§111.1. Implementation of Texas Essential Knowledge and Skills for Mathematics, Elementary, Adopted 2012.
Source: The provisions of this §111.1 adopted to be effective September 10, 2012, 37 TexReg 7109.
§111.6. Grade 4, Adopted 2012.

4.Intro.1  The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century. 

4.Intro.2  The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. 

4.Intro.3  For students to become fluent in mathematics, students must develop a robust sense of number. The National Research Council's report, "Adding It Up," defines procedural fluency as "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately." As students develop procedural fluency, they must also realize that true problem solving may take time, effort, and perseverance. Students in Grade 4 are expected to perform their work without the use of calculators. 

4.Intro.4  The primary focal areas in Grade 4 are use of operations, fractions, and decimals and describing and analyzing geometry and measurement. These focal areas are supported throughout the mathematical strands of number and operations, algebraic reasoning, geometry and measurement, and data analysis. In Grades 35, the number set is limited to positive rational numbers. In number and operations, students will apply place value and represent points on a number line that correspond to a given fraction or terminating decimal. In algebraic reasoning, students will represent and solve multistep problems involving the four operations with whole numbers with expressions and equations and generate and analyze patterns. In geometry and measurement, students will classify twodimensional figures, measure angles, and convert units of measure. In data analysis, students will represent and interpret data. 

4.Intro.5  Statements that contain the word "including" reference content that must be mastered, while those containing the phrase "such as" are intended as possible illustrative examples. 

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
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:

4.1B 
Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 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
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 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
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 IX. Communication and Representation

4.1E 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 IX. Communication and Representation

4.1F 
Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Mathematical relationships
 Connect and communicate mathematical ideas
 Conjectures and generalizations from sets of examples and nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:

4.1G 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 IX. Communication and Representation

4.2 
Number and operations. The student applies mathematical process standards to represent, compare, and order whole numbers and decimals and understand relationships related to place value. The student is expected to:

4.2A 
Interpret
THE VALUE OF EACH PLACEVALUE POSITION AS 10 TIMES THE POSITION TO THE RIGHT AND AS ONETENTH OF THE VALUE OF THE PLACE TO ITS LEFT
Including, but not limited to:
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 One billions place
 Hundred millions place
 Ten millions place
 One millions place
 Hundred thousands place
 Ten thousands place
 One thousands place
 Hundreds place
 Tens place
 Ones place
 Tenths place
 Hundredths place
 Base10 place value system
 A number system using ten digits 0 – 9
 Relationships between places are based on multiples of 10.
 Moving left across the places, the values are 10 times the position to the right.
 Ex:
 Moving right across the places, the values are onetenth the value of the place to the left.
 Ex:
 Place value relationships and relationships between the values of digits involving whole numbers through (less than or equal to) 1,000,000,000 and decimals to the hundredths (greater than or equal to 0.01)
 Ex:
 Place value relationships are based on multiples of ten whereas relationships between the values of digits may or may not be based on multiples of ten.
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 described the mathematical relationships found in the base10 place value system through the hundred thousands place.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate .
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation

4.2B 
Represent
THE VALUE OF THE DIGIT IN WHOLE NUMBERS THROUGH 1,000,000,000 AND DECIMALS TO THE HUNDREDTHS USING EXPANDED NOTATION AND NUMERALS
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Decimals (great than or equal to 0.01)
 Decimal number – a number in the base10 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
 Numeral – a symbol used to name a number
 Digit – any numeral from 0 – 9
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 One billions place
 Hundred millions place
 Ten millions place
 One millions place
 Hundred thousands place
 Ten thousands place
 One thousands place
 Hundreds place
 Tens place
 Ones place
 Tenths place
 Hundredths place
 Base10 place value system
 A number system using ten digits 0 – 9
 Relationships between places are based on multiples of 10.
 Moving left across the places, the values are 10 times the position to the right
 Ex:
 Multiplying a number by 10 increases the place value of each digit.
 Ex: 10 × [4(100) + 8(10) + 9 + 3(0.1) + 2 (0.01)] = 4(1,000) +8(100) + 9(10) + 3 + 2(0.1)
 Ex: 489.32 × 10 = 4893.2 × 1
 Moving right across the places, the values are onetenth the value of the place to the left.
 Ex:
 Dividing a number by 10 decreases the place value of each digit.
 Ex: [4(100) + 8(10) + 9 + 3 (0.1)] ÷ 10 = 4(10) + 8 + 9(0.1) + 3(0.01)
 Ex: 489. 32 ÷ 10 = 48.932 ÷ 1
 The magnitude (relative size) of whole number places through the billions place
 The magnitude of one billion
 1,000,000,000 can be represented as 10 hundred millions.
 1,000,000,000 can be represented as 100 ten millions.
 1,000,000,000 can be represented as 1,000 one millions.
 The magnitude (relative size) of decimal places through the hundredths
 The magnitude of onetenth
 0.1 can be represented as 1 tenth.
 0.1 can be represented as 10 hundredths.
 The magnitude of onehundredth
 0.01 can be represented as 1 hundredth.
 Standard form – the representation of a number using digits (e.g., 985,156,789.78)
 Period – a threedigit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
 Billions period is composed of the one billions place, ten billions place, and hundred billions place.
 Millions period is composed of the one millions place, ten millions place, and hundred millions place.
 Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
 Units period is composed of the ones place, tens place, and hundreds place.
 The word “billion” after the numerical value of the billions period is stated when read.
 A comma between the billions period and the millions period is recorded when written but not stated when read.
 The word “million” after the numerical value of the millions period is stated when read.
 A comma between the millions period and the thousands period is recorded when written but not stated when read.
 The word “thousand” after the numerical value of the thousands period is stated when read.
 A comma between the thousands period and the units period is recorded when written but not stated when read.
 The word “unit” after the numerical value of the units period is not stated when read.
 The word “hundred” in each period is stated when read.
 The words “ten” and “one” in each period are not stated when read.
 The tens place digit and ones place digit in each period are stated as a twodigit number when read.
 The whole part of a decimal number is recorded to the left of the decimal point when written and stated as a whole number.
 The decimal point is recorded to separate the whole part of a decimal number from the fractional part of a decimal number when written and is stated as “and” when read.
 The fractional part of a decimal number is recorded to the right of the decimal point when written.
 The fractional part of a decimal number is stated as a whole number with the label of the smallest decimal place value when read (e.g., 0.5 is read as 5 tenths; 0.25 is read as 25 hundredths; etc.).
 The “ths” ending denotes the fractional part of a decimal number.
 Zeros are used as place holders between digits of a number as needed, whole part and fractional part, to maintain the value of each digit (e.g., 400.05).
 Leading zeros in a decimal number are not commonly used in standard form, but are not incorrect and do not change the value of the decimal number (e.g., 0,037,564,215.55 equals 37,564,215.55).
 Trailing zeros after a fractional part of a decimal number may or may not be used and do not change the value of the decimal number (e.g., 400.50 equals 400.5).
 Ex:
 Word form – the representation of a number using written words (e.g., 985,156,789.78 as nine hundred eightyfive million, one hundred fiftysix thousand, seven hundred eightynine and seventyeight hundredths)
 The word “billion” after the numerical value of the billions period is stated when read and recorded when written.
 A comma between the billions period and the millions period is not stated when read but is recorded when written.
 The word “million” after the numerical value of the millions period is stated when read and recorded when written.
 A comma between the millions period and the thousands period is not stated when read but is recorded when written.
 The word “thousand” after the numerical value of the thousands period is stated when read and recorded when written.
 A comma between the thousands period and the units period is not stated when read but is recorded when written.
 The word “unit” after the numerical value of the units period is not stated when read and not recorded when written.
 The word “hundred” in each period is stated when read and recorded when written.
 The words “ten” and “one” in each period are not stated when read and not recorded when written.
 The tens place digit and ones place digit in each period are stated as a twodigit number when read and recorded using a hyphen, where appropriate, when written (e.g., twentythree, thirteen, etc.).
 The whole part of a decimal number is recorded the same as a whole number with all appropriate unit labels prior to recording the fractional part of a decimal number.
 The decimal point is recorded as the word “and” to separate the whole part of a decimal number from the fractional part of a decimal number when written and is stated as “and” when read.
 The fractional part of a decimal number followed by the label of the smallest decimal place value is recorded when written and stated when read.
 The “ths” ending denotes the fractional part of a decimal number.
 The zeros in a number are not stated when read and are not recorded when written (e.g., 854,091,005.26 in standard form is read and written as eight hundred fiftyfour million, ninetyone thousand, five and twentysix hundredths in word form).
 Ex:
 Place Value forms
 Expanded form – the representation of a number as a sum of place values (e.g., 985,156,789.78 as 900,000,000 + 80,000,000 + 5,000,000 + 100,000 + 50,000 + 6,000 + 700 + 80 + 9 + 0.7 + 0.08)
 Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 905,150,089.08 as 900,000,000 + 0 + 5,000,000 + 100,000 + 50,000 + 0 + 0 + 80 + 9 + 0.0 + 0.08 or as 900,000,000 + 5,000,000 + 100,000 + 50,000 + 80 + 9 + 0.08).
 Expanded form is written following the order of place value.
 The sum of place values written in random order is an expression but not expanded form.
 Ex:
 Expanded notation – the representation of a number as a sum of place values where each term is shown as a digit(s) times its place value (e.g., 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9 + 7(0.1) + 8(0.01) or 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9 + 7() + 8())
 Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 905,150,089.08 as (9 × 100,000,000) + (0 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (0 × 1,000) + (0 × 100) + (8 × 10) + (9 × 1) + (0 × 0.1) + (8 × 0.01) or as (9 × 100,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (8 × 10) + (9 × 1) + (8 × 0.01) or e.g., 905,150,089.08 as (9 × 100,000,000) + (0 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (0 × 1,000) + (0 × 100) + (8 × 10) + (9 × 1) + (0 × ) + (8 × ) or as (9 × 100,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (8 × 10) + (9 × 1) + (8 × )).
 Expanded notation is written following the order of place value.
 Expanded notation to represent the value of a digit(s) within a number
 Ex:
 Multiple representations of various forms of a number
 Ex:
 Equivalent relationships between place value of decimals through the hundredths (e.g., 0.2 is equivalent to 20 hundredths).
Note(s):
 Grade Level(s):
 Grade 3 composed and decomposed numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.
 Grade 4 introduces the millions and billions period.
 Grade 4 introduces representing the value of a decimal to the hundredths using expanded notation and numerals.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation

