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 TITLE : Unit 11: Making Connections SUGGESTED DURATION : 14 days

#### Unit Overview

Introduction
This unit bundles student expectations that address adding and subtracting whole numbers, decimal numbers, and fractions, solving one-, two-, or multi-step problems using all four operations, and solving problems involving input-output tables, numerical expressions, measurement concepts, and data representations. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Unit 02, students used addition or subtraction with whole numbers and decimals to tenths and hundredths to solve problems. In Unit 05, students applied problem-solving skills to solve one-, two-, or multi-step problems involving addition and subtraction of whole numbers and decimals to the hundredths place, multiplication of whole numbers up to two-digit factors and up to four-digit factors by one-digit factors, and division of whole numbers up to four-digit dividends by one-digit divisors with remainders in appropriate contexts. Students also represented real-world situations using input-output tables and numerical expressions to generate a number pattern that follows a given rule. These identified rules incorporated an algebraic understanding of the relationship of the values in the resulting sequence and their position in the sequence. In Unit 06, students represented and solved addition and subtraction of fractions with equal denominators. In Unit 07, students represented data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions. They examined the characteristics of each data representation, as well as compared the similarities and differences between them. Adequate understandings of these data representations allowed students to solve 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 Unit 08, students were formally introduced to formulas to determine the perimeter and area of rectangles and squares. Students used models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l × w). Students also solved problems involving length, problems that deal with measurements of intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.

During this Unit
Students revisit previously studied concepts, operations, multiple representations, and measurement. Students add and subtract whole numbers, decimals, and fractions with like denominators, solve problem situations involving the four operations, and represent those situations using strip diagrams and equations. Students represent real-world problem situations using input-output tables and numerical expressions to generate a number pattern that follows a given rule. Students solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate. They also solve problems related to perimeter and area of rectangles where dimensions are whole numbers. Students represent data in a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers, decimals, and fractions. Students are expected to solve one- and two-step problems, including situations involving the data representations.

Other considerations: Reference the Mathematics COVID-19 Implementation Tool Grade 4

After this Unit
In Grade 5, students will represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations. They will estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division. Students will multiply, with fluency, a three-digit number by a two-digit number using the standard algorithm and will solve, with proficiency, quotients up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm. In addition, students will represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity. They will generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph, and will be expected to recognize the difference between additive and multiplicative numerical patterns given in a table or graph. Grade 5 students will represent and solve problems involving perimeter and/or area, volume, and conversions within a measurement system, customary or metric. They will represent categorical data with bar graphs or frequency tables, and numerical data with dot plots or stem-and-leaf plots. Students will solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot.

Research
According to the National Research Council (2001), “when students fail to grasp the concepts that underlie procedures or cannot connect the concepts to the procedures, they frequently generate flawed procedures that result in systematic patterns of errors” (p. 196). Chapin and Johnson (2000), acknowledge that “when students explore patterns and generalize relationships among numbers, they are developing informal understanding of one of the most important topics in high school and college algebra – functions. A function is a relationship in which two sets are linked by a rule that pairs each element of the first set with exactly one element of the second set. We use functions every day without realizing it” (p. 205). The National Council of Teachers of Mathematics (2010) states that “One challenge for students is to understand the distinction between length and area. In fourth grade, students should have the opportunity to distinguish between the area and the perimeter of a shape and to realize that neither one determines the other” (p. 69). According to Van de Walle & Lovin(2006) “a focus in grades 3 – 5 should be to add to and refine the various forms of data representations that students have likely been exposed to in the early grades. Students should see that the primary purpose of data, either in graphical form or in numeric form, is to answer questions about the population from which the data are drawn” (p. 320).

Chapin, S & Johnson, A. (2000). Math matters: Understanding the math you teach. Sausalito, CA: Math Solutions Publications.
National Council of Teachers of Mathematics. (2010). Focus in grade 5: Teaching with curriculum focal points. 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, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3 – 5. Boston, MA: Pearson Education, Inc.

 Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Recognizing and understanding numerical patterns and operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (addition and subtraction of whole numbers and decimals; addition and subtraction of fractions with equal denominators; multiplication and division of whole numbers).
• How does the context of a problem situation affect the representation, operation(s), and/or solution strategy that could be used to solve the problem?
• How can representing a problem situation using …
• an equation(s) with a letter standing for the unknown
• objects and pictorial models that build to the number line and properties of operations
• strip diagrams or other pictorial models
• input-output tables and numerical expressions
… aid in problem solving?
• What patterns and relationships can be found within and between the words, pictorial models, number patterns, tables, rules, expressions, and equations used to represent a problem situation?
• How does understanding …
• relationships within and between operations
• properties of operations
• place value
• partial products
… aid in determining an efficient strategy or representation to investigate problem situations?
• What strategies can be used to determine …
• the sum
• the difference
• any unknown
… in an addition or subtraction situation involving …
• whole numbers and decimals?
• fractions with like denominators?
• What strategies can be used to determine …
• the product
• an unknown factor
… in a multiplication situation?
• What strategies can be used to determine …
• the quotient
• an unknown
… in a division situation?
• Why is it important to understand when and how to use standard algorithms?
• When adding or subtracting decimal numbers, why is it important to align the place values?
• Why is it important to be able to perform operations with whole numbers fluently?
• What relationships exist between …
• multiplication and division?
• subtraction and division?
• quotients and remainders?
• operations with whole numbers and operations with decimals?
• input-output tables, number patterns, numerical expressions, rules, and sequences?
• When using addition to solve a problem situation, why can the order of the addends be changed?
• When using subtraction to solve a problem situation, why can the order of the minuend and subtrahend not be changed?
• When using multiplication to solve a problem situation, why can the order of the factors be changed?
• When using division to solve a problem situation, why can the order of the dividend and divisor not be changed?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (addition and subtraction of whole numbers and decimals; addition and subtraction of fractions with equal denominators; multiplication and division of whole numbers).
• When adding two non-zero whole numbers and/or positive decimals, why is the sum always greater than each of the addends?
• When subtracting two non-zero whole numbers and/or positive decimals with the minuend greater than the subtrahend, why is the difference always less than the minuend?
• When multiplying two counting numbers greater than one, why is the product always greater than each of the factors?
• When dividing two counting numbers greater than one with the dividend greater than the divisor, why is the quotient always …
• less than the dividend?
• greater than one?
• Number and Operations
• Number
• Whole numbers
• Fractions
• Decimals
• Operations
• Subtraction
• Multiplication
• Division
• Problem Types
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Patterns and Relationships
• Input-output tables
• Representations
• Concrete models
• Pictorial models
• Expressions
• Equations
• 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.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? 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)
• Objects have unique measurable attributes that can be defined and described in order to make sense of their relationship to other objects in the world.
• How does the context of a problem situation affect the representation, operation(s), and/or solution strategy that could be used to solve the problem?
• What strategies can be used to solve problems involving …
• perimeter of a rectangle?
• area of a rectangle?
• length?
• intervals of time?
• liquid volumes?
• mass?
• money?
• How does understanding …
• relationships within and between operations
• properties of operations
• place value
• the relationship between addition and perimeter
• the relationship between multiplication and area
• measurement concepts
• the difference between length of time and time of day
… aid in determining an efficient strategy or representation to investigate problem situations?
• 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
• dot plot
• stem-and-leaf plot
… and how can it be used to organize data?
• How does the density of …
• dots in a dot plot
• leaves in a stem-and-leaf plot
… relate to the frequency and variability of the distribution of the 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, dot plots, 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?
• Measurement
• Geometric Relationships
• Perimeter
• Area
• Measureable Attributes
• Distance and length
• Capacity and liquid volume
• Mass
• Time
• Money
• Data Analysis
• Data
• Statistical Representations
• Frequency tables
• 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.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Underdeveloped Concepts:

• Some students may think that decimal numbers should be lined up vertically according to the maximum number of digits in order to use the standard algorithm rather than realizing that they must be lined up according to place values.
• Students may think that when adding fractions with equal denominators that they must add the numerators and then add the denominators.
• Some students may attempt to solve multistep problems by using only a one-step process.
• Some students may apply a rule for an additive numerical pattern to a multiplicative pattern.
• Some students may misinterpret values in an input-output table by comparing input values to other input values, or by comparing output values to other output values, rather than interpreting the relationship between input values and corresponding output values.
• Some students may use a formula to find area of a rectangle, but misunderstand that area could also be found by arranging an array of unit squares that sufficiently cover that rectangle.
• 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.
• Some students may confuse numerical data with a count or measure of the data.
• Some students may try to represent numerical data in a stem-and-leaf plot without first arranging the leaves for each stem in order.
• Some students may not transfer the understanding that 10 in any place value position (place) makes one (group) in the next place position or vice versa when adding or subtracting whole numbers to adding or subtracting decimals.
• Some students may have a procedural understanding of the standard algorithms for addition and/or subtraction while lacking conceptual understanding of the operations.
• Some students who work through the standard algorithm procedures may think about numbers as digits and ignore place value leading to an unreasonable amount rather than think about place value to help determine a reasonable amount.
• Some students may think that decimal numbers should be lined up vertically according to the maximum number of digits in order to use the standard algorithm rather than realizing that they must be lined up according to place values.
• Some students may misrepresent a problem situation such as “8 pages of an album with 56 baseball cards arranged equally on the pages results in 7 cards on each page” with the equation “8 ÷ 56 = 7”.
• Some students may be able to perform a symbolic procedure for multiplication or division with limited understanding of the multiplication or division concepts involved.
• Some students may be emergent thinkers when making connections among strip diagrams, equations, and problem situations.
• Some students may begin measuring at the end of the ruler instead of at zero, while some students may begin measuring beginning with the number 1 on the ruler without compensating for the missing unit.
• Some students may not recognize that the dots on a dot plot may represent more than one piece of data, as do the symbols in a pictograph.
• Although some students may be proficient at displaying data using different representations, they may lack the experience to solve problems by analyzing the data.

#### Unit Vocabulary

• Additive numerical pattern – a pattern that occurs when a constant non-zero value is added to an input value to determine the output value
• Area – the measurement attribute that describes the number of square units a figure or region covers
• Associative property of addition – if three or more addends are added, they can be grouped in any order, and the sum will remain the same; a + b + c = (a + b) + c = a + (b + c)
• Categorical data – data that represents the attributes of a group of people, events, or objects
• Commutative property of addition – if the order of the addends are changed, the sum will remain the same; a + b = c; therefore, b + a = c
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Data – information that is collected about people, events, or objects
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• 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 category or number occurs
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Expression – a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s)
• Factor – a number multiplied by another number to find a product
• Fluency – efficient application of procedures with accuracy
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• 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
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Mixed number – a number that is composed of a whole number and a fraction
• Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Perimeter – a linear measurement of the distance around the outer edge of a figure
• Product – the total when two or more factors are multiplied
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Rule – an expression describing the relationship between the input and output values in a pattern or sequence
• Sequence – a list of numbers or a collection of objects in a specific order that follows a particular pattern or rule
• 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.
• Strip diagram – a linear model used to illustrate number relationships
• Trailing zeros – a sequence of zeros in the decimal part of a number that follow the last non-zero digit, and whether recorded or deleted, does not change the value of the number
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Analog clock Attribute Beaker Capacity Cent Centimeter Composite figure Congruent Cup Customary Day Difference Digital clock Dollar Duration End time Equivalent Eye dropper Fluid ounce Foot Formula Gallon Graduated cylinder Gram Horizontal Hour Hundredth Inch Input Interval Key Kilogram Kiloliter Kilometer Label Length Liquid volume Liter Mass Measuring container or jar Measuring cup Measuring tape Meter Meter stick Metric Mile Milligram Milliliter Millimeter Minute Money Month Number line Ounce Output Pan balance Parallel Perpendicular Pint Place value Position Pound Quart Range Rectangle Remainder Right angle Rule/process Ruler Scale Second Side Square Square unit Standard algorithm Start time Stop watch Sum Tally mark Tenth Time Title Ton Triple beam balance Unit Unknown Value Vertex Vertical Volume Week Weight Yard Yardstick Year
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 Center 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 4 Mathematics TEKS

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE 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.

