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 TITLE : Unit 09: Patterns on a Coordinate Plane SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address additive and multiplicative numerical patterns as well as graphing in the first quadrant of the coordinate plane. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 3, students represented fractions of halves, fourths, and eighths as distances from zero on a number line. In Grade 4, students represented fractions and decimals to the tenths and hundredths as distances from zero on a number line and represented 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. In Grade 5 Unit 02, students represented and solved multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.

During this Unit
Students are introduced to the coordinate plane and its key attributes including the axes and origin. Students analyze the process for graphing ordered pairs in the first quadrant of the coordinate plane and identify the first number in an ordered pair as the x-coordinate that indicates the movement parallel to the x-axis starting at the origin and the second number in the ordered pair as the y-coordinate that indicates the movement parallel to the y-axis starting at the origin. Although graphing is limited to the Quadrant I of the coordinate plane, ordered pairs may include any positive rational number, including fractions and decimals. Students are expected to graph ordered pairs in the first quadrant of the coordinate plane that are generated from number patterns or an input-output table. Number patterns are examined closely as students recognize the difference between additive and multiplicative numerical patterns when given in a table or graph. Students use input-output tables and graphs to generate numerical patterns when given a rule in the form y = ax (multiplicative numerical pattern) or y = a + x (additive numerical pattern).

After this Unit
In Grade 6, students will compare the two rules verbally, numerically, graphically, symbolically in the form y = ax or y = a + x in order to differentiate between additive and multiplicative relationships. They will graph points in all four quadrants using ordered pairs of rational numbers. Students will also identify independent and dependent quantities from tables and graphs, write an equation that represents the relationship between independent and dependent quantities from a table, and represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.

Research
According to the National Research Council (2001), “Number is intimately connected with geometry, as illustrated in this chapter by our use of the number line and the area model of multiplication. Those same models of number can, of course, arise when measurement is introduced in geometry. The connection between number and algebra is illustrated in the chapter by our use of algebra to express properties of number systems and other general relationships between numbers. The links from number to geometry and to algebra are forged even more strongly when students are introduced to the coordinate plane, in which perpendicular number lines provide a system of coordinates for each point” (p. 107). Connecting algebraic thinking to the coordinate plane also allows students to have two representations of a pattern. They also state that “It is important for students to see that whatever relationships they discover, they exist in both forms. So if a relationship is found in a table, challenge students to see how that plays out in the physical version” (2001, p. 423). When exploring functions, Van de Walle, Karp, and Bay-Williams (2004) add, “The study of functions should focus on change relationships in contexts that are meaningful and interesting to students. An informal study of change relationships can begin as early as the third grade and continue through high school. Students should develop an understanding of the multiple methods of expressing real-world functional relationships using words, graphs, equations, and tables” (p 436).