4.2C 
Compare, Order
WHOLE NUMBERS TO 1,000,000,000
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Place value – the value of a digit as determined by its location in a number, such as ones, tens, hundreds, one thousands, ten thousands, etc.
 Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Ex: 100,050 is to the left of 100,500 on a number line, so 100,050 < 100,500; or 100,500 is to the right of 100,050 on a number line, so 100,500 > 100,050.
 Ex: 1,175,000 is to the left of 1,750,000 on a number line, so 1,175,000 < 1,750,000; or 1,750,000 is to the right of 1,175,000 on a number line, so 1,750,000 > 1,175,000.
 Compare two numbers using place value charts.
 Compare digits in the same place value positions beginning with the greatest value.
 If these digits are the same, continue to the next smallest place until the digits are different.
 Ex:
 Numbers that have common digits but are not equal in value (different place values)
 Ex:
 Numbers that have a different number of digits
 Ex:
 Compare two numbers using a number line.
 Number lines (horizontal/vertical)
 Proportionally scaled number lines (predetermined intervals with at least two labeled numbers)
 Open number lines (no marked intervals)
 Ex:
 Order numbers – to arrange a set of numbers based on their numerical value
 A set of numbers can be compared in pairs in the process of ordering numbers.
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Order a set of numbers on a number line.
 Ex:
 Order a set of numbers on an open number line.
 Ex:
 Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
 Ex:
 Ex:
 Ex:
Represent
COMPARISONS OF WHOLE NUMBERS TO 1,000,000,000 USING THE SYMBOLS >, <, OR =
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Comparative language and comparison symbols
 Inequality words and symbols
 Greater than (>)
 Less than (<)
 Ex:
 Equality words and symbol
 Equal to (=)
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 compared and ordered whole numbers up to 100,000 and represented comparisons using the symbols >, <, or =.
 Grade 5 will compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation

4.2D 
Round
WHOLE NUMBERS TO A GIVEN PLACE VALUE THROUGH THE HUNDRED THOUSANDS PLACE
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Rounding – a type of estimation with specific rules for determining the closest value
 Nearest 10; 100; 1,000; 10,000; or 100,000
 Number lines
 Proportionally scaled number lines (predetermined intervals)
 Open number lines (no marked intervals)
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Ex:
 Rounding to the nearest 10 on a number line
 Determine the two consecutive multiples of 10 that the number being rounded falls between.
 Begin with the value of the original tens place within the number and then identify the next highest value in the tens place.
 Determine the halfway point between the consecutive multiples of 10.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 10 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original tens place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest tens place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest tens place.
 Rounding to the nearest 100 on a number line
 Determine the two consecutive multiples of 100 that the number being rounded falls between.
 Begin with the value of the original hundreds place within the number and then identify the next highest value in the hundreds place.
 Determine the halfway point between the consecutive multiples of 100.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 100 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original hundreds place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest hundreds place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest hundreds place.
 Rounding to the nearest 1,000 on a number line
 Determine the two consecutive multiples of 1,000 that the number being rounded falls between.
 Begin with the value of the original thousands place within the number and then identify the next highest value in the thousands place.
 Determine the halfway point between the consecutive multiples of 1,000.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 1,000 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original thousands place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest thousands place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest thousands place.
 Rounding to the nearest 10,000 on a number line
 Determine the two consecutive multiples of 10,000 that the number being rounded falls between.
 Begin with the value of the original ten thousands place within the number and then identify the next highest value in the ten thousands place.
 Determine the halfway point between the consecutive multiples of 10,000.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 10,000 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original ten thousands place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest ten thousands place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest ten thousands place.
 Rounding to the nearest 100,000 on a number line
 Determine the two consecutive multiples of 100,000 that the number being rounded falls between.
 Begin with the value of the original hundred thousands place within the number and then identify the next highest value in the hundred thousands place.
 Determine the halfway point between the consecutive multiples of 100,000.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 100,000 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original hundred thousands place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest hundred thousands place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest hundred thousands place.
 Round a given number to the closest multiple of 10; 100; 1,000; 10,000; or 100,000 on a number line.
 Round a given number to the higher multiple of 10; 100; 1,000; 10,000; or 100,000 if it falls exactly halfway between the multiples on a number line.
 Rounding numerically based on place value
 Find the place to which you are rounding.
Look at the digit of the next lowest place value, the digit to the right of which you are rounding. If the digit in that place is less than 5, then the digit in the rounding place remains the same. If the digit in that place is greater than or equal to 5, then the digit in the rounding place increases by 1. The digit(s) to the right of the place of which you are rounding is replaced with “0”.
Note(s):
 Grade Level(s):
 Grade 3 introduced rounding to the nearest 10 or 100 or using compatible numbers to estimate solutions to addition and subtraction problems.
 Grade 5 will round decimals to the tenths or hundredths.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation

4.2E 
Represent
DECIMALS, INCLUDING TENTHS AND HUNDREDTHS, USING CONCRETE AND VISUAL MODELS AND MONEY
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 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
 Various concrete and visual models
 Number lines, decimal disks, decimal grids, base10 blocks, money, etc.
Note(s):
 Grade Level(s):
 Previous grade levels used the decimal point in money only.
 Grade 4 introduces representing decimals, including tenths and hundredths, using concrete and visual models and money.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation
 X. Connections

4.2F 
Compare, Order
DECIMALS USING CONCRETE AND VISUAL MODELS TO THE HUNDREDTHS
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 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
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
 Comparative language and comparison symbols
 Inequality words and symbols
 Greater than (>)
 Less than (<)
 Equality words and symbol
 Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Ex: 1.2 is to the left of 1.5 on a number line, so 1.2 < 1.5; or 1.5 is to the right of 1.2 on a number line, so 1.5 > 1.2.
 Ex: 2.37 is to the left of 2.73 on a number line, so 2.37 < 2.73 ; or 2.73 is to the right of 2.37 on a number line, so 2.73 > 2.37.
 Compare two decimals using place value charts.
 Compare digits in the same place value position beginning with the greatest place value.
 If these digits are the same, continue to the next smallest place until the digits are different.
 Ex:
 Numbers that have common digits but are not equal in value (different place values)
 Ex:
 Numbers that have a different number of digits
 Ex:
 Compare two decimals with various concrete and visual models.
 Number lines, decimal disks, decimal grids, base10 blocks, money, etc.
 Ex: Number lines (horizontal/vertical)
 Ex: Decimal grids
 Ex: Decimal disks
 Ex: Base10 blocks
 Ex: Money
 Order numbers – to arrange a set of numbers based on their numerical value
 Order three or more decimals with various concrete and visual models.
 Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
 Number lines, decimal disks, decimal grids, base10 blocks, money, etc.
 Ex: Number lines
 Ex: Decimal grids
 Ex: Decimal disks
 Ex: Base10 blocks
 Ex: Money
Note(s):
 Grade Level(s):
 Grade 4 introduces comparing and ordering decimals using concrete and visual models to the hundredths.
 Grade 5 will compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation
 X. Connections

4.2G 
Relate
DECIMALS TO FRACTIONS THAT NAME TENTHS AND HUNDREDTHS
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 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
 Fractions (proper, improper, and mixed numbers)
 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
 Various concrete and visual models
 Number line (horizontal/vertical)
 Number line representing values less than one
 Ex:
 Number line representing values greater than one
 Ex:
 Number line representing values between tick marks
 Ex:
 Area model (tenths and hundredths grids)
 Decimals and fractions of the same whole
 Ex:
 Decimals and fractions less than one
 Ex:
 Decimals and fractions greater than one
 Ex:
 Ex:
 Decimal disks
 Decimals and fractions of the same whole
 Ex:
 Decimals and fractions less than one
 Ex:
 Decimals and fractions greater than one
 Ex:
 Base10 blocks
 Decimals and fractions to same whole
 Ex:
 Decimals and fractions less than one
 Ex:
 Decimals and fractions greater than one
 Ex:
 Ex:
 Money
 Decimal and fraction relationships of a dollar
 Ex:
 Fraction language
 Ex:
Note(s):
 Grade Level(s):
 Grade 4 introduces relating decimals to fractions that name tenths and hundredths.
 Grade 6 will use equivalent fractions, decimals, and percents to show equal parts of the same whole.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation
 X. Connections

4.2H 
Determine
THE CORRESPONDING DECIMAL TO THE TENTHS OR HUNDREDTHS PLACE OF A SPECIFIED POINT ON A NUMBER LINE
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 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
 All decimals, to the tenths or hundredths place, can be located as a specified point on a number line.
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Ex: Proportionally scaled number lines (predetermined intervals with at least two labeled numbers)
 Characteristics of an open number line
 An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
 Numbers/positions are placed on the empty number line only as they are needed.
 When reasoning on an open number line, the position of zero is often not placed.
 When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 The placement of the first two numbers on an open number line determines the scale of the number line.
 Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
 The differences between numbers are approximated by the distance between the positions on the number line.
 Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
 Ex: Open number lines (with no marked intervals)
 Purpose of open number line
 Open number lines allow for the consideration of the magnitude of numbers and the placevalue relationships among numbers when locating a given whole number
 Number lines representing values less than one to the tenths place
 Ex:
 Number lines representing values greater than one to the tenths place
 Ex:
 Number lines representing values less than one to the hundredths place
 Ex:
 Number lines representing values greater than one to the hundredths place
 Ex:
 Number lines representing values between tick marks
 Relationship between tenths and hundredths
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 represented a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric Reasoning
 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:

4.3A 
Represent
A FRACTION AS A SUM OF FRACTIONS , WHERE a AND b ARE WHOLE NUMBERS AND b > 0, INCLUDING WHEN a > b
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
 Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a nonzero whole number
 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
 Represent an amount less than, equal to, or greater than 1 using a sum of unit fractions
 Ex: written as
 Ex: or 1 written as
 Multiple Representations
 Concrete models of whole objects
 Linear model
 Fraction bars, customary ruler, linking cube trains, folded paper strips, etc.
 Ex: Fraction bars
 Ex: Customary ruler
 Ex: Linking cube trains
 Ex: Folded paper strip
 Area models
 Fraction circles or squares, pattern blocks, etc.
 Ex: Fraction circles
 Ex: Pattern blocks
 Concrete models of a set of objects
 Pattern blocks, color tiles, counters, etc.
 Ex: Pattern blocks
 Ex: Color tiles
 Ex: Counters
 Pictorial models
 Fraction strips, fraction bar models, number lines, etc.
 Ex: Fraction strip or fraction bar models
 Ex: Number lines

4.3B 
Decompose
A FRACTION IN MORE THAN ONE WAY INTO A SUM OF FRACTIONS WITH THE SAME DENOMINATOR USING CONCRETE AND PICTORIAL MODELS AND RECORDING RESULTS WITH SYMBOLIC REPRESENTATIONS
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
 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
 Decompose fractions into smaller fractional parts represented by a sum of unit fractions or multiples of unit fractions with the same denominator
 Concrete models of whole objects
 Linear models
 Fraction bars, customary ruler, linking cube trains, folded paper strips, etc.
 Ex: Fraction bars
 Ex: Customary ruler
 Ex: Linking cube trains
 Ex: Folded paper strips
 Area models
 Fraction circles or squares, pattern blocks, etc.
 Ex: Fraction circles
 Ex: Pattern blocks
 Concrete models of a set of objects
 Pattern blocks, color tiles, counters, etc.
 Ex: Pattern blocks
 Ex: Color tiles
 Ex: Counters
 Pictorial models
 Fraction strips, bar models, number lines, etc.
 Ex: Fraction strips or bar models
 Ex: Area models
 Ex: Number lines
Note(s):
 Grade Level(s):
 Grade 3 composed and decomposed a fraction with a numerator greater than zero and less than or equal to b as a sum of parts .
 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

4.3C 
Note(s):
 Grade Level(s):
 Grade 3 explained that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model.
 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
 IX. Communication and Representation

4.3D 
Represent
THE COMPARISON OF TWO FRACTIONS WITH DIFFERENT NUMERATORS AND DIFFERENT DENOMINATORS USING THE SYMBOLS >, =, OR <
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 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}
 Fractions (proper, improper, or mixed numbers with equal or unequal 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
 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
 Comparative language and symbols
 Inequality words and comparison symbols
 Greater than (>)
 Less than (<)
 Ex:
 Equality words and symbol
 Equal to (=)
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 compared two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.
 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

4.3E 
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.)
 Ex:
 Fraction strips and other strip models
 Ex:
 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
 Ex:
Therefore,
 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.)
 Ex: Commutative property of addition
 Ex: Associatve property of addition
 Fraction strips and other strip models
 Ex: Commutative property of addition
 Ex: Associative property of addition
 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.)
 Ex:
 Fraction strips and other strip models
 Ex:
 Recognition of addition and subtraction in mathematical and realworld 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

4.3F 
Note(s):
 Grade Level(s):
 Grade 4 evaluates the reasonableness of sums and differences of fractions using benchmark fractions 0, and 1, referring to the same whole.
 Grade 5 will add and subtract positive rational numbers fluently.
 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

4.3G 
Represent
FRACTIONS AND DECIMALS TO THE TENTHS OR HUNDREDTHS AS DISTANCES FROM ZERO ON A NUMBER LINE
Including, but not limited to:
 Fractions (proper, improper, and mixed numbers)
 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
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 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
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Fractions or decimals to the tenths or hundredths as distances from zero on a number line
 Relationship between a fraction represented using a strip diagram to a fraction represented on a number line and the relationship between a decimal represented using a strip diagram to a decimal represented on a number line
 Strip diagram – a linear model used to illustrate number relationships
 Ex:
 Ex:
 Fractions or decimals as distances from zero on a number line greater than 1
 Point on a number line read as the number of whole units from zero and the fractional or decimal amount of the next whole unit
 Ex:
 Ex:
 Number line beginning with a number other than zero
 Distance from zero to first marked increment is assumed even when not visible on the number line.
 Ex:
 Ex:
 Relationship between fractions as distances from zero on a number line to fractional measurements as distances from zero on a customary ruler, yardstick, or measuring tape
 Ex:
 Measuring a specific length using a starting point other than zero on a customary ruler, yardstick, or measuring tape
 Distance from zero to first marked increment not counted
 Length determined by number of whole units and the fractional amount of the next whole unit
 Ex:
 Relationship between fractions and decimals as distances from zero on a number line to fractional and decimal measurements as distances from zero on a metric ruler, meter stick, or measuring tape
 Ex:
 Measuring a specific length using a starting point other than zero on a metric ruler, meter stick, or measuring tape
 Distance from zero to first marked increment not counted
 Length determined by number of whole units and the fractional amount of the next whole unit
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 represented fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.
 Grade 3 determined the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.
 Grade 6 will identify a number, its opposite, and its absolute value.
 Grade 6 will locate, compare, and order integers and rational numbers using a number line.
 Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I. Numeric 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, 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
 Ex:
 Decimals (less than or greater than one to the tenths and hundredths)
 Decimal number – a number in the base10 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.
 Ex: Tenths grids for addition
 Ex: Hundredths grids for addition
 Ex: Tenths number line for addition
 Ex: Hundredths number line for addition
 Ex: Base10 blocks for addition
 Ex: Tenths grids for subtraction
 Ex: Hundredths grids for subtraction
 Ex: Tenths number line for subtraction
 Ex: Hundredths number line for subtraction
 Ex: Base10 blocks for subtraction
 Trailing zeros – a sequence of zeros in the decimal part of a number that follow the last nonzero digit, and whether recorded or deleted, does not change the value of the number
 Ex:
 Ex:
 Standard algorithm
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 solved with fluency onestep and twostep 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 realworld 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

4.4B 
Determine
PRODUCTS OF A NUMBER AND 10 OR 100 USING PROPERTIES OF OPERATIONS AND PLACE VALUE UNDERSTANDINGS
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}
 Multiplication of whole numbers
 Product – the total when two or more factors are multiplied
 Factor – a number multiplied by another number to find a product
 Knowledge of patterns in place value to solve multiplication involving multiples of 10 or 100 (e.g., 98 × 10; 98 × 100; 980 × 10; 980 × 100; 9,800 × 10; 9,800 × 100; etc.)
 Properties of operations
 Distributive property of multiplication – if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together
 a × (b + c) = (a × b) + (a × c)
 Ex: 10 × 25 = 10 × (20 + 5) = (10 × 20) + (10 × 5) = 200 + 50 = 250
 Multiplying a number by 10 is equal to multiplying each place value digit by 10.
 Multiplying a number by 100 is equal to multiplying each place value digit by 100.
 Commutative property of multiplication – if the order of the factors are changed, the product will remain the same
 a × b = c; therefore, b × a = c
 Ex: 25 × 10 = 250 and 10 × 25 = 250
Therefore, 25 × 10 = 10 × 25
 Ex:
 Place value understanding
 When multiplying a number by 10, the product is 10 times larger meaning that each digit in the number shifts 1 place value position to the left, leaving a zero in the ones place to show groups of tens.
 When multiplying a number by 100, the product is 100 times larger meaning that each digit in the number shifts 2 place value positions to the left, leaving zeros in the ones place and tens place to show groups of hundreds.
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 represented multiplication facts by using a variety of approaches such as repeated addition, equalsized groups, arrays, area models, equal jumps on a number line, and skip counting.
 Grade 3 recalled facts to multiply up to 10 by 10 with automaticity and recalled the corresponding division facts.
 Grade 5 will multiply with fluency a threedigit number by a twodigit number using 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
 IX. Communication and Representation

4.4C 
Represent
THE PRODUCT OF 2 TWODIGIT NUMBERS USING ARRAYS, AREA MODELS, OR EQUATIONS, INCLUDING PERFECT SQUARES THROUGH 15 BY 15
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}
 Multiplication of whole numbers
 Product – the total when two or more factors are multiplied
 Factor – a number multiplied by another number to find a product
 Products of twodigit factors by twodigit factors
 Arrays
 Arrangement of a set of objects in rows and columns
 Ex:
 Area models
 Arrangement of squares/rectangles in a grid format
 Connect the factors as the length and width, and the product as the area
 Ex:
 Equations
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Factor × factor = product
 Product = factor × factor
 Multiplication is commutative
 14 × 18 = 252
 18 × 14 = 252
 252 = 14 × 18
 252 = 18 × 14
 Perfect squares (through 15 × 15)
 Factors of a perfect square are the same
 Models of perfect squares result in a square
 Ex:
 Ex:
 Equations of perfect squares
 Factor × same factor = product
 Product = factor × same factor
Note(s):
 Grade Level(s):
 Grade 3 used strategies and algorithms, including the standard algorithm, to multiply a twodigit number by a onedigit number. Strategies may have included mental math, partial products, and the commutative, associative, and distributive properties.
 Grade 5 will multiply with fluency a threedigit number by a twodigit number using 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
 II.D. Algebraic Reasoning – Representations
 IX. Communication and Representation

