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Enhanced TEKS Clarification

Mathematics

Grade 4

Grade 4

§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.1The 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.2The 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 problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. 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.3For 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.4The 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 3-5, 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 multi-step problems involving the four operations with whole numbers with expressions and equations and generate and analyze patterns. In geometry and measurement, students will classify two-dimensional figures, measure angles, and convert units of measure. In data analysis, students will represent and interpret data.
4.Intro.5Statements 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.

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:
    • X. Connections

4.1B

Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

  • Problem-solving model
    • Analyze given information
    • Formulate a plan or strategy
    • Determine a solution
    • Justify the solution
    • Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxRCFP:
    • 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.

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.

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 and use representations to organize, record, and communicate mathematical ideas.

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.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

  • Mathematical relationships
    • Connect and communicate mathematical ideas
      • Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
      • Current knowledge to new learning

Note(s):

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxRCFP:
    • Representing, applying, and analyzing proportional relationships
    • Using expressions and equations to describe relationships, including the Pythagorean Theorem
    • Making inferences from data
  • TxCCRS:
    • X. Connections

4.1G

Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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 place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left.
Supporting Standard

Interpret

THE VALUE OF EACH PLACE-VALUE POSITION AS 10 TIMES THE POSITION TO THE RIGHT AND AS ONE-TENTH 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
  • Base-10 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 one-tenth 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 base-10 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.
Readiness Standard

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
  • 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
  • Base-10 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 one-tenth 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 one-tenth
        • 0.1 can be represented as 1 tenth.
        • 0.1 can be represented as 10 hundredths.
      • The magnitude of one-hundredth
        • 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 three-digit 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 two-digit 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 eighty-five million, one hundred fifty-six thousand, seven hundred eighty-nine and seventy-eight 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 two-digit number when read and recorded using a hyphen, where appropriate, when written (e.g., twenty-three, 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 fifty-four million, ninety-one thousand, five and twenty-six 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 and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =.
Supporting Standard

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.
        4.2C1.jpg
      • 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.
        4.2C2.jpg
    • 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 (pre-determined intervals with at least two labeled numbers)
          4.2C6.jpg
        • Open number lines (no marked intervals)
          4.2C7.jpg 
      • 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:
        4.2C13.jpg
    • Equality words and symbol
      • Equal to (=)
      • Ex:
        4.2C14.jpg
    • 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.
Supporting Standard

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 (pre-determined intervals)
      4.2D1.jpg
    • Open number lines (no marked intervals)
      4.2D2.jpg
    • 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.
      • Ex:
      • Ex:
      • Ex:
      • Ex:
      • Ex:
    • 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.
      • 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”.
      • Ex:
        4.2D14.jpg
      • Ex:
        4.2D15.jpg
      • Ex:
        4.2D16.jpg
      • Ex:
        4.2D17.jpg
      • Ex:
        4.2D18.jpg

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.
Supporting Standard

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
  • Various concrete and visual models
    • Number lines, decimal disks, decimal grids, base-10 blocks, money, etc.
      • Ex:
        4.2E1.jpg
      • Ex:
        4.2E2.jpg
      • Ex:
        4.2E3.jpg
      • Ex:
        4.2E4.jpg
      • Ex:
        4.2E5.jpg
      • Ex:
         

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 and order decimals using concrete and visual models to the hundredths.
Supporting Standard

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
  • 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
      • Equal to (=)
  • 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.
        4.2F1.jpg
      • 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.
        4.2F2.jpg
    • 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, base-10 blocks, money, etc.
        • Ex: Number lines (horizontal/vertical)
        • Ex: Decimal grids
        • Ex: Decimal disks
        • Ex: Base-10 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, base-10 blocks, money, etc.
        • Ex: Number lines
        • Ex: Decimal grids
        • Ex: Decimal disks
        • Ex: Base-10 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.
Readiness Standard

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
  • 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:
          4.2GG.jpg
      • Number line representing values greater than one
        • Ex:
          4.2G1.jpg
      • Number line representing values between tick marks
        • Ex:
          4.2G2.jpg
    • Area model (tenths and hundredths grids)
      • Decimals and fractions of the same whole
        • Ex:
          4.2G3.jpg
      • 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:
          4.2G7.jpg
      • Decimals and fractions less than one
        • Ex:
      • Decimals and fractions greater than one
        • Ex:
    • Base-10 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:
          4.2G14.jpg
    • 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.
Supporting Standard

