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 Instructional Focus DocumentGrade 5 Mathematics
 TITLE : Unit 10: Representing Data SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address categorical and numerical data, and discrete paired data. According to the Texas Education Agency, mathematical process standards including application, 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.

Prior to this Unit
In Grade 4, students represented data with a frequency table, dot plot, and stem-and-leaf plot marked with whole numbers and fractions. Students also solved one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot. In Grade 5 Unit 09, students graphed ordered pairs in Quadrant I of the coordinate plane.

During this Unit
Students represent categorical data with bar graphs and frequency tables. Numerical data, including data sets of measurements in fractions or decimals, is represented with dot plots or stem-and-leaf plots. Students are introduced to scatterplots as a means to represent discrete paired data. Students utilize all of these graphical representations to solve one- and two-step problems.

After this Unit
In Grade 6, students will represent numeric data graphically with representations that include dot plots, stem-and-leaf plots, histograms, and box plots. Students will interpret numeric data summarized in these representations and distinguish between situations that yield data with and without variability. Categorical data will be represented with relative frequency tables and percent bar graphs. The shape, center, and spread of data distributions will be analyzed and described for both categorical and numerical data.

Additional Notes
In Grade 5, representing categorical data with bar graphs or frequency tables and numerical data, and representing discrete paired data on a scatterplot are identified as STAAR Supporting Standards 5.9A and 5.9B. Solving one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot is identified as STAAR Readiness Standard 5.9C. All of these standards are subsumed within the Grade 5 STAAR Reporting Category 4: Data Analysis and Personal Financial Literacy as well as the Grade 5 Texas Response to Curriculum Focal Points (TxRCFP): Organizing, representing, and interpreting sets of data. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): VI.A. Statistical Reasoning – Describe Data, VIII. Problem Solving and Reasoning, and IX. Communication and Representation.

Research
The National Research Council (2001) reports that, “Elementary school students have difficulty analyzing and interpreting data. In one study, 80% of the first and second graders interviewed gave idiosyncratic or incomplete responses when they attempted to analyze data from a line plot and a bar graph. In another study, almost all of the fourth and sixth graders could describe bar graphs, but fewer could interpret them, and many fewer still could use the graphs to predict” (pg. 291). The National Council of Teachers of Mathematics (2011) states, “Using carefully designed questioning practices, elementary teachers must guide students’ construction of these representations and their interpretation of results so that students are prepared to learn statistics in the middle grades” (p. 82). Van de Walle (2004) adds, “A big conceptual idea in data analysis can be referred to as the shape of data: a sense of how data are spread or grouped, what characteristics they have, and what they tell us in a global way about the population from which they are taken. Each of the graphical techniques…gives a visual picture of the shape of data. Students should learn that different graphs provide different snapshots of the data. For the particular question being answered, the choice of graphs is made around the notion of the shape of data” (pg. 389).

National Council of Teachers of Mathematics. (2011). Developing essential understanding of statistics grades 6 – 8. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., and Findell, B. (Eds.) Mathematics Learning Study Committee, Center for Education Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
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. (2004). Elementary and middle school mathematics: teaching developmentally. Pearson Education: Boston, MA.

 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 representations display the counts (frequencies) or measures of data values in an organized, visual format so that the data can be interpreted efficiently.
• What are the characteristics of a …
• frequency table
• bar graph
• dot plot
• stem-and-leaf plot
… and how can it be used to organize data?
• How does the …
• length of the bars in a bar graph
• density of dots in a dot plot
• density of leaves in a stem-and-leaf plot
… relate to the frequency and variability of the distribution of the data?
• How are categorical data and numerical data …
• alike?
• different?
• What relationships exist between graphs of data involving whole numbers and graphs of data involving fractions or decimals?
• What types of …
• conclusions can be drawn
• problems can be solved
… using data in a graph?
• What strategies can be used to solve problems using data in a graph?
• What is the purpose of an organized, visual format and how does it aid in the ability to efficiently answer questions and solve problems?
• Different data displays of the same data may appear different because of their unique display characteristics but the representations are equivalent in counts (frequencies) or measures of data values.
• How are frequency tables, dot plots, bar graphs, and stem-and-leaf plots …
• alike?
• different?
• What characteristics aid in determining if data representations show representations with equivalent data sets?
• Which representation is easier to interpret? Why?
• Why is it important to be able to use different display representations if they are equivalent in counts or data values?
• Data Analysis
• Data
• Interpretation
• Conclusions
• Statistical Representations
• Frequency tables
• Bar graphs
• Dot plots
• Stem-and-leaf 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 representations display the counts (frequencies) or measures of data values in an organized, visual format so that the data can be interpreted efficiently.
• What are the characteristics of a …
• frequency table
• scatterplot
… and how can it be used to organize data?
• What relationships exist between …
• a scatterplot and a coordinate plane?
• the axes in a scatterplot and number lines?
• the number pairs in a table and points on a scatterplot?
• the location of a data point on a scatterplot and the origin?
• What strategies can be used to …
• solve problems using data in a scatterplot?
• determine if a relationship exists between two sets of data?
• What is the purpose of an organized, visual format and how does it aid in the ability to efficiently answer questions and solve problems?
• Different data displays of the same data may appear different because of their unique display characteristics but the representations are equivalent in counts (frequencies) or measures of data values.
• How are frequency tables and scatterplots …
• alike?
• different?
• What characteristics aid in determining if data representations show representations with equivalent data sets?
• Which representation is easier to interpret? Why?
• Why is it important to be able to use different display representations if they are equivalent in counts or data values?
• Data Analysis
• Data
• Interpretation
• Conclusions
• Statistical Representations
• Frequency tables
• Scatterplots
• 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 transportation type to get to school, as well as representing numeric data such as shoe size, 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.
• Some students may try to represent numerical data in a stem-and-leaf plot without first arranging the leaves for each stem in order.

