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 Instructional Focus DocumentGrade 6 Mathematics
 TITLE : Unit 10: Data Analysis SUGGESTED DURATION : 20 days

Unit Overview

Introduction
This unit bundles student expectations that address data analysis, including representing, interpreting, and describing data distributions, summarizing numeric and categorical data, and distinguishing between situations that yield data with and without variability. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.”

Prior to this Unit
In Grade 3, students summarized a data set with multiple categories using frequency tables, dot plots, pictographs, or bar graphs with scaled intervals. Students solved one- and two-step problems using categorical data represented in frequency tables, dot plots, pictographs, or bar graphs with scaled intervals. In Grade 4, students represented data with frequency tables, dot plots, and stem-and-leaf plots, and solved one- and two-step problems with the representations and data in the form of whole numbers, decimals, and fractions. Students also represented discrete paired data with a scatterplot.

During this Unit
Students extend previous knowledge of data representations including dot plots and stem-and-leaf plots, and are formally introduced to histograms, box plots, and percent bar graphs. Students use graphical representations to describe the shape, center, and spread of the data distribution. Descriptions of shape, center, and spread include skewed right, skewed left, symmetric, mean, median, mode, range, and interquartile range. Students also summarize numeric data with numerical summaries, including the measures of center and the measures of spread. Categorical data is summarized numerically with the mode and a relative frequency table and summarized graphically with a percent bar graph. Students are required to distinguish between situations that yield data with and without variability.

After this Unit
In Grade 7, students will extend their knowledge of using numeric data to describe the center, spread, and shape of the data distribution to comparing two groups of numeric data. Comparative box plots and dot plots are used to compare two sets of data graphically. Students will be introduced to circle graphs as a representation of categorical data as students examine part-to-part and part-to-whole comparisons.

In Grade 6, representing numeric data graphically and using the graphical representation to describe the data distribution are identified as Supporting Standards 6.12A and 6.12B. Summarizing numerical data with numerical summaries and summarizing categorical data with numerical and graphical summaries are described as Readiness Standards 6.12C and 6.12D. Interpreting numeric data summarized in graphical representations is identified as Readiness Standard 6.13A. Distinguishing between situations that yield data is classified as Supporting Standard 6.13B. All of these standards are subsumed under the Grade 6 STAAR Reporting Category 4: Data Analysis and Personal Financial Literacy and part of the Grade 6 Texas Response to Curriculum Focal Points (TxRCFP): Understanding data representation. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, IV. Measurement Reasoning, VII. Statistical Reasoning, VIII. Problem Solving and Reasoning, and IX. Communication and Representation.

Research
According to Van De Walle, Bay-Williams, Lovin, and Karp (2014), “Statistical literacy is critical to understanding the world around us, essential for effective citizenship, and vital for developing the ability to question information presented in the media (Shaughnessy, 2007)” (p. 325-326). The National Council of Teachers of Mathematics (2003) states that, “in working with data, students encounter and apply ideas that connect directly with those in the other strands of the mathematics curriculum as well as with the mathematical ideas that they regularly meet in other school subjects and in daily life” (p.1).

National Council of Teachers of Mathematics. (2003). Navigating through Data Analysis in grades 6 – 8. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van De Walle, J., Bay-Williams, J., Lovin, L., & Karp, K., (2014). Teaching student-centered mathematics: Developmentally Appropriate Instruction for Grades 6 - 8. (2nd ed., Vol. 3). Boston, MA: Pearson.