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
  • 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 “zoom-in” 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 (pre-determined intervals with at least two labeled numbers)
        4.2H1.jpg
    • 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 “zoom-in” 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)
        4.2H2.jpg 
    • Purpose of open number line
      • Open number lines allow for the consideration of the magnitude of numbers and the place-value 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:
        4.2H7.jpg
      • 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 a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b.
Supporting Standard

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 non-zero 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
    • Ex: or 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
            4.3A6.jpg
    • Concrete models of a set of objects
      • Pattern blocks, color tiles, counters, etc.
        • Ex: Pattern blocks
          4.3A7.jpg
        • Ex: Color tiles
          4.3A8.jpg
        • Ex: Counters
          4.3A9.jpg
    • Pictorial models
      • Fraction strips, fraction bar models, number lines, etc.
        • Ex: Fraction strip or fraction bar models
          4.3A10.jpg
        • Ex: Number lines
          4.3A11.jpg

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 .
    • Grade 6 will extend representations for division to include fraction notation such as  represents the same number as a ÷ b where b ≠ 0.
    • 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

4.3B

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.
Supporting Standard

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

Determine if two given fractions are equivalent using a variety of methods.
Supporting Standard

Determine

IF TWO GIVEN FRACTIONS ARE EQUIVALENT USING A VARIETY OF METHODS

Including, but not limited to:

  • 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
  • Equivalent fractions – fractions that have the same value
  • Comparisons of fractions are only valid when referring to the same size whole.
  • Variety of methods to determine if two fractions are equivalent
    • Equivalency using a number line
      • Ex:
    • Equivalency using an area model
      • Ex:
    • Equivalency using a strip diagram
      • Strip diagram – a linear model used to illustrate number relationships
      • Ex:
    • Equivalency using a numeric approach
      • Multiply and/or divide the numeratorand denominator by the same non-zero whole number
      • Simply each fraction
      • Ex:
      • Ex:
    • Equivalency using numeric reasoning
      • Relationship between numerators and denominators within fractions being compared
        • Ex:

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

Compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <.
Readiness Standard

Compare

TWO FRACTIONS WITH DIFFERENT NUMERATORS AND DIFFERENT DENOMINATORS

Including, but not limited to:

  • Fractions (proper, improper, or mixed 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
  • Benchmarks
    • Comparisons of fractions are only valid when referring to the same size whole.
      • Ex:
      • Ex:
      • Ex:
  • Equivalent fractions to determine common denominator or common numerator prior to comparing fractions
    • Common denominators
      • Common denominators standardize the size of the pieces; therefore, compare the number of pieces (numerator)
        • Larger numerator → more equal-size fractional pieces → larger fraction
        • Smaller numerator → fewer equal-size fractional pieces → smaller fraction
      • Least common multiple (LCM) – the smallest multiple that two or more numbers have in common
      • Least common denominator (LCD) – the least common multiple of the denominators of two or more fractions
      • Ex:
    • Common numerators
      • Common numerators standardize the number of pieces; therefore, compare the size of each piece (denominator)
        • Larger denominator → smaller fractional piece → smaller fraction
        • Smaller denominator → larger fractional piece → larger fraction
      • Least common multiple (LCM) – the smallest multiple that two or more numbers have in common
      • Least common numerator – the least common multiple of the numerators of two or more fractions
      • Ex:
  • Concrete or pictorial models
    • Comparisons of fractions are only valid when referring to the same size whole.
      • Ex:
      • Ex:
      • Ex:
    • Shaded portions of models may or may not be adjacent.
      • Ex:

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:
        4.3D6.jpg
  • Equality words and symbol
    • Equal to (=)
    • Ex:
      4.3D7.jpg

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 and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.
Readiness Standard

Represent, Solve

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

Including, but not limited to:

  • Fractions (proper, improper, or mixed numbers with equal denominators)
    • Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
    • Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
    • Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
    • Mixed number – a number that is composed of a whole number and a fraction
  • Addition
    • Sums of fractions limited to equal denominators
  • Subtraction
    • Differences of fractions limited to equal denominators
  • Fractional relationships
    • Relationship between the whole and the part
      • Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
      • Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
    • Referring to the same whole
      • Fractions are relationships, and the size or the amount of the whole matters
      • Common whole is needed when adding or subtracting fractions
      • Equivalent fractions to simplify solutions
  • Concrete objects and pictorial models for addition of fractions with equal denominators that build to the number line
    • Pattern blocks and other shapes (circles, squares, rectangles, etc.)
      • 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)
          • Ex:
                 
            Therfore,
    • 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 real-world problem situations
    • Ex:
    • Ex:
    • Ex:
    • Ex:

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

Evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole.
Supporting Standard

Evaluate

THE REASONABLENESS OF SUMS AND DIFFERENCES OF FRACTIONS USING BENCHMARK FRACTIONS 0, AND 1, REFERRING TO THE SAME WHOLE

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
  • 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
  • Reasoning with fraction benchmarks
    • With and without models
    • Like and unlike denominators
    • Ex:
    • Ex:
    • Ex:
    • Ex:
  • Reasoning with fraction benchmarks with and without models within problem situations
    • Ex:
    • Ex:

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.
Supporting Standard

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
  • 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 “zoom-in” 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:
        4.3G3.jpg
      • Ex:
        4.3G4.jpg
    • 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:
      4.3G7.jpg
    • 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:
        4.3G8.jpg
  • 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:
      4.3G9.jpg
    • 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:
        4.3G10.jpg

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 and subtract whole numbers and decimals to the hundredths place using the standard algorithm.
Readiness Standard

Add, Subtract

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

Including, but not limited to:

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

Note(s):

  • Grade Level(s):
    • Grade 3 solved with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
    • Grade 4 extends adding and subtracting of whole numbers from 1,000 to 1,000,000 and introduces adding and subtracting decimals, including tenths and hundredths.
    • Grade 5 will estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
    • Understanding decimals and addition and subtraction of decimals
  • TxCCRS:
    • I. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

4.4B

Determine products of a number and 10 or 100 using properties of operations and place value understandings.
Supporting Standard

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, equal-sized 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 three-digit number by a two-digit 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 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15.
Supporting Standard

Represent

THE PRODUCT OF 2 TWO-DIGIT 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 two-digit factors by two-digit 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
      • Ex: 15 × 15 = 252
    • Product = factor × same factor
      • Ex: 252 = 15 × 15

Note(s):

  • Grade Level(s):
    • Grade 3 used strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may have included mental math, partial products, and the commutative, associative, and distributive properties.
    • Grade 5 will multiply with fluency a three-digit number by a two-digit 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 four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties.
Supporting Standard

Use

STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO MULTIPLY UP TO A FOUR-DIGIT NUMBER BY A ONE-DIGIT NUMBER AND TO MULTIPLY A TWO-DIGIT NUMBER BY A TWO-DIGIT 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 two-digit factors by two-digit factors and up to four-digit factors by one-digit 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 two-digit number by a one-digit number. Strategies may have included mental math, partial products, and the commutative, associative, and distributive properties.
    • Grade 5 will multiply with fluency a three-digit number by a two-digit 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 four-digit whole number divided by a one-digit whole number using arrays, area models, or equations.
Supporting Standard

Represent

THE QUOTIENT OF UP TO A FOUR-DIGIT WHOLE NUMBER DIVIDED BY A ONE-DIGIT 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 four-digit dividends by one-digit divisors
      • Quotients may include remainders
        • Ex: 145 ÷ 6 = 24 R1
  • 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 real-world 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 one-step and two-step 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 four-digit dividend by a two-digit divisor using strategies and the standard algorithm.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
  • TxCCRS:
    • I. Numeric Reasoning
    • II.D. Algebraic Reasoning – Representations
    • IX. Communication and Representation