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 describe a relationship of a scatterplot by stating, “It just looks right,” rather than using mathematical reasoning such as “Most of the dots cluster around this point.”

#### Unit Vocabulary

• Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
• 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
• Discrete paired data – data that involves only distinct values that are finite or countable
• 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
• Frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs
• Graph – a visual representation of the relationships between data collected
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Scatterplot – a graphical representation used to display the relationship between the data of two variables
• 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.

Related Vocabulary:

 Axis/axes Double bar graph Frequency Horizontal Interval Key Label Leaf Number line Range Scale Stem Subtitle Tally marks Title 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 5 Mathematics TEKS

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.
5.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
5.1A 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• X. Connections
5.1B 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
5.1C 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 REAL OBJECTS, MANIPULATIVES, 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
• Real objects
• Manipulatives
• 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
5.1D 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.1E 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.1F 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• X. Connections
5.1G 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:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.9 Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data. The student is expected to:
5.9A Represent 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.
Supporting Standard

Represent

CATEGORICAL DATA WITH BAR GRAPHS OR FREQUENCY TABLES

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
• Categorical data – data that represents the attributes of a group of people, events, or objects
• May include numbers or ranges of numbers
• Limitations
• Whole numbers
• Fractions (proper, improper, and mixed numbers)
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Data representations
• Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
• Characteristics of a bar graph
• Titles, subtitles, and labels
• Title represents the purpose of collected data
• Subtitles clarify the meaning of data represented on each axis
• Labels identify each category
• Representation of categorical data
• Bars
• Placed in a horizontal or vertical linear arrangement to represent data
• Solid bars that are equal in width
• Independent bars that do not touch
• Length of the bar represents the distance from zero on the axis scale
• Axis
• Represented as a number line
• Scale intervals proportionally displayed
• Intervals of one or more units
• Every piece of data represented using a one-to-one or scaled correspondence, as indicated by the intervals on the axis
• Value of the data represented by the bar
• Determined by reading the number on the scaled axis associated with the length of the bar
• Represents the frequency for that category
• Double bar graph
• Two adjacent bars used to compare two sets of data for each category
• Frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs
• Characteristics of a frequency table
• Titles and column headers
• Title represents the purpose of collected data
• Column headers clarify the meaning of data represented in the table
• Representation of categorical data
• Table format
• Each category label listed in a row of the table
• Tally marks used to record the frequency of each category
• Numbers used to represent the count of tally marks in each category
• Every piece of data represented using a one-to-one correspondence
• Value of the data in each category
• Determined by the count of tally marks in that category
• Represents the frequency for that category
• Connection between graphs
• Same data represented using a frequency table and bar graph

Represent

NUMERICAL DATA, INCLUDING DATA SETS OF MEASUREMENTS IN FRACTIONS OR DECIMALS, WITH DOT PLOTS OR STEM-AND-LEAF 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
• Whole numbers
• Fractions (proper, improper, and mixed numbers)
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• 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 the 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 the dots relates to the frequency distribution of the 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
• Connection between graphs
• Same data represented using a dot plot and stem-and-leaf plot

Note(s):

• Grade Level(s):
• Grade 1 represented data to with picture and bar-type graphs.
• Grade 2 represented data with pictographs and bar graphs with intervals of one.
• 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 6 will represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VI.A. Statistical Reasoning – Describe Data
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
5.9B Represent discrete paired data on a scatterplot.
Supporting Standard