 Statistical displays often reveal patterns within data that can be analyzed to interpret information, inform understanding, make predictions, influence decisions, and solve problems in everyday life with degrees of confidence. How does society use or make sense of the enormous amount of data in our world available at our fingertips? How can data and data displays be purposeful and powerful? Why is it important to be aware of factors that may influence conclusions, predictions, and/or decisions derived from data?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Data can be described and quantified using various methods in order to communicate and reason statistically about the entire data set.
• What are the characteristics of …
• dot plots?
• stem-and-leaf plots?
• histograms?
• box plots?
• What is the process to construct a …
• dot plot?
• stem-and-leaf plot?
• histogram?
• box plot?
• How can the …
• shape
• center
… of a set of data be described?
• What is the process to determine the …
• mean
• median
• range
• interquartile range
… of a set of numeric data?
• How can clusters of data affect the …
• measures of center?
• measures of spread?
• What aspects of a data distribution can be emphasized with …
• dot plots?
• stem-and-leaf plots?
• histograms?
• box plots?
• How can data …
• with variability
• without variability
… be summarized?
• What types of situations yield data …
• with variability?
• without variability?
• Measurement and Data
• Data
• Numeric data
• Numerical summaries
• Graphical summaries
• Conclusions and predictions
• Variability
• Graphical representations
• Dot plots
• Stem-and-leaf plots
• Histograms
• Box plots
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Statistical displays often reveal patterns within data that can be analyzed to interpret information, inform understanding, make predictions, influence decisions, and solve problems in everyday life with degrees of confidence. How does society use or make sense of the enormous amount of data in our world available at our fingertips? How can data and data displays be purposeful and powerful? Why is it important to be aware of factors that may influence conclusions, predictions, and/or decisions derived from data?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Data can be described and quantified using various methods in order to communicate and reason statistically about the entire data set.
• How can categorical data be described …
• graphically?
• numerically?
• How can a …
• relative frequency table
• percent bar graph
… be used to summarize a data distribution?
• What is the process to construct a …
• relative frequency table?
• percent bar graph?
• Measurement and Data
• Data
• Categorical data
• Numerical summaries
• Graphical summaries
• Conclusions and predictions
• Variability
• Graphical representations
• Relative frequency tables
• Percent bar graphs
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that categorical and numerical data can always be displayed by the same representations rather than realizing that the appropriate representation for a set of data depends on the type of question being asked about the data (e.g., Bar graphs and frequency tables can represent categories of data that do not have numeric ordering, such as favorite colors or type of transportation to get to school, as well as representing numeric data such as shoe size or number of family members.)
• Some students may determine that fractions and decimals cannot be represented on a stem-and-leaf plot rather than using the whole number as the stem and fraction or decimal amount as the leaf.
• Students may choose one of the two center numbers in an even number of values as the median instead of finding the average of the two middle values.

Underdeveloped Concepts:

• Although some students may be proficient at displaying data using different representations, they may lack the experience to solve problems by analyzing the data.
• Some students may have difficulty constructing graphs due to the scale on the number lines.

Unit Vocabulary

• Box plot (box and whisker plot) – a graphical representation showing the five-number summary of data (minimum, lower quartile, median, upper quartile, maximum)
• Categorical data – data that represents the attributes of a group of people, events, or objects
• Data – information that is collected about people, events, or objects
• Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) and an axis to show the frequency (number of times) that each number occurs
• Graph – a visual representation of the relationships between data collected
• Histogram – a graphical representation of adjacent bars with different heights or lengths used to represent the frequency of data in certain ranges of continuous and equal intervals
• Interquartile range (IQR) – difference between the first quartile and the third quartile of a set of numbers (IQR = Q3 – Q1)
• Mean – average of a set of data found by finding the sum of a set of data and dividing the sum by the number of pieces of data in the set
• Median – the middle number of a set of data that has been arranged in order from greatest to least or least to greatest
• Mode – most frequent piece of data in a set of data
• Mode of categorical data (modal category) – most frequent category in a set of data
• Mode of numeric data – most frequent value in a set of data
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Percent – a part of a whole expressed in hundredths
• Percent bar graph – a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs as a percentage as compared to the related part(s) or to the whole
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Range – the difference between the greatest number and least number in a set of data
• Relative frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs as a percentage
• Stem-and-leaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating the digits in numerical values based on place value. The left digit(s) of the data form the stems and the remaining digit(s) or fraction form the leaves that correspond with each stem, as designated by a key.
• Variability – measure of the spread of a set of data

Related Vocabulary:

 Asymmetric Average Cluster Frequency Horizontal Interval Key Leaf Lower Quartile Measures of center Measures of spread Number line Outlier Peak Quartile Range Scale Shape Skewed Stem Symmetric Tally marks Title Upper Quartile Value Vertical
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 6 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
6.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
6.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• X. Connections
6.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• VIII. Problem Solving and Reasoning
6.1C

Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Process Standard

Select

TOOLS, INCLUDING PAPER AND PENCIL AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• VIII. Problem Solving and Reasoning
6.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• IX. Communication and Representation
6.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• IX. Communication and Representation
6.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• X. Connections
6.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• IX. Communication and Representation
6.12 Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to analyze problems. The student is expected to:
6.12A Represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots.
Supporting Standard

Represent

NUMERIC DATA GRAPHICALLY, INCLUDING DOT PLOTS, STEM-AND-LEAF PLOTS, HISTOGRAMS AND BOX PLOTS

Including, but not limited to:

• Graph – a visual representation of the relationships between data collected
• Organization of data used to interpret data, draw conclusions, and make comparisons
• Data – information that is collected about people, events, or objects
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Can be counted (discrete) or measured (continuous)
• Limitations
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents
• Data representations
• Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) and an axis to show the frequency (number of times) that each number occurs
• Characteristics of a dot plot
• Titles, subtitles, and labels
• Title represents the purpose of collected data
• Subtitle clarifies the meaning of number line
• Labels identify each numerical increment below the line
• Representation of numerical data
• Dots (or Xs)
• Placed in a horizontal or vertical linear arrangement
• Vertical graph beginning at the bottom and progressing up above the line
• Horizontal graph beginning at the left and progressing to the right of the line
• Spaced approximately equal distances apart within each category
• Axis
• Numerical data represented by a number line labeled with proportional increments
• Every piece of data represented using a one-to-one or scaled correspondence, as indicated by the key
• Dots (or Xs) generally represent one count
• May represent multiple counts if indicated with a key
• Value of the data in each category
• Determined by the number of dots (or Xs) or total value of dots (or Xs), as indicated by the key if given
• Represents the frequency for that category
• Density of dots relates to the frequency distribution of the data.
• Shape of the dot plot may be used to compare shape, spread, and center of data
• Stem-and-leaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating the digits in numerical values based on place value. The left digit(s) of the data form the stems and the remaining digit(s) or fraction form the leaves that correspond with each stem, as designated by a key.
• Characteristics of a stem-and-leaf plot
• Titles and column headers
• Title represents the purpose of collected data
• Column headers indicate stems and leaves
• Representation of numerical data
• Vertical line, such as in a T-chart, separates stems from their corresponding leaves
• Stems listed to the left of the vertical line with their corresponding leaves listed in a row to the right of the vertical line
• Determination of place value(s) that represents stems versus place value(s) that represents leaves is dependent upon how to best display the distribution of the entire data set and then indicated by a key
• Left digit(s) of the data forms the stems and remaining digit(s) or fraction forms the leaves that correspond with each stem, as indicated by the key
• Every piece of data represented using a one-to-one correspondence, including repeated values
• Stem represents one or more pieces of data in the set
• Leaf represents only one piece of data in the set
• Leaves provide frequency counts for the range of numbers included in that row of the stem-and-leaf plot
• Density of leaves relates to the frequency distribution of the data
• Histogram – a graphical representation of adjacent bars with different heights or lengths used to represent the frequency of data in certain ranges of continuous and equal intervals
• Characteristics of a histogram
• Titles and subtitles
• Title represents the purpose of collected data
• Subtitles clarify the meaning of data represented on each axis
• Representation of numerical data
• Bars
• Placed in a horizontal or vertical linear arrangement to represent data
• Solid bars that are equal in width
• Bars touch each other without overlap to indicate the intervals of the numeric data are continuous
• Length of the bar represents the distance from zero on the axis scale
• Axes
• Represented as number lines
• Scale intervals proportionally displayed
• Intervals of one or more units
• Individual data points cannot be determined from a histogram
• Value of the data represented by bar
• Determined by reading the number on the scaled axis associated with the length of the bar
• Represents the frequency for that range
• Box plot (box and whisker plot) – a graphical representation showing the five-number summary of data (minimum, lower quartile, median, upper quartile, maximum)
• Titles and subtitles
• Title represents the purpose of collected data
• Subtitle clarifies the meaning of the data represented on number line
• Representation of numerical data
• Axis
• Numerical data represented by a number line labeled with proportional increments
• Vertical or horizontal arrangement
• Does not display each element in the data set
• Data is divided into quartiles using the five-number summary.
• Minimum or lower extreme
• Quartile 1 (Q1): median of lower 50% of the data
• For data sets with an odd number of elements, the lower 50% of the data will not include the median of the entire data set.
• For data sets with an even number of elements, the lower 50% of the data will include the first half of the entire data set.
• Median of the entire data set
• Quartile 3 (Q3): median of the upper 50% of the data
• For data sets with an odd number of elements, the upper 50% of the data will not include the median of the entire data set.
• For data sets with an even number of elements, the upper 50% of the data will include the last half of the entire data set.
• Maximum or upper extreme
• Interquartile range (IQR) – difference between the first quartile and the third quartile of a set of numbers (IQR = Q3 – Q1)
• Outliers may or may not exist.
• Outliers calculated as any data point that falls outside of range of 1.5 times the IQR (Outliers = 1.5IQR) from Q1 and Q3
• From the lower quartile: Q1 – 1.5IQR
• From the upper quartile: Q3 + 1.5IQR
• Density of quartiles represents the frequency distribution of the data
• Shape of the box related to the spread (variability) of the data
• Connection between graphs
• Same data represented using a dot plot, stem-and-leaf plot, histogram, and box plot including the five-number summary