Specificity 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.
TEKS# SE# TEKS SPECIFICITY
4.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
4.1A 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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.
4.1B 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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 problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
4.1C 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1D 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
4.1E 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1F 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1G 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.3 Number and operations. The student applies mathematical process standards to represent and generate fractions to solve problems. The student is expected to:
4.3E Represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.

Represent, Solve

ADDITION AND SUBTRACTION OF FRACTIONS WITH EQUAL DENOMINATORS USING OBJECTS AND PICTORIAL MODELS THAT BUILD TO THE NUMBER LINE AND PROPERTIES OF OPERATIONS

Including, but not limited to:

• Fractions (proper, improper, or mixed numbers with equal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Sums of fractions limited to equal denominators
• Subtraction
• Differences of fractions limited to equal denominators
• Fractional relationships
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Referring to the same whole
• Fractions are relationships, and the size or the amount of the whole matters
• Common whole is needed when adding or subtracting fractions
• Equivalent fractions to simplify solutions
• Concrete objects and pictorial models for addition of fractions with equal denominators that build to the number line
• Pattern blocks and other shapes (circles, squares, rectangles, etc.)
• Fraction strips and other strip models
• Relationships between concrete objects and pictorial models for addition of fractions with equal denominators, number lines, and properties of operations
• Properties of operations
• Commutative property of addition – if the order of the addends are changed, the sum will remain the same
• a + b = c; therefore, b + a = c
• Associative property of addition – if three or more addends are added, they can be grouped in any order, and the sum will remain the same
• a + b + c = (a + b) + c = a + (b + c)
• Pattern blocks and other shapes (circles, squares, rectangles, etc.)
• Fraction strips and other strip models
• Concrete objects and pictorial models for subtraction of fractions with equal denominators that build to the number line
• Pattern blocks and other shapes (circles, squares, rectangles, etc.)
• Fraction strips and other strip models
• Recognition of addition and subtraction in mathematical and real-world problem situations

Note(s):

• Grade 4 introduces representing and solving addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.
• Grade 5 will represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
4.4 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations and decimal sums and differences in order to solve problems with efficiency and accuracy. The student is expected to:
4.4A Add and subtract whole numbers and decimals to the hundredths place using the standard algorithm.

WHOLE NUMBERS AND DECIMALS TO THE HUNDREDTHS PLACE USING THE STANDARD ALGORITHM

Including, but not limited to:

• Whole numbers
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Addition and subtraction of whole numbers
• Connection between place value and the standard algorithm
• Standard algorithm
• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Addition and subtraction of decimals
• Relate addition and subtraction of decimals to the hundredths place using concrete objects and pictorial models to the standard algorithm for adding and subtracting decimals.
• Trailing zeros – a sequence of zeros in the decimal part of a number that follow the last non-zero digit, and whether recorded or deleted, does not change the value of the number
• Standard algorithm

Note(s):

• Grade 3 solved with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
• Grade 4 extends adding and subtracting of whole numbers from 1,000 to 1,000,000 and introduces adding and subtracting decimals, including tenths and hundredths.
• Grade 5 will estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
4.4H Solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.

Solve

WITH FLUENCY ONE- AND TWO-STEP PROBLEMS INVOLVING MULTIPLICATION AND DIVISION, INCLUDING INTERPRETING REMAINDERS

Including, but not limited to:

• Whole numbers
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Fluency – efficient application of procedures with accuracy
• Standard algorithms for the four operations
• Automatic recall of basic facts
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Division
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Quotients up to four-digit dividends by one-digit divisors
• Quotients may include remainders
• Remainder dependent upon the mathematical or real-world situation
• Various ways to record remainder
• Ignore the remainder
• Add one to the quotient
• Remainder recorded as a fraction
• One- and two-step problem situations
• One-step problems
• Recognition of multiplication and division in mathematical and real-world problem situations
• Two-step problems
• Two-step problems must have one-step in the problem that involves multiplication and/or divison; however, the other step in the problem can involve addition and/or subtraction.
• Recognition of multiplication and division in mathematical and real-world problem situations
• Equation(s) to reflect solution process

Note(s):

• Grade 4 introduces solving with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.
• Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
• Grade 5 will solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm.
• Various mathematical process standards will be applied to this student expectation as appropriate
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
4.5 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:
4.5A Represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.