4.4D 
Use
STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO MULTIPLY UP TO A FOURDIGIT NUMBER BY A ONEDIGIT NUMBER AND TO MULTIPLY A TWODIGIT NUMBER BY A TWODIGIT NUMBER. STRATEGIES MAY INCLUDE MENTAL MATH, PARTIAL PRODUCTS, AND THE COMMUTATIVE, ASSOCIATIVE, AND DISTRIBUTIVE PROPERTIES
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}
 Multiplication of whole numbers
 Product – the total when two or more factors are multiplied
 Factor – a number multiplied by another number to find a product
 Products of twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Strategies and algorithms for multiplication
 Mental math
 Accurate computation without the aid of paper, pencil, or other tools
 Ex:
 Partial products
 Decomposing the factor(s) into smaller parts, multiplying the parts, and combining the intermittent parts
 Ex:
 Properties of operations
 Commutative property of multiplication – if the order of the factors are changed, the product will remain the same
 a × b = c; therefore, b × a = c
 Ex: 25 × 10 = 250 and 10 × 25 = 250
Therefore, 25 × 10 = 10 × 25
 Associative property of multiplication – if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same
 a × b × c = (a × b) × c = a × (b × c)
 Ex: 25 × 10 × 2
(25 × 10) × 2 = 250 × 2 = 500 or 25 × (10 × 2) = 25 × 20 = 500 Therefore, 25 × 10 × 2 = (25 × 10) × 2 = 25 × (10 × 2)
 Distributive property of multiplication – if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together
 a × (b + c) = (a × b) + (a × c)
 Ex: 10 × 25 = 10 × (20 + 5) = (10 × 20) + (10 × 5) = 200 + 50 = 250
 Ex: 27 × 25 = (25 + 2) × 25 = (25 × 25) + (2 × 25) = 625 + 50 = 675
 Standard algorithm
 Standardized steps or routines used in computation
 Ex:
 Connections between strategies and operations
 Ex:
 Equation(s) to reflect solution process
Note(s):
 Grade Level(s):
 Grade 3 used strategies and algorithms, including the standard algorithm, to multiply a twodigit number by a onedigit number. Strategies may have included mental math, partial products, and the commutative, associative, and distributive properties.
 Grade 5 will multiply with fluency a threedigit number by a twodigit number using 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
 II.D. Algebraic Reasoning – Representations
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

4.4E 
Represent
THE QUOTIENT OF UP TO A FOURDIGIT WHOLE NUMBER DIVIDED BY A ONEDIGIT WHOLE NUMBER USING ARRAYS, AREA MODELS, OR EQUATIONS
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}
 Division of whole numbers
 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 fourdigit dividends by onedigit divisors
 Quotients may include remainders
 Relationships between multiplication and division to help in solution process
 a ÷ b = c, so b × c = a
 Ex: 1,107 ÷ 9 = 123, so 123 × 9 = 1,107
 Recognition of division in mathematical and realworld problem situations
 Representations of quotients
 Arrays
 Arrangement of a set of objects in rows and columns
 Ex:
 Area models
 Arrangement of squares/rectangles in a grid format
 Connect the factors as the length and width, and the product as the area
 Ex:
 Ex:
 Equations
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Dividend ÷ divisor = quotient
 Quotient = dividend ÷ divisor
 Division is not commutative even though multiplication is commutative.
 Ex: 14 × 18 = 252
18 × 14 = 252 252 ÷ 18 = 14; however, 18 ÷ 252 ≠ 14
Note(s):
 Grade Level(s):
 Grade 3 solved onestep and twostep problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.
 Grade 5 will solve with proficiency for quotients of up to a fourdigit dividend by a twodigit 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
 II.D. Algebraic Reasoning – Representations
 IX. Communication and Representation

4.4F 
Use
STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO DIVIDE UP TO A FOURDIGIT DIVIDEND BY A ONEDIGIT DIVISOR
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}
 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 fourdigit dividends by onedigit divisors
 Quotients may include remainders
 Recognition of division in mathematical and realworld problem situations
 Automatic recall of basic facts
 Relationships between multiplication and division to help in solution process
 a ÷ b = c, so b x c = a
 Ex: 1,107 ÷ 9 = 123, so 123 × 9 = 1,107
 Division structures
 Partitive division
 Total amount known
 Number of groups known
 Size or measure of each group unknown
 Ex:
 Quotative division (also known as Measurement division)
 Total amount known
 Size or measure of each group known
 Number of groups unknown
 Ex:
 Relationship between division and multiples of 10
 When the value of the dividend increases by a multiple of 10 and the value of the divisor remains the same, then the value of the quotient is multiplied by the same multiple of 10.
 Ex:
 Strategies and algorithms for division
 Decomposing division problem situations into partial quotients (using numbers that are compatible with the divisor)
 Ex:
 Ex:
 Standard algorithm using the distributive method
 Record steps that relate to the algorithm used including distributing the value in the quotient according to place value.
 Ex:
 Standard algorithm
 Ex:
 Equation(s) to reflect solution process
Note(s):
 Grade Level(s):
 Grade 4 introduces using strategies and algorithms, including the standard algorithm, to divide up to a fourdigit dividend by a onedigit divisor.
 Grade 5 will solve with proficiency for quotients of up to a fourdigit dividend by a twodigit 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

4.4G 
Round
TO THE NEAREST 10, 100, OR 1,000 TO ESTIMATE SOLUTIONS INVOLVING WHOLE NUMBERS
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
 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 twodigit factors by twodigit factors and up to fourdigit factors by onedigit 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 fourdigit dividends by onedigit divisors
 Recognition of operations in mathematical and realworld problem situations
 Estimation – reasoning to determine an approximate value
 Rounding – a type of estimation with specific rules for determining the closest value
 To the nearest 10; 100; or 1,000
 Number lines
 Proportionally scaled number lines (predetermined intervals)
 Open number line (no marked intervals)
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Ex:
 Rounding to the nearest 10 on a number line
 Determine the two consecutive multiples of 10 that the number being rounded falls between.
 Begin with the value of the original tens place within the number and then identify the next highest value in the tens place.
 Determine the halfway point between the consecutive multiples of 10.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 10 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original tens place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest tens place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest tens place.
 Rounding to the nearest 100 on a number line
 Determine the two consecutive multiples of 100 that the number being rounded falls between.
 Begin with the value of the original hundreds place within the number and then identify the next highest value in the hundreds place.
 Determine the halfway point between the consecutive multiples of 100.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 100 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original hundreds place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest hundreds place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest hundreds place.
 Rounding to the nearest 1,000 on a number line
 Determine the two consecutive multiples of 1,000 that the number being rounded falls between.
 Begin with the value of the original thousands place within the number and then identify the next highest value in the thousands place.
 Determine the halfway point between the consecutive multiples of 1,000.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 1,000 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original thousands place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest thousands place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest thousands place.
 Round a given number to the closest multiple of 10; 100; or 1,000 on a number line.
 Round a given number to the higher multiple of 10; 100; or 1,000 if it falls exactly halfway between the multiples on a number line.
 Round numbers to a common place then compute.
 If not designated, find the greatest common place value of all numbers in the problem to determine the place value to which you are rounding (e.g., round to the nearest 10 if only twodigit numbers are being considered in the problem; round to the nearest 100 if only threedigit numbers are being considered in the problem; round to the nearest 1,000 if only fourdigit numbers are being considered; round to the nearest 10 if both twodigit and threedigit numbers are being considered in the problem; round to the nearest 100 if both threedigit and fourdigit numbers are being considered; etc.).
 Vocabulary indicating estimation in mathematical and realworld problem situations (e.g., about, approximately, estimate, etc.)
 Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
 Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
 Ex:
 Ex:
 Ex:
 Ex:
 Ex:
 Rounding numerically based on place value
 Find the place to which you are rounding.
Look at the digit of the next lowest place value, the digit to the right of which you are rounding. If the digit in that place is less than 5, then the digit in the rounding place remains the same. If the digit in that place is greater than or equal to 5, then the digit in the rounding place increases by 1. The digit(s) to the right of the place of which you are rounding is replaced with “0”.
 Round numbers to a common place then compute.
 If not designated, find the greatest common place value of all numbers in the problem to determine the place value to which you are rounding (e.g., round to the nearest 10 if only twodigit numbers are being considered in the problem; round to the nearest 100 if only threedigit numbers are being considered in the problem; round to the nearest 1,000 if only fourdigit numbers are being considered; round to the nearest 10 if both twodigit and threedigit numbers are being considered in the problem; round to the nearest 100 if both threedigit and fourdigit numbers are being considered; etc.).
 Vocabulary indicating estimation in mathematical and realworld problem situations (e.g., about, approximately, estimate, etc.)
 Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
 Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
 Ex:
 Ex:
 Ex:
 Ex:
 x:
Use
COMPATIBLE NUMBERS TO ESTIMATE SOLUTIONS INVOLVING WHOLE NUMBERS
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
 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 twodigit factors by twodigit factors and up to fourdigit factors by onedigit 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 fourdigit dividends by onedigit divisors
 Recognition of operations in mathematical and realworld problem situations
 Estimation – reasoning to determine an approximate value
 Compatible numbers – numbers that are slightly adjusted to create groups of numbers that are easy to compute mentally
 Determine compatible numbers then compute.
 Vocabulary indicating estimation in mathematical and realworld problem situations (e.g., about, approximately, estimate, etc.)
 Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
 Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
 Ex:
 Ex:
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 rounded to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems.
 Grade 5 will round decimals to tenths or hundredths.
 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