Note(s):

• Grade 3 represented categorical data with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
• Grade 4 represented data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions.
• Grade 5 represented categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding data representation
• TxCCRS:
• I. Numeric Reasoning
• IV. Measurement Reasoning
• VII. Statistical Reasoning
• IX. Communication and Representation
6.12B Use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution.
Supporting Standard

Use

THE GRAPHICAL REPRESENTATION OF NUMERIC DATA TO DESCRIBE THE CENTER, SPREAD, AND SHAPE OF THE DATA DISTRIBUTION

Including, but not limited to:

• Graph – a visual representation of the relationships between data collected
• Organization of data used to describe and summarize data
• Data – information that is collected about people, events, or objects
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Can be counted (discrete) or measured (continuous)
• Limitations
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Center of the data distribution from a graphical representation
• Mean – average of a set of data found by finding the sum of a set of data and dividing the sum by the number of pieces of data in the set
• Median – the middle number of a set of data that has been arranged in order from greatest to least or least to greatest
• Mode – most frequent piece of data in a set of data
• Mean or median may describe the data distribution if the shape of the data is symmetrical
• Median may describe the data distribution if the shape of the data is skewed (asymmetrical)
• Outlier does not describe the numerical summary, although it may alter the relationship between the mean and the median
• Spread of the data distribution from a graphical representation
• Range – the difference between the greatest number and least number in a set of data
• May be expressed as a single value or as a range of numbers
• Interquartile range (IQR) – difference between the first quartile and the third quartile of a set of numbers (IQR = Q3 – Q1)
• Usually used only for box plots
• Outlier does not describe the numerical summary, although it may alter the relationship between the range and IQR.
• Shape of the data distribution from a graphical representation
• Symmetrical distribution
• Mean, median, and mode usually approximately the same
• Most data points clustered around the middle value of the range within the distribution
• Peak of the data usually occurs around the middle of the distribution
• Shape of the data resembles a bell curve when graphed
• Asymmetrical Distribution
• Skewed right
• Mean usually greater than the median, and median greater than the mode
• Median considered the better representation of the center of the distribution
• Most data points clustered towards the left, or the low end, of the range within the data distribution
• Peak occurs towards the left, or the low end, of the range within the data distribution
• Shape of the data has a tail to the right when graphed
• Skewed left
• Mean usually less than the median, and median less than the mode
• Median considered the better representation of the center of the distribution
• Most data points clustered towards the right, or the high end, of the range within the data distribution
• Peak occurs towards the right, or the high end, of the range within the data distribution
• Shape of the data has a tail to the left when graphed
• Not symmetrical or skewed
• No general statement about the median, median, or mode
• No general statement about the shape of the data
• Clusters of points usually do not indicate a pattern or trend within the data distribution

Note(s):

• Grade 6 introduces using the graphical representation of numeric data to describe the center, spread, and shape of the data distribution.
• Grade 7 will compare two groups of numeric data using dot plots or box plots by comparing their shapes, centers, and spreads.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding data representation
• TxCCRS:
• I. Numeric Reasoning
• IV. Measurement Reasoning
• VII. Statistical Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
6.12C Summarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution.

