 15: More Practice with Box Plots

Student Outcomes• Given a box plot, students summarize the data

by the 5-number summary (Min, Q1, Median, Q3, Max.)

• Students describe a set of data using the 5-number summary and the interquartile range.

• Students construct a box plot from a 5-number summary.



Lesson 16: Understanding Box Plots

Student Outcomes• Students summarize a data set using box plots,

the median, and the interquartile range. • Students use box plots to compare two data

distributions.





•Discuss and complete all the parts of exercises 6 to 10 from Lesson 15 of this module. (S100)




Interpreting Box Plots



Interpreting Box Plots



Interpreting box plots exercise 10 (p.101)



Agenda




Topic D: Summarizing and Describing DistributionsLesson 17:Developing a Statistical Project Lesson 18: Connecting Graphical Representations and Numerical Summaries Lesson 19:Comparing Data Distributions Lesson 20:Describing Center, Variability, and Shape of a Data Distribution from a Graphical Representation Lesson 21:Summarizing a Data Distribution by Describing Center, Variability, and Shape Lesson 22:Presenting a Summary of a Statistical Project 6 Instructional Days



Topic D: Summarizing and Describing DistributionsIn Topic D, students integrate what they have learned about graphical and numerical data summaries in the previous topics. They match dot plots and histograms to numerical measures of center and variability. Students estimate means and medians from graphical representations of data distributions. They also estimate mean absolute deviation (MAD) and interquartile range (IQR) from graphical representations based on an understanding of data distributions in terms of shape, center, and variability.



Lesson 18:Connecting Graphical Representations and Numerical Summaries

Student Outcomes• Students match the graphical representations

and numerical summaries of a distribution. Matches involve dot plots, histograms, and summary statistics.



Example from Lesson 18



Lesson 19:Comparing Data Distributions

• Student Outcomes• Given box plots of at least two data sets,

students will comment on similarities and differences in the distributions.



Lesson 20:Describing Center, Variability, and Shape of a Data Distribution from a Graphical Representation Student Outcomes• Given a frequency histogram, students are able to

describe the data collected, including the number of responses, an estimate of the mean or median, and an estimate of the interquartile range (IQR) or the mean absolute deviation (MAD).



Lesson 21: Summarizing a Data Distribution by Describing Center, Variability, and Shape Student Outcomes• Given a data set, students are able to describe

the data collected, including the number of responses, mean or median, and the MAD or the interquartile range (IQR).





•Discuss and complete all the parts of problems 1 to 11 from Lesson 20 of this module. (S134)




Ex 1: The Great Lakes Yellow Perch

Scientists collected data from many samples of yellow perch because they were concerned about the survival of the yellow perch. What data do you think researchers might want to collect about the perch? Scientists captured yellow perch from a lake in this region. They recorded data on each fish, and then returned each fish to the lake. Consider the following histogram of data on the length (in centimeters) for a sample of yellow perch.



Yellow Perch



Lesson 17 and 22:Developing a Statistical Project Two of the lessons in this topic (Lessons 17 and 22) allow students to experience the four-step process described at the beginning of this module through completion of a project. In this project, students experience the four-step investigative process by (1) formulating a statistical question, (2) designing and implementing a plan to collect data, (3) summarizing collected data graphically and numerically, and (4) using the data to answer the question posed.



Agenda




End of Module Assessment









Agenda




Summary of the 4 Topics

In Topic A, students begin to think and reason statistically, first by recognizing a statistical question as one that can be answered by collecting data (6.SP.A.1). Students learn that the data collected to answer a statistical question has a distribution that is often summarized in terms of center, variability, and shape (6.SP.A.2). Beginning in Topic A, and throughout the module, students see and represent data distributions using dot plots and histograms (6.SP.B.4).



Summary of the 4 Topics

In Topic B, students study mean as a measure of center and mean absolute deviation as a measure of variability. Students learn that these measures are preferred when the shape of the distribution is roughly symmetric.



Summary of the 4 topics

in Topic C, students study median as a measure of center and interquartile range as a measure of variability. Students learn that these measures are preferred when the shape of the distribution is skewed. Students develop in Topic B, and reinforce in Topic C, the idea that a measure of center provides a summary of all its values in a single number, while a measure of variation describes how values vary, also with a single number (6.SP.A.3).



Summary of the 4 topics

In Topic D, students synthesize what they have learned as they connect the graphical, verbal, and numerical summaries to each other within situational contexts, culminating with a major project (6.SP.B.4, 6.SP.B.5).



Questions and/or Comments

Documents

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 6 – Module 6 Statistics (Lessons