4.4F

Use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor.
Supporting Standard

Use

STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO DIVIDE UP TO A FOUR-DIGIT DIVIDEND BY A ONE-DIGIT 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 four-digit dividends by one-digit divisors
      • Quotients may include remainders
        • Ex: 145 ÷ 6 = 24 R1
  • Recognition of division in mathematical and real-world 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:
        4.4F3.jpg
  • 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 four-digit dividend by a one-digit divisor.
    • Grade 5 will solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
  • TxCCRS:
    • I. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

4.4G

Round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers.
Supporting Standard

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
    • Sums of whole numbers
  • Subtraction
    • Differences of whole numbers
  • Multiplication
    • Product – the total when two or more factors are multiplied
    • Factor – a number multiplied by another number to find a product
    • Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
  • Division
    • Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
    • Dividend – the number that is being divided
    • Divisor – the number the dividend is being divided by
    • Quotients up to four-digit dividends by one-digit divisors
  • Recognition of operations in mathematical and real-world problem situations
    • Multi-step problems
  • 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 (pre-determined intervals)
      4.4G1.jpg
    • Open number line (no marked intervals)
      4.4G2.jpg
    • 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.
      • Ex:
      • Ex:
      • Ex:
    • 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.
      • Ex:
      • Ex:
      • Ex:
    • 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 two-digit numbers are being considered in the problem; round to the nearest 100 if only three-digit numbers are being considered in the problem; round to the nearest 1,000 if only four-digit numbers are being considered; round to the nearest 10 if both two-digit and three-digit numbers are being considered in the problem; round to the nearest 100 if both three-digit and four-digit numbers are being considered; etc.).
      • Vocabulary indicating estimation in mathematical and real-world 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”.
      • Ex:
        4.4G15.jpg
      • Ex:
        4.4G16.jpg
      • Ex:
        4.4G17.jpg
    • 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 two-digit numbers are being considered in the problem; round to the nearest 100 if only three-digit numbers are being considered in the problem; round to the nearest 1,000 if only four-digit numbers are being considered; round to the nearest 10 if both two-digit and three-digit numbers are being considered in the problem; round to the nearest 100 if both three-digit and four-digit numbers are being considered; etc.).
      • Vocabulary indicating estimation in mathematical and real-world 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:
        4.4G19.jpg
      • Ex:
        4.4G20.jpg
      • Ex:
        4.4G21.jpg
      • 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
    • Sums of whole numbers
  • Subtraction
    • Differences of whole numbers
  • Multiplication
    • Product – the total when two or more factors are multiplied
    • Factor – a number multiplied by another number to find a product
    • Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
  • Division
    • Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
    • Dividend – the number that is being divided
    • Divisor – the number the dividend is being divided by
    • Quotients up to four-digit dividends by one-digit divisors
  • Recognition of operations in mathematical and real-world problem situations
    • Multi-step problems
  • 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 real-world 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:
      4.4G23.jpg
    • Ex:
      4.4G24.jpg
    • Ex:
    • Ex:
      4.4G26.jpg 

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 two-step problems involving multiplication and division, including interpreting remainders.
Readiness Standard

Solve

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

Including, but not limited to:

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

Note(s):

  • Grade Level(s):
    • Grade 4 introduces solving with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.
    • Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
    • Grade 5 will solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm.
    • Various mathematical process standards will be applied to this student expectation as appropriate
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
  • TxCCRS:
    • I. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

4.5

Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:

4.5A

Represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.
Readiness Standard

Represent

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

Including, but not limited to:

  • Whole numbers
    • Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
    • Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
  • Addition
    • Sums of whole numbers
  • Subtraction
    • Differences of whole numbers
  • Multiplication
    • Product – the total when two or more factors are multiplied
    • Factor – a number multiplied by another number to find a product
    • Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
  • Division
    • Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
    • Dividend – the number that is being divided
    • Divisor – the number the dividend is being divided by
    • Quotients of whole numbers up to four-digit dividends by one-digit divisors
      • Quotients may include remainders
  • Representations of an unknown quantity in an equation
    • Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
    • Any single letter to represent the unknown quantity (e.g., 24 – 8 = y, etc.)
    • Equal sign at beginning or end and unknown in any position
      • 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 real-world 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:
      • Part-part-whole part unknown
        • Ex:
      • Part-Part-whole 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:
  • Multi-step 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 two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations.
    • Grade 3 represented and solved one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations.
    • Grade 3 determined the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product.
    • Grade 5 will represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
  • TxCCRS:
    • I. Numeric Reasoning
    • II.D. Algebraic Reasoning – Representations
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