Represent

MULTI-STEP PROBLEMS INVOLVING THE FOUR OPERATIONS WITH WHOLE NUMBERS USING STRIP DIAGRAMS AND EQUATIONS WITH A LETTER STANDING FOR THE UNKNOWN QUANTITY

Including, but not limited to:

• Whole numbers
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Sums of whole numbers
• Subtraction
• Differences of whole numbers
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Division
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Quotients of whole numbers up to four-digit dividends by one-digit divisors
• Quotients may include remainders
• Representations of an unknown quantity in an equation
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Any single letter to represent the unknown quantity (e.g., 24 – 8 = y, etc.)
• Equal sign at beginning or end and unknown in any position
• Multi-step problem situations involving the four operations in a variety of problem structures
• Recognition of addition, subtraction, multiplication, and/or division in mathematical and real-world problem situations
• Representation of problem situations with strip diagrams and equations with a letter standing for the unknown
• Strip diagram – a linear model used to illustrate number relationships
• Relationship between quantities represented and problem situation

Note(s):

• Grade 3 represented one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations.
• Grade 3 represented and solved one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations.
• Grade 3 determined the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product.
• Grade 5 will represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• 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.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.
• 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.
4.5B Represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence.

Represent

PROBLEMS USING AN INPUT-OUTPUT TABLE AND NUMERICAL EXPRESSIONS TO GENERATE A NUMBER PATTERN THAT FOLLOWS A GIVEN RULE REPRESENTING THE RELATIONSHIP OF THE VALUES IN THE RESULTING SEQUENCE AND THEIR POSITION IN THE SEQUENCE

Including, but not limited to:

• Whole numbers
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Sums of whole numbers
• Subtraction
• Differences of whole numbers
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Division
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Quotients of whole numbers up to four-digit dividends by one-digit divisors
• Data sets of whole numbers
• Sets may or may not begin with 1
• Sets may or may not be listed in sequential order
• Sequence – a list of numbers or a collection of objects in a specific order that follows a particular pattern or rule
• Expression – a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s)
• Rule – an expression describing the relationship between the input and output values in a pattern or sequence
• Various representations of problem situations
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Relationship between input-output tables and number patterns
• When the input is the position in the sequence, then the output is the value in the sequence.
• When the input is the value in the sequence, then the output is the position in the sequence.
• Relationship between values in a number pattern
• Additive numerical pattern – a pattern that occurs when a constant non-zero value is added to an input value to determine the output value
• Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value
• Relationship between numerical expressions and rules to create input-output tables representing the relationship between each position in the sequence and the value in the sequence

Note(s):

• Grade 3 represented real-world relationships using number pairs in a table and verbal descriptions.
• Grade 5 will generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph.
• Grade 5 will recognize the difference between additive and multiplicative numerical patterns given in a table or graph.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• V.B. Statistical Reasoning – Describe data
• V.B.4. Describe patterns and departure from patterns in the study of data.
• 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.
4.5D Solve problems related to perimeter and area of rectangles where dimensions are whole numbers.