4.4H 
Solve
WITH FLUENCY ONE AND TWOSTEP 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 twodigit factors by twodigit factors and up to fourdigit factors by onedigit 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 fourdigit dividends by onedigit divisors
 Quotients may include remainders
 Remainder dependent upon the mathematical or realworld situation
 Various ways to record remainder
 Ignore the remainder
 Ex:
 Add one to the quotient
 Ex:
 Remainder is the answer
 Ex:
 Remainder recorded as a fraction
 Ex:
 One and twostep problem situations
 Onestep problems
 Recognition of multiplication and division in mathematical and realworld problem situations
 Ex:
 Ex:
 Twostep problems
 Twostep problems must have onestep 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 realworld problem situation
 Equation(s) to reflect solution process
Note(s):
 Grade Level(s):
 Grade 4 introduces solving with fluency one and twostep problems involving multiplication and division, including interpreting remainders.
 Grade 5 will multiply with fluency a threedigit number by a twodigit number using the standard algorithm.
 Grade 5 will solve with proficiency for quotients of up to a fourdigit dividend by a twodigit 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
MULTISTEP 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
 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 twodigit factors by twodigit factors and up to fourdigit factors by onedigit 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 fourdigit dividends by onedigit 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
 Ex: g = 6 + 4; 6 + 4 = g
 Ex: x = 10 – 4; 10 – 4 = x
 Ex: 10 = x + 4; x + 4 = 10
 Ex: r = 6 × 4; 6 × 4 = r
 Ex: p = 24 ÷ 4; 24 ÷ 4 = p
 Ex: 24 = 6 × z; z × 6 = 24
 Recognition of addition, subtraction, multiplication, and/or division in mathematical and realworld problem situations
 Representation of problem situations with strip diagrams and equations
 Strip diagram – a linear model used to illustrate number relationships
 Relationship between quantities represented and problem situation
 Types of problem structures
 Addition and subtraction problem structures
 Join start unknown
 Ex:
 Join change unknown
 Ex:
 Join result unknown
 Ex:
 Separate start unknown
 Ex:
 Separate change unknown
 Ex:
 Separate result unknown
 Ex:
 Partpartwhole part unknown
 Ex:
 PartPartwhole whole unknown
 Ex:
 Additive comparison difference unknown
 Ex:
 Additive comparison compare quantity (larger quantity) unknown
 Ex:
 Additive comparison referent (smaller quantity) unknown
 Ex:
 Multiplicative structures
 Multiplication product unknown
 Ex:
 Multiplication factor unknown
 Ex:
 Division structures
 Partitive division
 Total amount known
 Number of groups known
 Size or measure of each group unknown
 Ex:
 Quotative division (also known as Measurement division)
 Total amount known
 Size or measure of each group known
 Number of groups unknown
 Ex:
 Multistep problem situations involving the four operations in a variety of problem structures
 Ex:
 Ex:
 Ex:
 Ex:
 Ex:
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 represented one and twostep 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 twostep 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 multistep 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 INPUTOUTPUT 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
 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 twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Data sets of whole numbers
 Sets may or may not begin with 1
 Sets may or may not be listed in sequential order
 Various representations of problem situations
 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)
 Inputoutput table – a table which represents how the application of a rule on a value, input, results in a different value, output
 Input – position in the sequence
 Output – value in the sequence
 Relationship between values in a number pattern
 Additive numerical pattern – a pattern that occurs when a constant nonzero value is added to an input value to determine the output value
 Multiplicative numerical pattern – a pattern that occurs when a constant nonzero value is multiplied by an input value to determine the output value
 Ex:
 Ex:
 Sequence – an ordered list of numbers, usually set apart by commas, such as {2, 4, 6, 8, 10, 12, …}
 Relationship between inputoutput tables and sequences
 Input – position in the sequence
 Output – value in the sequence
 Ex:
 Relationship between numerical expressions and rules to create inputoutput tables representing the relationship between each position in the sequence (input) and the value in the sequence (output)
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 represented realworld 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

4.5C 
Use
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 x w)
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 an additive onedimensional linear measure
 Models to determine formulas for perimeter
 Rectangle (P = l + w + l + w or P = 2l + 2w)
 Square (P = 4s)
 Area – the measurement attribute that describes the number of square units a figure or region covers
 Area is a multiplicative twodimensional square unit measure.
 Models to determine formulas for area
 Rectangle (A = l × w)
 Square (A = s × s)
Note(s):
 Grade Level(s):
 Grade 4 introduces use 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).
 Grade 5 will use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l × w × h, V = s × s × s, and V = Bh).
 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.C. Measurement Reasoning – Measurement involving geometry and algebra
 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
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 onedimensional linear measure.
 Whole number side lengths
 Recognition of perimeter embedded in mathematical and realworld problem situations
 Ex: How much lace is needed to go around the edge of the rectangular tablecloth?
 Ex: How much fencing is needed to enclose a garden?
 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
 Ex:
 Ex:
 Determine perimeter by measuring to determine side lengths
 Ruler, STAAR Grade 4 Mathematics Reference Materials ruler, yardstick, meter stick, measuring tape, etc.
 Ex:
 Determine missing side length when given perimeter and remaining side length
 Ex:
 Perimeter of composite figures
 Ex:
 Area – the measurement attribute that describes the number of square units a figure or region covers
 Area is a twodimensional square unit measure.
 Whole number side lengths
 Recognition of area embedded in mathematical and realworld problem situations
 Ex: How much fabric is needed to cover a bulletin board?
 Ex: How much carpet is needed to cover the living room floor?
 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
 Ex:
 Ex:
 Determine area by measuring to determine side lengths
 Ruler, STAAR Grade 4 Mathematics Reference Materials ruler, yardstick, meter stick, measuring tape, etc.
 Ex:
 Determine missing side length when given area and remaining side length
 Ex:
 Area of composite figures
 Ex:
 Multiple ways to decompose a composite figure to determine perimeter and/or area
 Ex:
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

4.6 
Geometry and measurement. The student applies mathematical process standards to analyze geometric attributes in order to develop generalizations about their properties. The student is expected to:

4.6A 
Identify
POINTS, LINES, LINE SEGMENTS, RAYS, ANGLES, AND PERPENDICULAR AND PARALLEL LINES
Including, but not limited to:
 Point – a specific location in space
 Has no dimension and is usually represented by a small dot
 Ex:
 Line – a set of points that form a straight path that goes in opposite directions without ending
 Line labels
 Lines named according to two points on a line
 Ex:
 Lines named by one lower case cursive letter
 Ex:
 Parallel lines – lines that lie in the same plane, never intersect, and are always the same distance apart
 Various orientations including vertical, horizontal, diagonal, and parallel lines of even, uneven, or offset lengths
 Ex:
 Notation may be given using chevrons or arrows to represent parallel lines.
 If more than one set of parallel lines are present, the number of chevrons or arrows distinguishes the sets of parallel lines.
 Ex:
 Intersecting lines – lines that meet or cross at a point
 Various orientations including vertical, horizontal, diagonal, and intersecting lines of even, uneven, or offset lengths
 Ex:
 Perpendicular lines – lines that intersect at right angles to each other to form square corners
 Various orientations including vertical, horizontal, diagonal, and perpendicular lines of even, uneven, or offset lengths
 Notation is given as a box in the angle corner to represent a 90° angle.
 Ex:
 Lines in pictorial models and polygons
 Ex:
 Extending lines beyond pictorial models
 Ex:
 Ex:
 Line segment – part of a line between two points on the line, called endpoints of the segment
 Ex:
 Ray – part of a line that begins at one endpoint and continues without end in one direction
 Relationships between line segments, rays, and lines
 A line segment is part of a ray and part of a line
 A ray is part of a line
 Ex:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle – two rays with a common endpoint (the vertex)
 Angle labels for a single angle
 Angle label with one letter, the letter of the vertex
 Angle label with three letters, where the middle letter is the vertex of the angle
 Angle label with a number or letter designated within the angle
 Angle symbol with one letter, the letter of the vertex
 Angle symbol with three letters, where the middle letter is the vertex of the angle
 Angle symbol with a number or letter designated within the angle
 Ex:
 Angle labels for adjacent angles
 Adjacent angles – two nonoverlapping angles that share a common vertex and exactly one ray
 Ex:
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute – an angle that measures less than 90°
 Right – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse – an angle that measures greater than 90° but less than 180°
 Straight – an angle that measures 180° (a straight line)
 Ex:
 Angles in pictorial models and polygons
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Measuring angles
 Grade Level Connections (reinforces previous learning and/or provides development for future learning)
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

4.6B 
Identify, Draw
ONE OR MORE LINES OF SYMMETRY, IF THEY EXIST, FOR A TWODIMENSIONAL FIGURE
Including, but not limited to:
 Line of symmetry – line dividing an image into two congruent parts that are mirror images of each other
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Twodimensional figures and realworld figures
 Shapes with more than one line of symmetry
 Ex:
 Shapes with no lines of symmetry
 Ex:
 Shapes on which lines of symmetry have not been drawn
 Ex:
 Across a vertical line, across a horizontal line, or across a diagonal line of symmetry
 Ex:
 A line of reflection exists for a figure if for every point on one side of the line of reflection, there is a corresponding point the same distance from the line.
Note(s):
 Grade Level(s):
 Grade 3 used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Grade Level Connections (reinforces previous learning and/or provides development for future learning)
 TxCCRS:
 IX. Communication and Representation

