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:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

6.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

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 nonexamples, 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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

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, stemandleaf plots, histograms, and box plots.
Supporting Standard

Represent
NUMERIC DATA GRAPHICALLY, INCLUDING DOT PLOTS, STEMANDLEAF 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 onetoone 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
 Stemandleaf 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 stemandleaf 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 Tchart, 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 onetoone 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 stemandleaf 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 fivenumber 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 fivenumber 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, stemandleaf plot, histogram, and box plot including the fivenumber summary
Note(s):
 Grade Level(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 stemandleaf 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 stemandleaf plots.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding data representation
 TxCCRS:
 V.B. Statistical Reasoning – Describe data
 V.B.2. Construct appropriate visual representations of data.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.1. Analyze data sets using graphs and summary statistics.

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 Level(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:
 V.B. Statistical Reasoning – Describe data
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

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.
Readiness Standard

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 Level(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:
 V.B. Statistical Reasoning – Describe data
 V.B.1. Classify types of data.
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.1. Analyze data sets using graphs and summary statistics.
 V.C.3. Make predictions using summary statistics.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

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.
Readiness Standard

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 parttopart relationships or parttowhole 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 Level(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:
 V.B. Statistical Reasoning – Describe data
 V.B.1. Classify types of data.
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.1. Analyze data sets using graphs and summary statistics.
 V.C.3. Make predictions using summary statistics.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

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, stemandleaf plots, histograms, and box plots.
Readiness Standard

Interpret
NUMERIC DATA SUMMARIZED IN DOT PLOTS, STEMANDLEAF 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
 Stemandleaf 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 fivenumber summary of data (minimum, lower quartile, median, upper quartile, maximum)
Note(s):
 Grade Level(s):
 Grade 5 solved one and twostep problems using data from a frequency table, dot plot, bar graph, stemandleaf 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:
 V.B. Statistical Reasoning – Describe data
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.1. Analyze data sets using graphs and summary statistics.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

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 Level(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:
 V.B. Statistical Reasoning – Describe data
 V.B.1. Classify types of data.
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.