Solve

PROBLEMS RELATED TO PERIMETER AND AREA OF RECTANGLES WHERE DIMENSIONS ARE WHOLE NUMBERS

Including, but not limited to:

• Rectangle
• 4 sides
• 4 vertices
• Opposite sides congruent
• 2 pairs of parallel sides
• 4 pairs of perpendicular sides
• 4 right angles
• Square (a special type of rectangle)
• 4 sides
• 4 vertices
• All sides congruent
• 2 pairs of parallel sides
• 4 pairs of perpendicular sides
• 4 right angles
• Perimeter – a linear measurement of the distance around the outer edge of a figure
• Perimeter is a one-dimensional linear measure.
• Whole number side lengths
• Recognition of perimeter embedded in mathematical and real-world problem situations
• Formulas for perimeter from STAAR Grade 4 Mathematics Reference Materials
• Square
• P = 4s, where s represents the side length of the square
• Rectangle
• P = l + w + l + w or P = 2l + 2w, where l represents the length of the rectangle and w represents the width of the rectangle
• Determine perimeter when given side lengths with or without models
• Determine perimeter by measuring to determine side lengths
• Ruler, STAAR Grade 4 Mathematics Reference Materials ruler, yardstick, meter stick, measuring tape, etc.
• Determine missing side length when given perimeter and remaining side length
• Perimeter of composite figures
• Area – the measurement attribute that describes the number of square units a figure or region covers
• Area is a two-dimensional square unit measure.
• Whole number side lengths
• Recognition of area embedded in mathematical and real-world problem situations
• Formulas for area from STAAR Grade 4 Mathematics Reference Materials
• Square
• A = s × s, where s represents the side length of the square
• Rectangle
• A = l × w, where l represents the length of the rectangle and w represents the width of the rectangle
• Determine area when given side lengths with and without models
• Determine area by measuring to determine side lengths
• Ruler, STAAR Grade 4 Mathematics Reference Materials ruler, yardstick, meter stick, measuring tape, etc.
• Determine missing side length when given area and remaining side length
• Area of composite figures
• Multiple ways to decompose a composite figure to determine perimeter and/or area

Note(s):

• Grade 4 introduces solving problems related to perimeter and area of rectangles where dimensions are whole numbers.
• Grade 5 will represent and solve problems related to perimeter and/or area and related to volume.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.1. Find the perimeter and area of two-dimensional figures.
4.8 Geometry and measurement. The student applies mathematical process standards to select appropriate customary and metric units, strategies, and tools to solve problems involving measurement. The student is expected to:
4.8C Solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.

Solve

PROBLEMS THAT DEAL WITH MEASUREMENTS OF LENGTH, INTERVALS OF TIME, LIQUID VOLUMES, MASS, AND MONEY USING ADDITION, SUBTRACTION, MULTIPLICATION, OR DIVISION AS APPROPRIATE

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Products of two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Quotients up to four-digit dividends by one-digit divisors
• Decimals (greater than one and less than one)
• Addition and subtraction of money amounts up to hundredths
• Conversions limited to multiples of halves (e.g., 0.5, 1.5, 4.5, etc.)
• Determined using reasoning that half of any value is that value divided by 2
• Fractions (proper, improper, and mixed numbers)
• Addition and subtraction of fractions with like denominators
• Conversions limited to multiples of halves (e.g., etc.)
• Determined using reasoning that half of any value is that value divided by 2
• Typically used customary and metric units
• Customary
• Length: miles, yards, feet, inches
• Volume (liquid volume) and capacity: gallons, quarts, pints, cups, fluid ounces
• Weight: tons, pounds, ounces
• Metric
• Length: kilometer, meter, centimeters, millimeters
• Volume (liquid volume) and capacity: kiloliter, liter, milliliter
• Mass: kilogram, gram, milligram
• Based on prefixes attached to base unit
• Base units include meter for length, liter for volume and capacity, and gram for weight and mass.
• Kilo: one thousand base units
• Deci: one-tenth of a base unit
• Centi: one-hundredth of a base unit
• Milli: one-thousandth of a base unit
• Typically used measurement tools
• Customary
• Length: rulers, yardsticks, measuring tapes
• Volume (liquid volume) and capacity: measuring cups, measuring containers or jars
• Metric
• Length: rulers, meter sticks, measuring tapes
• Volume (liquid volume) and capacity: beakers, graduated cylinders, eye droppers, measuring containers or jars
• Mass: pan balances, triple beam balances
• Problem situations that deal with measurements of length
• Addition, subtraction, multiplication, and/or division of measurements of length with or without conversion
• May or may not include using measuring tools to determine length
• Problem situations that deal with intervals of time (clocks: hours, minutes, seconds)
• Addition and subtraction of time intervals in minutes
• Such as a 1 hour and 45-minute event minus a 20-minute event equals 1 hour 25 minutes
• Time intervals given
• Pictorial models and tools
• Measurement conversion tables
• Analog clock with gears, digital clock, stop watch, number line, etc.
• Time conversions
• 1 hour = 60 minutes; 1 minute = 60 seconds
• Fractional values of time limited to multiples of halves
• Elapsed time
• Finding the end time
• Finding the start time
• Finding the duration
• Problem situations that deal with intervals of time (calendar: years, months, weeks, days)
• Time conversions
• 1 year = 12 months; 1 year = 52 weeks; 1 week = 7 days; 1 day = 24 hours
• Fractional values of time limited to multiples of halves
• Problem situations that deal with measurements of volume (liquid volume) and capacity
• Addition, subtraction, multiplication, and/or division of measurements of volume (liquid volume) and capacity with or without conversion
• May or may not include using measuring tools to determine volume (liquid volume) and capacity
• Problem situations that deal with measurements of mass
• Addition, subtraction, multiplication, and/or division of measurements of mass with or without conversion
• May or may not include using measuring tools to determine mass
• Problem situations that deal with money
• Addition and subtraction may include whole number or decimal amounts
• Multiplication and division limited to amounts expressed as cents or dollars with no decimal values
• Comparison of money amounts
• Making change
• Range of dollar amounts