4.6C 
Apply
KNOWLEDGE OF RIGHT ANGLES TO IDENTIFY ACUTE, RIGHT, AND OBTUSE TRIANGLES
Including, but not limited to:
 Angle – two rays with a common endpoint (the vertex)
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute – an angle that measures less than 90°
 Right – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse – an angle that measures greater than 90° but less than 180°
 Triangle – a polygon with three sides and three vertices
 Triangles are named based on their largest angle.
 Acute triangle – a triangle in which each of the three angles is acute (less than 90 degrees)
 Right triangle – a triangle with one right angle (exactly 90 degrees) and two acute angles
 Obtuse triangle – a triangle that has one obtuse angle (greater than 90 degrees) and two acute angles
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories.
 Grade 4 introduces formal and symbolic geometric language for lines, line segments, rays, and angles.
 Grade 5 will classify twodimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

4.6D 
Classify
TWODIMENSIONAL FIGURES BASED ON THE PRESENCE OR ABSENCE OF PARALLEL OR PERPENDICULAR LINES OR THE PRESENCE OR ABSENCE OF ANGLES OF A SPECIFIED SIZE
Including, but not limited to:
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Regular figure – a polygon with all sides and angles congruent
 Irregular figure – a polygon with sides and/or angles that are not all congruent
 Classify – applying an attribute to categorize a sorted group
 Angle – two rays with a common endpoint (the vertex)
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute – an angle that measures less than 90°
 Right – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse – an angle that measures greater than 90° but less than 180°
 Line – a set of points that form a straight path that goes in opposite directions without ending
 Parallel lines – lines that lie in the same plane, never intersect, and are always the same distance apart
 means is parallel to .
 Notation may be given using chevrons or arrows to represent parallel lines.
 If more than one set of parallel lines are present, the number of chevrons or arrows distinguishes the sets of parallel lines.
 Ex:
 Perpendicular lines – lines that intersect at right angles to each other to form square corners
 means is perpendicular to .
 Notation is given as a box in the angle corner to represent a 90° angle.
 Ex:
 Sides of twodimensional figures are composed of line segments, the part of a line between two points on the line
 Congruent – of equal measure, having exactly the same size and same shape
 Angle congruency marks – angle marks indicating angles of the same measure
 m∠A ≅ m∠C means ∠A is congruent to ∠C.
 Ex:
 Side congruency marks – side marks indicating side lengths of the same measure
 means is congruent to .
 Ex:
 Types of twodimensional figures
 Circle
 A figure formed by a closed curve with all points equal distance from the center
 No straight sides
 No vertices
 No parallel or perpendicular sides
 Ex:
 Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
 Ex:
 Types of polygons
 Triangle
 3 sides
 3 vertices
 No parallel sides
 Types of triangles
 Triangles are named based on their largest angle.
 Scalene triangle
 3 sides
 3 vertices
 No congruent sides
 No parallel sides
 Up to one possible pair of perpendicular sides
 Right triangle with two sides that are perpendicular to form a right angle and three different side lengths
 Ex:
 No congruent angles
 Right triangle with one 90° angle and two other angles each of different measures
 Ex:
 Isosceles triangle
 3 sides
 3 vertices
 At least 2 congruent sides
 No parallel sides
 Up to one possible pair of perpendicular sides
 Right triangle with two sides that are perpendicular to form a right angle and are each of the same length
 Ex:
 At least 2 congruent angles
 Right triangle with one 90° angle and two other angles each of the same measure
 Ex:
 Obtuse triangle with two angles of the same measure and one angle greater than 90°
 Ex:
 Acute triangle with all angles measuring less than 90° and at least two of the angles of the same measure
 Ex:
 Equilateral triangle/Equiangular triangle (a special type of isosceles triangle)
 3 sides
 3 vertices
 All sides congruent
 No parallel or perpendicular sides
 All angles congruent
 Acute triangle with all angles measuring 60°
 Ex:
 Quadrilateral
 4 sides
 4 vertices
 Types of quadrilaterals
 Trapezoid
 4 sides
 4 vertices
 Exactly one pair of parallel sides
 Up to two possible pairs of perpendicular sides
 Ex:
 Parallelogram
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 Opposite angles congruent
 Ex:
 Types of parallelograms
 Rectangle
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Ex:
 Rhombus
 4 sides
 4 vertices
 All sides congruent
 2 pairs of parallel sides
 Opposite angles congruent
 Ex:
 Square (a special type of rectangle and a special type of rhombus)
 4 sides
 4 vertices
 All sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Ex:
 Pentagon
 5 sides
 5 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 Hexagon
 6 sides
 6 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 7gon (heptagon)
 7 sides
 7 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 Octagon
 8 sides
 8 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 9gon (nonagon)
 9 sides
 9 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 Decagon
 10 sides
 10 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 11gon (hendecagon)
 11 sides
 11 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 12gon (dodecagon)
 12 sides
 12 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Ex:
 Classification of twodimensional figures based on attributes of sides and angles
Note(s):
 Grade Level(s):
 Grade 3 classified and sorted two and threedimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.
 Grade 5 will classify twodimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

4.7 
Geometry and measurement. The student applies mathematical process standards to solve problems involving angles less than or equal to 180 degrees. The student is expected to:

4.7A 
Illustrate
THE MEASURE OF AN ANGLE AS THE PART OF A CIRCLE WHOSE CENTER IS AT THE VERTEX OF THE ANGLE THAT IS "CUT OUT" BY THE RAYS OF THE ANGLE. ANGLE MEASURES ARE LIMITED TO WHOLE NUMBERS.
Including, but not limited to:
 Ray – part of a line that begins at one endpoint and continues without end in one direction
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle – two rays with a common endpoint (the vertex)
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute – an angle that measures less than 90°
 Right – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse – an angle that measures greater than 90° but less than 180°
 Straight – an angle that measures 180° (a straight line)
 Angle measures limited to whole numbers, 0° to 180°
 Center of the circle – the point equidistant from all points on the circle
 Circle
 A figure formed by a closed curve with all points equal distance from the center
 No straight sides
 No vertices
 No parallel or perpendicular sides
 A circle measures 360° for one full rotation around the center of the circle.
 Ex:
 Representation of an angle measure as a “turn” around the center point of a circle “cut out” by the rays of the angle where the vertex of the angle is aligned to the center of the circle.
 Ex:
Note(s):
 Grade Level(s):
 Grade 4 introduces illustrating the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers.
 Foundational for work with central angles in Geometry and radian measures in Precalculus.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IX. Communication and Representation

4.7B 
Note(s):
 Grade Level(s):
 Grade 4 introduces illustrating degrees as the units used to measure an angle, where of any circle is one degree and an angle that "cuts" out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers.
 Leads to the development of radian measures in Precalculus.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

4.7C 
Determine
THE APPROXIMATE MEASURES OF ANGLES IN DEGREES TO THE NEAREST WHOLE NUMBER USING A PROTRACTOR
Including, but not limited to:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle measures limited to whole numbers, 0° to 180°
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute – an angle that measures less than 90°
 Right – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse – an angle that measures greater than 90° but less than 180°
 Straight – an angle that measures 180° (a straight line)
 Protractor – a tool used to determine the measure of an angle
 Two sets of measures from 0° to 180° going in opposite directions
 Ex:
 Relationships between a protractor and a circle
 One protractor is a semicircle, 180°
 Two protractors make a complete circle, 360°
 Ex:
 Measurement or “m” notation indicates the measure of the angle in degrees (e.g., m1 = 50°)
 Measure angles with a ray aligned at zero degrees.
 When aligning the ray to zero degrees on the right side of the protractor, read the angle measurement using the inner set of measures from right to left.
 When aligning the ray to zero degrees on the left side of the protractor, read the angle measurement using the outer set of measures from left to right.
 Ex:
 Measure angles whose rays may lie between numerically marked intervals.
 Relate to reading unmarked whole number intervals on a number line.
 Ex:
 Measure angles where a ray of the angle does not lie on zero degrees.
 Read measure of both rays using either the inner or the outer set of measures, then subtract smaller measure from larger measure to determine angle measure.
 Ex:
 Measure angles within twodimensional figures.
 Treat the sides of the figure that form the angle as rays.
 Ex:
 Use a right angle, 90°, as a benchmark to determine angle classifications (acute, obtuse, and right) to determine reasonableness of angle measures.
 Ex:
Note(s):
 Grade Level(s):
 Grade 4 introduces determining the approximate measures of angles in degrees to the nearest whole number using a protractor.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IV.A. Measurement Reasoning – Measurement involving physical and natural attributes
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

4.7D 
Draw
AN ANGLE WITH A GIVEN MEASURE
Including, but not limited to:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle measures limited to whole numbers, 0° to 180°
 Angle – two rays with a common endpoint (the vertex)
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute – an angle that measures less than 90°
 Right – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse – an angle that measures greater than 90° but less than 180°
 Straight – an angle that measures 180° (a straight line)
 Protractor – a tool used to determine the measure of an angle
 Use a protractor to draw an angle of a given measure
 Use the straight edge of the protractor to draw a ray.
 Place the vertex of the protractor on the endpoint of the ray.
 Align the vertex and the 0° mark on the protractor to the ray.
 Use the scale beginning with 0 and mark the given angle measure.
 Use the straightedge of the protractor to draw a ray from the vertex to the angle mark.
 Ex:
Note(s):
 Grade Level(s):
 Grade 4 introduces drawing an angle with a given measure.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 IV.A. Measurement Reasoning – Measurement involving physical and natural attributes
 X. Communication and Representation

4.7E 
Note(s):
 Grade Level(s):
 Grade 4 introduces determining the measure of an unknown angle formed by two nonoverlapping adjacent angles given one or both angle measures.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric Reasoning – Figures and their properties
 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:

4.8A 
Identify
RELATIVE SIZES OF MEASUREMENT UNITS WITHIN THE CUSTOMARY AND METRIC SYSTEMS
Including, but not limited to:
 Relative size – size in relation to a measure
 Sizes within a single system of measurement (e.g., sizes within customary or sizes within metric systems)
 Typically used units of measure and their relative sizes in words and abbreviations
 Metric unit names are based on prefixes attached to base unit.
 Base units include meter for length, liter for volume and capacity, and gram for mass.
 Kilo: one thousand base units
 Deci: onetenth of a base unit
 Centi: onehundredth of a base unit
 Milli: onethousandth of a base unit
 Length – the measurement attribute that describes a continuous distance from end to end
 Customary units typically used for length
 Inch (in.)
 12 inches (in.) = 1 foot (ft)
 Foot (ft)
 1 foot (ft) = 12 inches (in.)
 3 feet (ft) = 1 yard (yd)
 Yard (yd)
 1 yard (yd) = 3 feet (ft)
 1,760 yards (yd) = 1 mile (mi)
 Mile (mi)
 1 mile (mi) = 1,760 yards (yd)
 Measurement tools typically used for customary length
 Rulers, yardsticks, measuring tapes
 Relative size of customary units of length in realworld context
 Ex:
 Ex:
 Metric units typically used for length
 Millimeter (mm)
 10 millimeters (mm) = 1 centimeter (cm)
 Centimeter (cm)
 1 centimeter (cm) = 10 millimeters (mm)
 100 centimeters (cm) = 1 meter (m)
 Decimeter (dm)
 1 decimeter (dm) = 100 millimeters (mm)
 1 decimeter (dm) = 10 centimeters (cm)
 Meter (m)
 1 meter (m) = 100 centimeters (cm)
 1,000 meters (m) = 1 kilometer (km)
 Kilometer (km)
 1 kilometer (km) = 1,000 meters (m)
 Measurement tools typically used for metric length
 Rulers, meter sticks, measuring tapes
 Relative size of metric units of length in realworld context
 Ex:
 Ex:
 Liquid volume – the measurement attribute that describes the amount of space that a liquid or dry, pourable material takes up, typically measured using standard units of capacity
 Customary units typically used for liquid volume (capacity)
 Fluid ounce (fl oz)
 8 fluid ounces (fl oz) = 1 cup (c)
 Cup (c)
 1 cup (c) = 8 fluid ounces (fl oz)
 2 cups (c) = 1 pint (pt)
 Pint (pt)
 1 pint (pt) = 2 cups (c)
 2 pints (pt) = 1 quart (qt)
 Quart (qt)
 1 quart (qt) = 2 pints (pt)
 4 quarts (qt) = 1 gallon (gal)
 Gallon (gal)
 1 gallon (gal) = 4 quarts (qt)
 Measurement tools typically used for customary liquid volume
 Measuring cups, measuring containers or jars
 Relative size of customary units of liquid volume (capacity) in realworld context
 Ex:
 Ex:
 Metric units typically used for liquid volume (capacity)
 Milliliter (mL)
 1,000 milliliters (mL) = 1 liter (L)
 Liter (L)
 1 liter (L) = 1,000 milliliters (mL)
 1,000 liters (L) = 1 kiloliter (kL)
 Kiloliter (kL)
 1 kiloliter (kL) = 1,000 liters (L)
 Measurement tools typically used for metric liquid volume
 Beakers, graduated cylinders, eye droppers, measuring containers or jars
 Relative size of metric units of liquid volume (capacity) in realworld context
 Ex:
 Ex:
 Weight – the measurement attribute that describes how heavy an object is, determined by the pull of gravity on the object (weight depends upon location)
 Customary units typically used for weight
 Ounce (oz)
 16 ounces (oz) = 1 pound (lb)
 Pound (lb)
 1 pound (lb) = 16 ounces (oz)
 2,000 pounds (lb) = 1 ton (T)
 Ton (T)
 1 ton (T) = 2,000 pounds (lb)
 Measurement tools typically used for weight
 Spring scales, kitchen scales, bathroom scales
 Relative size of customary units of weight in realworld context
 Ex:
 Ex:
 Mass – the measurement attribute that describes the amount of matter in an object (mass remains constant, regardless of location)
 Metric units typically used for mass
 Milligram (mg)
 1,000 milligrams (mg) = 1 gram (g)
 Gram (g)
 1 gram (g) = 1,000 milligrams (mg)
 1,000 grams (g) = 1 kilogram (kg)
 Kilogram (kg)
 1 kilogram (kg) = 1,000 grams (g)
 Measurement tools typically used for mass
 Pan balances, triple beam balances
 Relative size of metric units of mass in realworld context
 Ex:
 Ex:
 Recognition of appropriate unit of measure in realworld context
Note(s):
 Grade Level(s):
 Grade 4 introduces identifying relative sizes of measurement units within the customary and metric systems.
 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:
 IX. Communication and Representation

4.8B 
Convert
MEASUREMENTS WITHIN THE SAME MEASUREMENT SYSTEM, CUSTOMARY OR METRIC, FROM A SMALLER UNIT INTO A LARGER UNIT OR A LARGER UNIT INTO A SMALLER UNIT WHEN GIVEN OTHER EQUIVALENT MEASURES REPRESENTED IN A TABLE
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Products of twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Quotients up to fourdigit dividends by onedigit divisors
 Decimals (less than one or greater than one)
 Limited to multiples of halves (e.g., 0.5, 1.5, 4.5, etc.)
 Determined by reasoning that half of any value is that value divided by 2
 Fractions (proper, improper, and mixed numbers
 Limited to multiples of halves (e.g.,, etc.)
 Determined by reasoning that half of any value is that value divided by 2
 Onestep conversions from a smaller unit to a larger unit or from a larger unit to a smaller unit
 Onestep conversions from a smaller unit to a larger unit or from a larger unit to a smaller unit
 Conversion – a change from one measurement unit to another measurement unit without changing the amount
 Typically used units of measure
 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: onetenth of a base unit
 Centi: onehundredth of a base unit
 Milli: onethousandth of a base unit
 Relationship between converting units
 Converting within the same measurement system, customary or metric
 Multiplication converts larger units to smaller units.
 Division converts smaller units to larger units.
 Ex: Length
 Ex: Volume (liquid volume) and capacity
 Ex: Weight and mass
 Convert measurements within the customary measurement system from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table.
 Length
 Rule/process column given in a table
 Ex:
 Rule/process column not given in a table
 Ex:
 Volume (liquid volume) and capacity
 Rule/process column given in a table
 Ex:
 Rule/process column not given in a table
 Ex:
 Weight
 Rule/process column given in a table
 Ex:
 Rule/process column not given in a table
 Ex:
 Convert measurements within the metric measurement system from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table.
 Length
 Rule/process column given in a table
 Ex:
 Rule/process column not given in a table
 Ex:
 Volume (liquid volume) and capacity
 Rule/process column given in a table
 Ex:
 Rule/process column not given in a table
 Ex:
 Mass
 Rule/process column given in a table
 Ex:
 Rule/process column not given in a table
 Ex:
 Equivalent measures in tables may have missing information in one or both columns.
 Ex:
Note(s):
 Grade Level(s):
 Grade 4 introduces converting measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table.
 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

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
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Products of twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Quotients up to fourdigit dividends by onedigit 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: onetenth of a base unit
 Centi: onehundredth of a base unit
 Milli: onethousandth 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
 Ex:
 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 45minute event minus a 20minute 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
 Ex: Half of a minute or of a minute = 60 seconds ÷ 2 = 30 seconds
 Ex: 1 hour 30 minutes or hours = 60 minutes + (60 minutes ÷ 2) = 60 minutes + 30 minutes = 90 minutes
 Ex:
 Elapsed time
 Finding the end time
 Ex:
 Finding the start time
 Ex:
 Finding the duration
 Ex:
 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
 Ex: Half of a day or of a day = 24 hours ÷ 2 = 12 hours
 Ex: 1 year 6 months or years = 12 months + (12 months ÷ 2) = 12 months + 6 months = 18 months
 Ex:
 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
 Ex:
 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
 Ex:
 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
 Comparision of money amounts
 Ex: 4 quarters = $1.00; 2 half dollars = $1.00; $5.00 = 4 one dollar bills and 4 quarters; etc.
 Ex:
 Making change
 Ex:
 Range of dollar amounts
 Ex:
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 15minute event plus a 30minute 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 STEMANDLEAF PLOT MARKED WITH WHOLE NUMBERS AND FRACTIONS
Including, but not limited to:
 Whole numbers
 Fractions (proper, improper, and mixed numbers)
 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
 Ex: What is your favorite color?
 Ex. Do you have a brother?
 Ex: Which sporting event do you prefer?
 Categorical data may represent numbers or ranges of numbers.
 Ex: How many pets do you have?
 Ex: How many letters are in your name?
 Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
 Numerical data can be counted or measured.
 Ex: How many hours do you spend studying each night?
 Ex: How old were you when you lost your first tooth?
 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 clarifies the meaning of the data represented.
 Categorical data is represented with labels.
 Data can represent an item, a category, a number, or a range of numbers.
 Tally marks are used to record frequencies.
 Numbers are used to represent the count of tally marks in each category.
 Count of tally marks represents the frequency of how often a category occurs.
 Ex:
 Ex:
 Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) to show the frequency (number of times) that each category or number occurs
 Characteristics of a dot plot
 Title clarifies the meaning of the data represented.
 Categorical data is represented with labels.
 When categorical data is used it is orderly and not arbitrary.
 Numerical data is represented with labels and may be whole numbers or fractions.
 Data can represent an item, a category, a number, or a range of numbers.
 Categories are represented by a line, or number line, labeled with categories.
 Counts related to numbers represented by a number line.
 Dots (or Xs) recorded vertically above the line to represent the frequency of each category or number.
 Dots (or Xs) generally represent one count.
 Dots (or Xs) may represent multiple counts if indicated with a key.
 Value of the category is determined by the number of dots (or Xs) drawn.
 Density of dots relates to the frequency of distribution of the data.
 Ex:
 Ex:
 Stemandleaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating one place value from another place value of a data set. The larger of the two place values is called the stem and the smaller of the two place values is called the leaf.
 Characteristics of a stemandleaf plot
 Title clarifies the meaning of the data represented.
 Numerical data is represented with labels and may be whole numbers or fractions.
 The place value of the stem and leaf is dependent upon the values of data in the set.
 For fractions, usually the whole number is the stem and fractional values are the leaves.
 For sets of data close in value, usually the stem is represented by the place value of a number before the last digit and the leaves are represented by the last digit in the number.
 The stem represents one or more pieces of data in the set.
 The leaf represents only one piece of data in the set.
 The leaves provide the frequency counts for the range of numbers included in that row of the stemandleaf plot.
 Density of leaves relates to the frequency of distribution of the data.
 Ex:
Note(s):
 Grade Level(s):
 Grade 1 represented data to with picture and bartype graphs.
 Grade 2 represented data with pictographs and bar graphs with intervals of one.
 Grade 3 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 stemandleaf 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 stemandleaf plots.
 Grade 6 will represent numeric data graphically, including dot plots, stemandleaf plots, histograms, and box plots.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 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 TWOSTEP PROBLEMS USING DATA IN WHOLE NUMBER, DECIMAL, AND FRACTION FORM IN A FREQUENCY TABLE, DOT PLOT, OR STEMANDLEAF PLOT
Including, but not limited to:
 Whole numbers
 Decimals (less than or greater than one to the tenths and hundredths)
 Fractions (proper, improper, and mixed numbers)
 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 twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Division
 Quotients of whole numbers up to fourdigit dividends by onedigit divisors
 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
 Ex: What is your favorite color?
 Ex. Do you have a brother?
 Ex: Which sporting event do you prefer?
 Categorical data may represent numbers or ranges of numbers.
 Ex: How many pets do you have?
 Ex: How many letters are in your name?
 Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
 Numerical data can be counted or measured.
 Ex: How many hours do you spend studying each night?
 Ex: How old were you when you lost your first tooth?
 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) to show the frequency (number of times) that each category or number occurs
 Stemandleaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating one place value from another place value of a data set. The larger of the two place values is called the stem and the smaller of the two place values is called the leaf.
 One and twostep problem situations using graphical representations
 Ex:
 Ex:
Note(s):
 Grade Level(s):
 Grade 3 solved one and twostep problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
 Grade 5 will solve one and twostep problems using data from a frequency table, dot plot, bar graph, stemandleaf 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

