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Instructional Focus Document
Grade 4 Mathematics
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.

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.

Additional Notes
In Grade 4, solving with fluency one- and two-step problems involving multiplication and division, representing multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity, representing problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule and representing the relationship of the values in the resulting sequence and their position in the sequence are addressed by STAAR Readiness Standards 4.4H, 4.5A, and 4.5B. All of these standards are subsumed under the Grade 4 Texas Response to Curriculum Focal Points (TxRCFP): Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems. 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 is STAAR Readiness Standard 4.3E and within the Grade 4 Focal Point: Building foundations for addition and subtraction of fractions (TxRCFP). Adding and subtracting whole numbers and decimals to the hundredths place using the standard algorithm is STAAR Readiness Standard 4.4A and part of the Grade 4 Focal Points: Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems and Understanding decimals and addition and subtraction of decimals (TxRCFP). All of these standards are all included in the Grade 4 STAAR Reporting Category 2: Computations and Algebraic Relationships. Solving problems related to perimeter and area of rectangles where dimensions are whole numbers and solving problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate are STAAR Readiness Standards 4.5D and 4.8C. These two standards are part of the Grade 4 Focal Point: Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems as well as the Grade 4 STAAR Reporting Category 3: Geometry and Measurement. Representing data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions is STAAR Readiness Standard. 4.9A. Solving 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 is STAAR Supporting Standard 4.9B. These two standards are part of the Grade 4 STAAR Reporting Category 4: Data Analysis and Personal Financial Literacy and encompassed within the Grade 4 Focal Points: Building foundations for addition and subtraction of fractions, Understanding decimals and addition and subtraction of decimals, and Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, II.D. Algebraic Reasoning – Representations, III.A. Geometric Reasoning – Figures and their properties, IV.A. Measurement Reasoning – Measurement involving physical and natural attributes, IV.C. Measurement Reasoning – Measurement involving geometry and algebra, VI.B. Statistical Reasoning – Describe Data, VIII. Problem Solving and Reasoning, IX. Communication and Representation, X. Connections.

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 …
      • addition and subtraction?
      • addition and multiplication?
      • 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
      • Addition
      • 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 A over B.png 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 A over B.png 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 A over B.png 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 Creator if your district has granted access to that tool.

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

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

 

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

 

Texas Education Agency – Mathematics Curriculum

 

Texas Education Agency – STAAR Mathematics Resources

 

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

 

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

 

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

 

Texas Education Agency Texas Gateway – Resources Aligned to Grade 4 Mathematics TEKS


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

Legend:

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

Legend:

  • Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
  • Unit-specific clarifications are in italicized, blue text.
  • Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
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:
    • X. Connections
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:
    • VIII. Problem Solving and Reasoning
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:
    • VIII. Problem Solving and Reasoning
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:
    • IX. Communication and Representation
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:
    • IX. Communication and Representation
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:
    • X. Connections
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:
    • IX. Communication and Representation
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.
Readiness Standard

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
  • Addition
    • 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 Level(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. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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.
Readiness Standard

Add, Subtract

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 Level(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. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
4.4H Solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.
Readiness Standard

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 is the answer
            • 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 Level(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. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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.
Readiness Standard

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}
  • Addition
    • 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 Level(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:
    • I. Numeric Reasoning
    • II.D. Algebraic Reasoning – Representations
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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.
Readiness Standard

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}
  • Addition
    • 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 Level(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:
    • I. Numeric Reasoning
    • II.D. Algebraic Reasoning – Representations
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
4.5D Solve problems related to perimeter and area of rectangles where dimensions are whole numbers.
Readiness Standard

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 Level(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:
    • III.A. Geometric Reasoning – Figures and their properties
    • IV.A. Measurement Reasoning – Measurement involving physical and natural attributes
    • IV.C. Measurement Reasoning – Measurement involving geometry and algebra
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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.
Readiness Standard

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 Level(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. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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.
Readiness Standard

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
        • Titles and column headers
          • 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
        • Titles and column headers
          • Title represents the purpose of collected data
          • Column headers indicate stems and leaves
        • Representation of numerical data
          • Vertical line, such as in a T-chart, separates stems from their corresponding leaves
          • Stems listed to the left of the vertical line with their corresponding leaves listed in a row to the right of the vertical line
        • Determination of place value(s) that represents stems versus place value(s) that represents leaves is dependent upon how to best display the distribution of the entire data set and then indicated by a key
          • Left digit(s) of the data forms the stems and remaining digit(s) or fraction forms the leaves that correspond with each stem, as indicated by the key
        • Every piece of data represented using a one-to-one correspondence, including repeated values
          • Stem represents one or more pieces of data in the set
          • Leaf represents only one piece of data in the set
        • Leaves provide frequency counts for the range of numbers included in that row of the stem-and-leaf plot
        • Density of 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 Level(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:
    • VI.B. Statistical Reasoning – Describe Data
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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
    • Addition
      • 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 Level(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:
    • I. Numeric Reasoning
    • VI.B. Statistical Reasoning – Describe Data
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
Click here to collapse or expand this section.
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 03/01/2019
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