Note(s):

• Grade 3 determined solutions to problems involving addition and subtraction of time intervals in minutes using pictorial models or tools such as a 15-minute event plus a 30-minute event equals 45 minutes.
• Grade 4 introduces solving problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.
• Grade 5 will solve problems by calculating conversions within a measurement system, customary or metric.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
4.9 Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data. The student is expected to:
4.9A Represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions.

Represent

DATA ON A FREQUENCY TABLE, DOT PLOT, OR STEM-AND-LEAF PLOT MARKED WITH WHOLE NUMBERS AND FRACTIONS

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
• Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
• Can be counted or measured.
• Limitations
• Whole numbers
• Fractions (proper, improper, and mixed numbers)
• Data representations
• 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
• Title represents the purpose of collected data
• Column headers clarify the meaning of the data represented in the table
• Representation of categorical or numerical 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
• 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 category or number occurs
• Characteristics of a dot plot
• Titles, subtitles, and labels
• Title represents the purpose of collected data
• Subtitle clarifies the meaning of categories or number line
• Labels identify each category or numerical increment below the line
• Representation of categorical or 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
• Categorical data represented by a line segment labeled with categories
• 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
• 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 the leaves relates to the frequency distribution of the data
• Connection between graphs representing the same data
• Dot plot to stem-and-leaf plot
• Stem-and-leaf plot to dot plot
• Same data represented using a frequency table, dot plot, or stem-and-leaf plot

Note(s):

• Grade 1 represented data with picture and bar-type graphs.
• Grade 2 represented data with pictographs and bar graphs with intervals of one.
• Grade 3 summarized a data set with multiple categories using a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
• Grade 4 introduces representing data on a stem-and-leaf plot.
• Grade 5 will 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.
• 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• V.B. Statistical Reasoning – Describe data
• V.B.2. Construct appropriate visual representations of data.
•  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.
4.9B Solve 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.
Supporting Standard

Solve

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

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
• Limitations
• One- or two-step problems
• Sums of whole numbers
• Sums of decimals up to the hundredths
• Sums of fractions limited to equal denominators
• Subtraction
• Differences of whole numbers
• Differences of decimals with values limited to the hundredths
• Differences of fractions limited to equal denominators
• Multiplication
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Division
• Quotients of whole numbers up to four-digit dividends by one-digit divisors
• Data Representations
• Frequency table – a table to organize data that lists categories and 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 category or 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.
• Solve problems using data represented in frequency tables, dot plots, or stem-and-leaf plots

Note(s):

• 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 5 will solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.