4.5B

Represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence.
Readiness Standard

Represent

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

Including, but not limited to:

  • Whole numbers
    • Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
    • Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
  • Addition
    • Sums of whole numbers
  • Multiplication
    • Product – the total when two or more factors are multiplied
    • Factor – a number multiplied by another number to find a product
    • Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
  • 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)
    • Input-output 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 non-zero value is added to an input value to determine the output value
        • Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value
    • Ex:
    • Ex:
  • Sequence – an ordered list of numbers, usually set apart by commas, such as {2, 4, 6, 8, 10, 12, …}
  • Relationship between input-output tables and sequences
    • Input – position in the sequence
    • Output – value in the sequence
    • Ex:
  • Relationship between numerical expressions and rules to create input-output 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 real-world relationships using number pairs in a table and verbal descriptions.
    • Grade 5 will generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph.
    • Grade 5 will recognize the difference between additive and multiplicative numerical patterns given in a table or graph.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
  • TxCCRS:
    • I. Numeric Reasoning
    • II.D. Algebraic Reasoning – Representations
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

4.5C

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).

Use

MODELS TO DETERMINE THE FORMULAS FOR THE PERIMETER OF A RECTANGLE (l + w + l + w OR2l + 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 one-dimensional linear measure
  • Models to determine formulas for perimeter
    • Rectangle (P = l + w + l + w or P = 2l + 2w)
      • Ex:
    • Square (P = 4s)
      • Ex:
        4.5C2.jpg
  • Area – the measurement attribute that describes the number of square units a figure or region covers
    • Area is a multiplicative two-dimensional square unit measure.
  • Models to determine formulas for area
    • Rectangle (A = l x w)
      • Ex:
        4.5C3.jpg
    • Square (A = s x s)
      • Ex:
        4.5C4.jpg

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 x 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 x w x h, V = s x s x 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.
Readiness Standard

Solve

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

Including, but not limited to:

  • Rectangle
    • 4 sides
    • 4 vertices
    • Opposite sides congruent
    • 2 pairs of parallel sides
    • 4 pairs of perpendicular sides
    • 4 right angles
  • Square (a special type of rectangle)
    • 4 sides
    • 4 vertices
    • All sides congruent
    • 2 pairs of parallel sides
    • 4 pairs of perpendicular sides
    • 4 right angles
  • Perimeter – a linear measurement of the distance around the outer edge of a figure
    • Perimeter is a one-dimensional linear measure.
    • Whole number side lengths
  • Recognition of perimeter embedded in mathematical and real-world problem situations
    • 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 two-dimensional square unit measure.
    • Whole number side lengths
  • Recognition of area embedded in mathematical and real-world 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 x s, where s represents the side length of the square
    • Rectangle
      • A = l x 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.
Supporting Standard

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:
      4.6A1.jpg
  • 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:
          4.6A2.jpg
      • 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 off-set 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 off-set 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 off-set lengths
      • Notation is given as a box in the angle corner to represent a 90° angle.
        • Ex:
    • Lines in pictorial models and polygons
      • Ex:
        4.6A7.jpg
    • 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 (°)
      • Ex: 90 degrees or 90°
  • 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 non-overlapping 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 and draw one or more lines of symmetry, if they exist, for a two-dimensional figure.
Supporting Standard

Identify, Draw

ONE OR MORE LINES OF SYMMETRY, IF THEY EXIST, FOR A TWO-DIMENSIONAL FIGURE

Including, but not limited to:

  • Line of symmetry – line dividing an image into two congruent parts that are mirror images of each other
  • Two-dimensional figure – a figure with two basic units of measure, usually length and width
  • Two-dimensional figures and real-world figures
  • Shapes with more than one line of symmetry
    • Ex:
      4.6B1.jpg
  • Shapes with no lines of symmetry
    • Ex:
      4.6B2.jpg
  • Shapes on which lines of symmetry have not been drawn
    • Ex:
      4.6B3.jpg
  • 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.
Supporting Standard

