
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
strikethrough.

Legend:  Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
 Unitspecific 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:

5.1B 
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:
 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 nonexamples, 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:

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 stemandleaf 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 onetoone 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 onetoone 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 STEMANDLEAF 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 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 the dots relates to the frequency distribution of the 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
 Connection between graphs
 Same data represented using a dot plot and stemandleaf plot
Note(s):
 Grade Level(s):
 Grade 1 represented data to with picture and bartype 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 stemandleaf plot marked with whole numbers and fractions.
 Grade 6 will represent numeric data graphically, including dot plots, stemandleaf 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 xaxis and yaxis
 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 realworld problems, including those generated by number patterns or found in an inputoutput 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, nonlinear, 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 twostep problems using data from a frequency table, dot plot, bar graph, stemandleaf plot, or scatterplot.
Readiness Standard

Solve
ONE AND TWOSTEP PROBLEMS USING DATA FROM A FREQUENCY TABLE, DOT PLOT, BAR GRAPH, STEMANDLEAF 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 twostep 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 threedigit factors by twodigit factors
 Products of decimals limited to threedigit factors by twodigit 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 fourdigit dividends and twodigit divisors
 Quotients of decimals limited to fourdigit dividends and twodigit 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
 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
 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.
 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, stemandleaf plots, or scatterplots
Note(s):
 Grade Level(s):
 Grade 1 drew conclusions and generated and answered questions using information from picture and bartype graphs.
 Grade 2 wrote and solved onestep 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 oneand twostep problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
 Grade 4 solved one and twostep problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stemandleaf plot.
 Grade 6 will interpret numeric data summarized in dot plots, stemandleaf 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