Summarize

NUMERIC DATA WITH NUMERICAL SUMMARIES, INCLUDING THE MEAN AND MEDIAN (MEASURES OF CENTER) AND THE RANGE AND INTERQUARTILE RANGE (IQR) (MEASURES OF SPREAD)

Including, but not limited to:

• Graph – a visual representation of the relationships between data collected
• Organization of data used to describe and summarize data
• Data – information that is collected about people, events, or objects
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Can be counted (discrete) or measured (continuous)
• Limitations
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents
• Measures of center
• Mean – average of a set of data found by finding the sum of a set of data and dividing the sum by the number of pieces of data in the set
• Median – the middle number of a set of data that has been arranged in order from greatest to least or least to greatest
• Mode – most frequent piece of data in a set of data
• Measures of spread
• Range – the difference between the greatest number and least number in a set of data
• May be expressed as a single value or as a range of numbers
• Interquartile range (IQR) – difference between the first quartile and the third quartile of a set of numbers (IQR = Q3 – Q1)
• Usually used only for box plots

Use

NUMERICAL SUMMARIES TO DESCRIBE THE CENTER, SPREAD, AND SHAPE OF THE DATA DISTRIBUTION

Including, but not limited to:

• Center of the data distribution from numerical summaries
• Mean
• Median
• Mode
• Mean or median may describe the data distribution if the shape of the data is symmetrical
• Median may describe the data distribution if the shape of the data is skewed (asymmetrical)
• Outlier does not describe the numerical summary, although it may alter the relationship between the mean and the median
• Spread of the data distribution from numerical summaries
• Range
• May be expressed as a single value or as a range of numbers
• Interquartile range (IQR)
• The smaller the spread, the closer the data values are to each other.
• The larger the spread, the farther the data values are from each other.
• Outlier does not describe the numerical summary, although it may alter the relationship between the range and IQR
• Shape of the data distribution from numerical summaries
• Symmetrical
• Mean, median, and mode
• Usually approximately the same
• Tend to be in the middle category(ies) in a histogram
• Tend to be in the middle column(s) in a dot plot
• Tend to be in the middle row(s) in a stem and leaf plot
• Asymmetrical
• Mean, median, and mode
• Tend not to be in the middle category(ies) in a histogram
• Tend not to be in the middle column(s) in a dot plot
• Tend not to be in the middle row(s) in a stem and leaf plot
• Skewed right
• Mean usually greater than the median, and median greater than the mode
• Median considered the better representation of the center of the distribution
• Skewed left
• Mean usually less than the median, and median less than the mode
• Median considered the better representation of the center of the distribution

Note(s):

• Grade 6 introduces summarizing numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and using these summaries to describe the center, spread, and shape of the data distribution.
• Grade 7 will compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding data representation
• TxCCRS:
• I. Numeric Reasoning
• IV. Measurement Reasoning
• VII. Statistical Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
6.12D Summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution.

Summarize

CATEGORICAL DATA WITH NUMERICAL AND GRAPHICAL SUMMARIES, INCLUDING THE MODE, THE PERCENT OF VALUES IN EACH CATEGORY (RELATIVE FREQUENCY TABLE), AND THE PERCENT BAR GRAPH

Including, but not limited to:

• Graph – a visual representation of the relationships between data collected
• Organization of data used to describe and summarize data
• Data – information that is collected about people, events, or objects
• Categorical data – data that represents the attributes of a group of people, events, or objects
• May represent numbers or ranges of numbers
• Limitations
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents
• Percent – a part of a whole expressed in hundredths
• Mode of categorical data (modal category) – most frequent category in a set of data
• Data representations
• Relative frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs as a percentage
• Characteristics of a relative frequency table
• Titles and labels
• Title represents the purpose of collected data
• Column headers and row labels clarify meaning of the data represented in the table
• Representation of categorical data
• Table format
• May include frequency count of each category
• Includes the frequency of each category as a percentage of the total frequency for all categories
• Data values
• Calculated by dividing the number of observations in a specific category by the total number of observations
• Represented as percents, with the total of all categories representing 100%
• Percent bar graph – a graphical representation to organize data that uses solid bars that do not touch each other to show the frequency (number of times) that each category occurs as a percentage as compared to the related part(s) or to the whole
• Characteristics of a percent bar graph
• Titles, subtitles, and labels
• Title represents the purpose of collected data
• Labels identify each category
• Representation of categorical data
• Percent bars
• Placed in a linear horizontal or vertical arrangement to represent data
• Scale of the axis
• Intervals of one or more units
• Scaled intervals proportionally displayed
• Represented as a number line from 0% to 100%
• Bars of graph
• Represent the relative frequency (as a percentage) for each category
• May represent part-to-part relationships or part-to-whole relationships
• Length of the bar represents
• Percentage of data points for a given category
• Distance from zero on the scale of the axis
• Value of the data represented by the bar
• Determined by reading its associated number (the intervals) on the axis scale

Use

NUMERICAL AND GRAPHICAL SUMMARIES TO DESCRIBE THE DATA DISTRIBUTION

Including, but not limited to:

• Summaries of data distribution
• Numerical summary
• Mode appears as the greatest percent for each category in a relative frequency table
• Graphical summary
• Comparative heights or lengths of the category bars can be used to draw conclusions about the data represented
• Mode appears as the tallest or longest bar in a percent bar graph

Note(s):

• Grade 6 introduces summarizing categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and using these summaries to describe the data distribution.
• Grade 7 will compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding data representation
• TxCCRS:
• I. Numeric Reasoning
• IV. Measurement Reasoning
• VII. Statistical Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
6.13 Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to solve problems. The student is expected to:
6.13A Interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots.

Interpret

NUMERIC DATA SUMMARIZED IN DOT PLOTS, STEM-AND-LEAF PLOTS, HISTOGRAMS, AND BOX PLOTS

Including, but not limited to:

• Data – information that is collected about people, events, or objects
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Can be counted (discrete) or measured (continuous)
• Limitations
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents
• Numeric summaries
• Mean – average of a set of data found by finding the sum of a set of data and dividing the sum by the number of pieces of data in the set
• Median – the middle number of a set of data that has been arranged in order from greatest to least or least to greatest
• Mode of numeric data – most frequent value in a set of data
• Range – the difference between the greatest number and least number in a set of data
• May be expressed as a single value or as a range of numbers
• Interquartile range (IQR) – difference between the first quartile and the third quartile of a set of numbers (IQR = Q3 – Q1)
• Usually used only for box plots
• Data representations
• Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) and a scale axis to show the frequency (number of times) that each number occurs
• Stem-and-leaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating the digits in numerical values based on place value. The left digit(s) of the data form the stems and the remaining digit(s) or fraction form the leaves that correspond with each stem, as designated by a key.
• Histogram – a graphical representation of adjacent bars with different heights or lengths used to represent the frequency of data in certain ranges of continuous and equal intervals
• Box plot (box and whisker plot) – a graphical representation showing the five-number summary of data (minimum, lower quartile, median, upper quartile, maximum)

Note(s):

• Grade 5 solved one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot.
• Grade 6 introduces box plots and histograms.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding data representation
• TxCCRS:
• I. Numeric Reasoning
• IV. Measurement Reasoning
• VII. Statistical Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
6.13B Distinguish between situations that yield data with and without variability.
Supporting Standard

Distinguish

BETWEEN SITUATIONS THAT YIELD DATA WITH AND WITHOUT VARIABILITY

Including, but not limited to:

• Variability – measure of the spread of a set of data
• Data with variability may occur as data is recorded at different time periods.
• Data with variability may occur as data is recorded at a single time for different or many subjects.
• Data with variability can be summarized with a range.
• Data without variability may be recorded from an individual or event at a specific time.
• Data without variability can be summarized with a single value.

Note(s):

• Grade 6 introduces distinguishing between situations that yield data with and without variability.
• Grade 8 will determine the mean absolute deviation, which is a measure of variability for quantitative data.
• Statistics will distinguish among different sources of variability, including measurement, natural, induced, and sampling variability.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding data representation
• TxCCRS:
• IV. Measurement Reasoning
• VII. Statistical Reasoning
• IX. Communication and Representation 