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 two-dimensional 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:
    • Measuring angles
  • TxCCRS:
    • III.A. Geometric Reasoning – Figures and their properties
    • IX. Communication and Representation

4.6D

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.
Readiness Standard

Classify

TWO-DIMENSIONAL 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:

  • Two-dimensional 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 two-dimensional 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
      • mAmC 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 two-dimensional 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:
        4.6D3.jpg 
    • 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:
                      4.6D5.jpg
                • No congruent angles
                  • Right triangle with one 90° angle and two other angles each of different measures
                    • Ex:
                      4.6D6.jpg
              • 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:
                      4.6D7.jpg
                • At least 2 congruent angles
                  • Right triangle with one 90° angle and two other angles each of the same measure
                    • Ex:
                      4.6D8.jpg
                  • Obtuse triangle with two angles of the same measure and one angle greater than 90°
                    • Ex:
                      4.6D9.jpg
                  • Acute triangle with all angles measuring less than 90° and at least two of the angles of the same measure
                    • Ex:
                      4.6D10.jpg
              • 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:
                      4.6D11.jpg
        • 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:
        • Heptagon or 7-gon
          • 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:
        • Nonagon or 9–gon
          • 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:
        • Hendecagon or 11–gon
          • 11 sides
          • 11 vertices
          • Possible parallel and/or perpendicular sides
          • Possible acute, obtuse, and/or right angles
          • Ex:
        • Dodecagon or 12–gon
          • 12 sides
          • 12 vertices
          • Possible parallel and/or perpendicular sides
          • Possible acute, obtuse, and/or right angles
          • Ex:
  • Classification of two-dimensional figures based on attributes of sides and angles
    • Ex:

Note(s):

  • Grade Level(s):
    • Grade 3 classified and sorted two- and three-dimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.
    • Grade 5 will classify two-dimensional 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:
    • Measuring angles
  • 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.

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 (°)
      • Ex: 90 degrees or 90°
  • 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:
      4.6E1.jpg 
  • 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 central angles in Geometry (G.12) and radian measures in Precalculus (P.4).
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Measuring angles
  • TxCCRS:
    • III.A. Geometric Reasoning – Figures and their properties
    • IX. Communication and Representation

4.7B

Illustrate degrees as the units used to measure an angle, where 1/360 of any circle is one degree and an angle that "cuts" n/360 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.

Illustrate

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

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 (°)
      • Ex: 90 degrees or 90°
    • Angle measures limited to whole numbers, 0° to 360°
  • Angle – two rays with a common endpoint (the vertex)
  • 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.
  • Representations of the  “cuts” out of a circle as degrees of angle measures
    • Ex:

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.
    • Foundational for central angles in Geometry (G.12) and radian measures in Precalculus (P.4).
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Measuring angles
  • 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.
Readiness Standard

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 (°)
      • Ex: 90 degrees or 90°
    • 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:
        4.7C1.jpg
    • Relationships between a protractor and a circle
      • One protractor is a semi-circle, 180º
      • Two protractors make a complete circle, 360º
      • Ex:
  • Measurement or “m” notation indicates the measure of the angle in degrees (e.g., m1 = 50º)
  • Measure angles with a ray aligned at zero degrees (right and/or left).
    • Ex:
  • Measure angles where a ray of the angle does not lie on zero degrees.
    • Ex:
  • Measure angles whose rays may lie between numerically marked intervals.
    • Ex:
  • Use a right angle, 90°, as a benchmark to determine angle classifications (acute, obtuse, and right) to determine reasonableness of angle measures.
  • Measure angles within two-dimensional figures.
    • 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:
    • Measuring angles
  • 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.
Supporting Standard

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 (°)
      • Ex: 90 degrees or 90°
    • 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: Use a protractor to draw a 150º angle.
        4.7D1.jpg 

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:
    • Measuring angles
  • 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

Determine the measure of an unknown angle formed by two non-overlapping adjacent angles given one or both angle measures.
Supporting Standard