4.10 
Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security. The student is expected to:

4.10A 
Distinguish
BETWEEN FIXED AND VARIABLE EXPENSES
Including, but not limited to:
 Expense – payment for goods and services
 Fixed expenses – expenses that are consistent from month to month
 Allows for greater planning in spending
 Often associated with necessary spending
 Often reflects needs
 Sometimes reflects wants
 Ex: Rent or car payment
 Variable expenses – expenses that vary in cost from month to month
 Allows for greater personal control in spending
 Often associated with discretionary spending
 Often reflects wants
 Sometimes reflects needs
 Ex: Groceries or gas
 Relationship between fixed and variable expenses
 Some expenses do not change from month to month and some expenses do change each month
 Some expenses that may be fixed for you may be variable for others depending on the situation
Note(s):
 Grade Level(s):
 Grade 3 explained the connection between human capital/labor and income.
 Grade 5 will define income tax, payroll tax, sales tax, and property tax.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 IX. Communication and Representation
 X. Connections

4.10B 
Calculate
PROFIT IN A GIVEN SITUATION
Including, but not limited to:
 Whole numbers
 Decimals (less than or greater than one to the tenths and hundredths)
 Addition
 Sums of whole numbers
 Sums of decimals up to the hundredths
 Subtraction
 Differences of whole numbers
 Differences of decimals with values limited to the hundredths
 Multiplication
 Products of whole numbers up to twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Division
 Quotients of whole numbers up to fourdigit dividends by onedigit divisors
 Income – money earned or received
 Income in a business also called revienue
 Expense – payment for goods and services
 Expenses in a business also called costs
 Profit – money that is made in a business after all the costs and expenses are paid
 Profit is calculated by subtracting expenses (costs) from income (revenue).
 Income – expenses = profit
 Revenues – cost = profit
 Determining profit from a single source for income and/or expenses
 Ex:
 Ex:
 Determining profit from multiple sources for incomes and/or expenses
 Ex:
 Ex:
 Relationship between income, expenses, and profit
 When income is greater than expenses there is a profit.
 When income is less than expenses, there is no profit or the costs exceed the income.
Note(s):
 Grade Level(s):
 Grade 3 described the relationship between the availability or scarcity of resources and how that impacts cost.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

4.10C 
Compare
THE ADVANTAGES AND DISADVANTAGES OF VARIOUS SAVINGS OPTIONS
Including, but not limited to:
 Savings – money set aside for future use
 Interest – money received for saving money in a bank account; money paid for borrowing money or making purchases on credit
 Interest earned from saving
 Used to encourage people to put money in a bank or credit union or to invest money
 Factors that affect the interest earned in a savings account
 Amount of money deposited in the account
 Interest rate
 Length of time the money is in the account
 Interest rate – price paid for using someone else’s money; the price paid to you for someone else to use your money
 Savings options (choices)
 Piggy bank
 Advantages of saving using a piggy bank
 Disadvantages of piggy bank
 Does not earn interest
 Low risk of theft or loss
 Interest bearing account – a bank or credit union account in which the money deposited earns interest so there will be more money in the future than originally deposited
 Saving accounts Advantages of savings accounts
 Advantages of savings accounts
 Money is easy to access and withdrawal usually does not incur a penalty
 No risk
 Earns interest
 Disadvantages of savings accounts
 Interest rate is usually low
 Certificates of deposit (CDs) and bonds are common types of investment accounts
 Advantages of CDs and bonds
 Low to almost no risk
 Interest rte is usually slightly higher than a savings account
 Earns interest
 Disadvantages of CDs and bonds
 Access to money without penalty occurs on maturity date
 Withdrawal prior to maturity date usually incurs a penalty
 Investing – purchasing something of value (e.g., stocks, bonds, real estate, etc.) with the goal of earning money over time if the value increases
 Advantages of investing
 Potential for profit is higher than an interest bearing account
 Disadvantages of investing
 Money is sometimes hard to access and/or a penalty is charged for withdrawal
 Low to high risk depending on the type of investment
 Potential loss due to economic situations
Note(s):
 Grade Level(s):
 Grade 3 listed reasons to save and explained the benefit of a savings plan, including for college.
 Grade 7 will calculate and compare simple interest and compound interest earning.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 IX. Communication and Representation
 X. Connections

4.10D 
Describe
HOW TO ALLOCATE A WEEKLY ALLOWANCE AMONG SPENDING; SAVING, INCLUDING FOR COLLEGE; AND SHARING
Including, but not limited to:
 Process to allocate (assign or distribute) weekly allowance
 Set a goal every week for both spending and saving.
 Calculate fixed and variable expenses for each week.
 Calculate the desired amount for savings each week.
 The remaining money, after expenses and savings, is allocated for personal spending and/or sharing.
 Reasons to allocate (assign or distribute) weekly allowance
 Predetermined spending amounts
 Ability to earn interest on savings
 Saving to pay for college
 Saving to purchase future wants and needs
 Saving to cover unexpected future expenses
Note(s):
 Grade Level(s):
 Grade 2 distinguished between a deposit and a withdrawal.
 Grade 5 will develop a system for keeping and using financial records.
 Grade 8 will estimate the cost of a twoyear and fouryear college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 IX. Communication and Representation
 X. Connections

4.10E 
Describe
THE BASIC PURPOSE OF FINANCIAL INSTITUTIONS, INCLUDING KEEPING MONEY SAFE, BORROWING MONEY, AND LENDING
Including, but not limited to:
 Financial institution – an establishment that focuses on dealing with financial transactions such as investments, loans, and deposits
 Ex: Banks, savings and loans, credit unions, and other investment companies
 Purposes of financial institutions
 Take in funds (deposits), pool that money, and lend that money to those who need funds.
 Keep deposits safe and regulate accounts and transactions according to federal and/or state laws.
 Provide a place where individuals, businesses, and governments can deposit and borrow money.
 Serve as agents for depositors (who lend money to the bank) and borrowers (to whom the bank lends money).
 Depositors and borrowers can be individuals and households, financial and nonfinancial firms, or national and local governments.
 Keep individual funds available on demand (e.g., checking accounts) or with some restrictions (e.g., savings or investments).
 Process payments to and from account holders and other financial institutions.
Note(s):
 Grade Level(s):
 Grade 3 explained that credit is used when wants or needs exceed the ability to pay and that it is the borrower's responsibility to pay it back to the lender, usually with interest.
 Grade 5 will identify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 IX. Communication and Representation
 X. Connections

Bibliography: Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from: http://www.thecb.state.tx.us/collegereadiness/crs.pdf Texas Education Agency. (2013). Introduction to the revised mathematics TEKS – kindergartenalgebra I vertical alignment. Retrieved from: https://www.texasgateway.org/resource/verticalalignmentchartsrevisedmathematicsteks Texas Education Agency. (2013) Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from: https://www.texasgateway.org/resource/txrcfptexasresponsecurriculumfocalpointsk8mathematicsrevised2013 Texas Education Agency. (2016). Mathematics TEKS – supporting information grade 4. Retrieved from: https://https://www.texasgateway.org/resource/mathematicstekssupportinginformation

Bold black text in italics: Knowledge and Skills Statement (TEKS); Bold black text: Student Expectation (TEKS) Bold red text in italics: Student Expectation identified by TEA as a Readiness Standard for STAAR Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR Blue text: Supporting information / Clarifications from TCMPC (Specificity) Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