Represent

DISCRETE PAIRED DATA ON A SCATTERPLOT

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
• Discrete paired data – data that involves only distinct values that are finite or countable
• Limitations
• Various forms of positive rational numbers within related data pairs
• Counting (natural) numbers
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Fractions (positive proper, improper, and mixed numbers)
• Data representation
• Scatterplot – a graphical representation used to display the relationship between the data of two variables
• Characteristics of a scatterplot
• Titles and subtitles
• Title represents the purpose of collected data
• Subtitles clarify the meaning of the data represented on each axis
• First quadrant of coordinate plane
• Number lines form x-axis and y-axis
• Proportional increments
• Intervals of one or more
• Break between 0 and the first marked interval indicated in one or both axes to accommodate large numbers if necessary
• Ordered pairs
• Pairs of data form each ordered pair
• Points not connected by a line
• Data pairs are analyzed to find possible relationships between the two sets of data.
• Pairs of numbers collected to determine if a relationship exists between the two sets of data
• Relationship between each data pair is discrete although the data itself could be either continuous or discrete in nature

Note(s):

• Grade Level(s):
• Grade 5 introduces graphing in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table.
• Grade 5 introduces representing discrete paired data on a scatterplot.
• Grade 8 will construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VI.A. Statistical Reasoning – Describe Data
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
5.9C Solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot.
Readiness Standard

Solve

ONE- AND TWO-STEP PROBLEMS USING DATA FROM A FREQUENCY TABLE, DOT PLOT, BAR GRAPH, STEM-AND-LEAF PLOT, OR SCATTERPLOT

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
• Categorical data – data that represents the attributes of a group of people, events, or objects
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Discrete paired data – data that involves only distinct values that are finite or countable
• Limitations
• One- or two-step problems
• Addition
• Sums of whole numbers
• Sums of decimals up to the thousandths
• Sums of fractions with equal and unequal denominators
• Subtraction
• Differences of whole numbers
• Differences of decimals with values limited to the thousandths
• Differences of fractions with equal and unequal denominators
• Multiplication
• Products of whole numbers up to three-digit factors by two-digit factors
• Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths
• Multiply tenths by tenths (e.g., 0.3 × 0.7 = 0.21, 1.2 × 1.2 = 1.44, 14.3 × 1.3 = 18.59, etc.)
• Multiply tenths by hundredths or vice versa (e.g., 0.5 × 0.12 = 0.06, 1.4 × 0.15 = 0.21, 21.4 × 0.45 = 9.63, etc.)
• Multiply tenths by thousandths or vice versa (e.g., 0.4 × 0.125 = 0.05, 0.125 × 8.4 = 1.05, etc.)
• Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 × 1.3 = 3.9, 42 × 7.45 = 312.9, 7.02 × 78 = 547.56, 6 × 0.125 = 0.75, etc.)
• Products of fractions where factors are limited to a fraction and a whole number
• Division
• Whole numbers with quotients up to four-digit dividends and two-digit divisors
• Quotients of decimals limited to four-digit dividends and two-digit whole number divisors, with quotients to the hundredths
• Dividend to the tenths and whole number divisor (e.g., 1.2 ÷ 24 = 0.05, 358.8 ÷ 23 = 15.6, 721.7 ÷ 14 = 51.55, etc.)
• Dividend to the hundredths and whole number divisor (e.g., 8.68 ÷ 4 = 2.17, 8.25 ÷ 15 = 0.55, 62.76 ÷ 12 = 5.23, etc.)
• Whole number dividends and whole number divisors that yield decimal quotients to the hundredths (e.g., 3 ÷ 4 = 0.75, 10 ÷ 8 = 1.25, 1000 ÷ 16 = 62.5, etc.)
• Quotients of fractions where dividend and divisors are limited to whole numbers by unit fractions and unit fractions by whole numbers
• Data representations
• Frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs
• Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
• Double bar graph
• 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
• 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.
• Scatterplot – a graphical representation used to display the relationship between the data of two variables
• Solve problems using data represented in frequency tables, dot plots, bar graphs, stem-and-leaf plots, or scatterplots

Note(s):

• Grade Level(s):
• Grade 1 drew conclusions and generated and answered questions using information from picture and bar-type graphs.
• Grade 2 wrote and solved one-step word problems involving addition or subtraction using data represented within pictographs and bar graphs with intervals of one, and drew conclusions and made predictions from information in a graph.
• Grade 3 solved one-and two-step problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
• Grade 4 solved one and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot.
• Grade 6 will interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots.
• Grade 6 will use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VI.A. Statistical Reasoning – Describe Data
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills. The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum. School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency. Choose appropriate ELPS to support instruction. Click here to collapse or expand this section. ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.1A use prior knowledge and experiences to understand meanings in English ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment ELPS.c.1F use accessible language and learn new and essential language in the process ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations. ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs. ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency ELPS.c.3E share information in cooperative learning interactions ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired ELPS.c.3I adapt spoken language appropriately for formal and informal purposes ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment. ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to: ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs ELPS.c.4H read silently with increasing ease and comprehension for longer periods ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs. ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to: ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as: ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
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