Determine

THE MEASURE OF AN UNKNOWN ANGLE FORMED BY TWO NON-OVERLAPPING ADJACENT ANGLES GIVEN ONE OR BOTH ANGLE MEASURES

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 (°)
      • Ex: 90 degrees or 90°
    • Angle measures limited to whole numbers, 0° to 180°
  • Angle – two rays with a common endpoint (the vertex)
  • Adjacent angles – two non-overlapping angles that share a common vertex and exactly one ray
    • Ex:
  • Complementary angles – two angles whose degrees measures have a sum of 90 degrees
  • Supplementary angles – two angles whose degrees measures have a sum of 180 degrees
  • Congruent angles – angles whose angle measurements are equal
  • Angle congruency marks – angle marks indicating angles of the same measure
  • Decompose and compose angle measures
    • Angle measures up to 360 degrees
    • The angle measure of the whole is the sum of the angle measure of the parts
      • Given the measure of one angle, and the whole, find the measure of the other angle.
        • Ex:
      • Given the measure of two angles, find the measure of the whole angle.
        • Ex:
      • Given the measure of the whole angle divided equally, find the measure of the equal sized angles
        • Ex:
      • Multiple steps to find a missing measure
        • Ex:
      • Adjacent angles within two-dimensional figures
        • Ex:
          4.7E6.jpg
      • Angles in context without graphics
        • Ex:

Note(s):

  • Grade Level(s):
    • Grade 4 introduces determining the measure of an unknown angle formed by two non-overlapping adjacent angles given one or both angle measures.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Measuring angles
  • 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.
Supporting Standard

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
    • 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 real-world 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 real-world 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 real-world 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 real-world 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 real-world 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 real-world context
        • Ex:
        • Ex:
    • Metric units
      • 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: one-tenth of a base unit
        • Centi: one-hundredth of a base unit
        • Milli: one-thousandth of a base unit

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.
Supporting Standard

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 two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
    • Quotients up to four-digit dividends by one-digit divisors
  • Decimals (less than or greater than one, limited to multiples of halves; e.g., 1.5, 0.5, 4.5, etc.)
  • Fractions (proper, improper, and mixed numbers, limited to multiples of halves; e.g.,, etc.)
  • One-step 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: one-tenth of a base unit
        • Centi: one-hundredth of a base unit
        • Milli: one-thousandth 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
        4.8B1.jpg
      • Ex: Volume (liquid volume) and capacity
        4.8B2.jpg
      • Ex: Weight and mass
        4.8B3.jpg
  • 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:
    • Volumn (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:
          4.8B11.jpg
    • 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.
Readiness Standard

Solve

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

Including, but not limited to:

  • Whole numbers (0 – 1,000,000,000)
    • Products of two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
    • Quotients up to four-digit dividends by one-digit divisors
  • Decimals (values greater than and less than one, limited to multiples of halves; e.g., 1.5, 0.5, 4.5, etc.)
    • Addition and subtraction of money amounts up to hundredths
  • Fractions (proper, improper, and mixed numbers, limited to multiples of halves; e.g., , etc.)
  • Typically used customary and metric units
    • Customary
      • Length: miles, yards, feet, inches
      • Volume (liquid volume) and capacity: gallons, quarts, pints, cups, fluid ounces
      • Weight: tons, pounds, ounces
    • Metric
      • Length: kilometer, meter, centimeters, millimeters
      • Volume (liquid volume) and capacity: kiloliter, liter, milliliter
      • Mass: kilogram, gram, milligram
      • Based on prefixes attached to base unit
        • Base units include meter for length, liter for volume and capacity, and gram for weight and mass.
        • Kilo: one thousand base units
        • Deci: one-tenth of a base unit
        • Centi: one-hundredth of a base unit
        • Milli: one-thousandth of a base unit
  • Typically used measurement tools
    • Customary
      • Length: rulers, yardsticks, measuring tapes
      • Volume (liquid volume) and capacity: measuring cups, measuring containers or jars
    • Metric
      • Length: rulers, meter sticks, measuring tapes
      • Volume (liquid volume) and capacity: beakers, graduated cylinders, eye droppers, measuring containers or jars
      • Mass: pan balances, triple beam balances
  • Problem situations that deal with measurements of length
    • Addition, subtraction, multiplication, and/or division of measurements of length with or without conversion
    • May or may not include using measuring tools to determine length
    • 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 45-minute event minus a 20-minute event equals 1 hour 25 minutes
    • Time intervals given
    • Pictorial models and tools
      • Measurement conversion tables
      • Analog clock with gears, digital clock, stop watch, number line, etc.
    • Time conversions
      • 1 hour = 60 minutes; 1 minute = 60 seconds
      • Fractional values of time
        • Ex: Half of a minute or  of a minute = 30 seconds; 1 hour 30 minutes or 1 hours = 90 minutes; etc.
      • 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
        • Ex: Half of a day or  of a day = 12 hours; 1 year 3 months = 15 months; 1 year 6 months or 1 years = 18 months; etc.
      • 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:
        4.8C9.jpg
    • Range of dollar amounts
      • Ex:

Note(s):

  • Grade Level(s):
    • Grade 4 introduces solving problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.
    • Grade 5 will solve problems by calculating conversions within a measurement system, customary or metric.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
  • TxCCRS:
    • I. Numeric Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections

4.9

Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data. The student is expected to:

4.9A

Represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions.
Readiness Standard

Represent

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

Including, but not limited to:

  • 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:
    • Stem-and-leaf 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 stem-and-leaf 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 stem-and-leaf 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 bar-type graphs.
    • Grade 2 represented data with pictographs and bar graphs with intervals of one.
    • Grade 3 summarized a data set with multiple categories using a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
    • Grade 4 introduces representing data on a stem-and-leaf plot.
    • Grade 5 will represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots.
    • Grade 6 will represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
    • Understanding decimals and addition and subtraction of decimals
    • Building foundations for addition and subtraction of fractions
  • TxCCRS:
    • VI.B. Statistical Reasoning – Describe Data
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

4.9B

Solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot.
Supporting Standard

Solve

ONE- AND TWO-STEP PROBLEMS USING DATA IN WHOLE NUMBER, DECIMAL, AND FRACTION FORM IN A FREQUENCY TABLE, DOT PLOT, OR STEM-AND-LEAF PLOT

Including, but not limited to:

  • 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 two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
  • Division
    • Quotients of whole numbers up to four-digit dividends by one-digit divisors
  • Data – 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
    • Stem-and-leaf 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 two-step problem situations using graphical representations
    • Ex:
    • Ex:

Note(s):

  • Grade Level(s):
    • Grade 3 solved one- and two-step problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
    • Grade 5 will solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
    • Understanding decimals and addition and subtraction of decimals
    • Building foundations for addition and subtraction of fractions
  • TxCCRS:
    • I. Numeric Reasoning
    • VI.B. Statistical Reasoning – Describe Data
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation

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.
Supporting Standard

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
      • Ex: Allowance

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:
    • Financial Literacy
  • TxCCRS:
    • IX. Communication and Representation
    • X. Connections

4.10B

Calculate profit in a given situation.
Supporting Standard

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 two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
  • Division
    • Quotients of whole numbers up to four-digit dividends by one-digit divisors
  • 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:
    • Financial Literacy
  • TxCCRS:
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections

4.10C

Compare the advantages and disadvantages of various savings options.

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
        • Money is easy to access
      • 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:
    • Financial Literacy
  • TxCCRS:
    • IX. Communication and Representation
    • X. Connections

4.10D

Describe how to allocate a weekly allowance among spending; saving, including for college; and sharing.

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
    • Pre-determined 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.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxRCFP:
    • Financial Literacy
  • TxCCRS:
    • IX. Communication and Representation
    • X. Connections

4.10E

Describe the basic purpose of financial institutions, including keeping money safe, borrowing money, and lending.
Supporting Standard

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:
    • Financial Literacy
  • 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 – kindergarten-algebra I vertical alignment. Retrieved from:
https://www.texasgateway.org/resource/vertical-alignment-charts-revised-mathematics-teks

Texas Education Agency. (2013) Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from:
https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

Texas Education Agency. (2016). Mathematics TEKS – supporting information grade 4. Retrieved from:
https://https://www.texasgateway.org/resource/mathematics-teks-supporting-information

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)

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