National Research Council. (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., and Findell, B. (Eds.) Mathematics Learning Study Committee, Center for Education Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding the components of the coordinate plane and how to graph points (ordered pairs) leads to the development of relational thinking about coordinates and their location.
• What attributes exist in a coordinate plane?
• What is represented by the …
• first number
• second number
… in an ordered pair?
• What relationships exist between the …
• axes on a coordinate plane and number lines?
• axes and the origin?
• axes and an ordered pair?
• x-coordinate and movement along the x-axis?
• y-coordinate and movement along the y-axis?
• Geometry
• Coordinate Plane
• Ordered pairs
• Location
• Associated Mathematical Processes
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Recognizing and understanding numerical patterns and relationships leads to efficient, accurate, and flexible representations.
• What process can be used to generate a numerical pattern when given a …
• rule in the form y = ax?
• rule in the form y = a + x?
• graph?
• What are the characteristics of an additive numerical pattern in a …
• table?
• graph?
• What are the characteristics of a multiplicative numerical pattern in a …
• table?
• graph?
• What process can be used to determine if a numerical pattern is additive or multiplicative when given a …
• table?
• graph?
• How can …
• representing
• identifying
… a numerical relationship in a(n) …
• table
• graph
• equation
… aid in problem solving?
• Understanding the components of the coordinate plane and how to graph points (ordered pairs) leads to the development of relational thinking about coordinates and their location.
• What attributes exist in a coordinate plane?
• What is represented by the …
• first number
• second number
… in an ordered pair?
• How are ordered pairs generated from …
• number patterns?
• an input-output table?
• How are the axes used to plot points on a coordinate plane?
• What process can be used to …
• graph
• identify
… a point on a coordinate plane?
• Why is it important to recognize the intervals of the x- and y-axes when graphing ordered pairs on a coordinate plane?
• What relationships exist between the …
• axes on a coordinate plane and number lines?
• axes on a coordinate plane and the origin?
• axes on a coordinate plane and an ordered pair?
• x-coordinate and movement along the x-axis?
• y-coordinate and movement along the y-axis?
• origin and the location of an ordered pair on the coordinate plane?
• x- and y-coordinates in an ordered pair and an input-output table?
• number pattern in an input-output table and the location of the points on a coordinate plane?
• In what situations might someone need to navigate a coordinate plane?
• Algebraic Reasoning
• Patterns and Relationships
• Input-output tables
• Multiplicative
• Representations
• Equations
• Geometry
• Coordinate Plane
• Ordered pairs
• Location
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may reverse the names of the axis rather than always calling the horizontal axis the x-axis and the vertical axis the y-axis.
• Some students may not recognize the paired number (0, 0) with the special name of origin.
• Some students may think they can use either number in an ordered pair to graph a point on coordinate plane rather than always associating the first number in an ordered pair to the x-axis and the second number to the y-axis.
• Some students may not follow the convention of recording an ordered pair by always writing the x-coordinate first, followed by a comma and the y-coordinate inside parentheses.
• Some students may think that if the first pair of numbers in an input-output table follows a rule, the rule is true for the remainder of the values in the table, rather than verifying to make sure the rule holds true for all pairs of numbers in the table.
• Some students may not relate the input and output values from an input-output table to an ordered pair where the input represents the x-coordinate and the output represents the y-coordinate.
• Some students may confuse an additive pattern with a multiplicative pattern by considering the changes within only the input or only the output values rather than considering the change between the input value and its related output value.

#### Unit Vocabulary

• Additive numerical pattern – a pattern that occurs when a constant non-zero value is added to an input value to determine the output value (y = x + a)
• Axes – the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane
• Coordinate plane (coordinate grid) – a two-dimensional plane on which to plot points, lines, and curves
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• 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
• Factor – a number multiplied by another number to find a product
• 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.
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Intersecting lines – lines that meet or cross at a point
• Mixed number – a number that is composed of a whole number and a fraction
• Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value (y = ax)
• Origin – the starting point in locating points on a coordinate plane
• Perpendicular lines – lines that intersect at right angles to each other to form square corners
• Product – the total when two or more factors are multiplied
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Quadrants – any of the four areas created by dividing a plane with an x-axis and y-axis
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Coordinate system Comma Down Graph Horizontal Input Intersection Interval Left Line Numerical pattern Ordered pair Output Parentheses Point Quadrant I Right Rule Set Scale Sequential Substitute Table Unit Up Value Vertical x-axis x-coordinate x-value y-axis y-coordinate y-value
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

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

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 5 Mathematics TEKS

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

Legend:

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

Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
5.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
5.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• X. Connections
5.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
5.1C

Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Process Standard

Select

TOOLS, INCLUDING 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
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
5.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• X. Connections
5.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.4 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:
5.4C Generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph.

Generate

A NUMERICAL PATTERN WHEN GIVEN A RULE IN THE FORM y = ax OR y = x + a AND GRAPH

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}
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• 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 (positive 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
• Sums of whole numbers
• Sums of decimals up to the thousandths
• Sums of fractions with equal and unequal denominators
• 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 three-digit factors by two-digit factors
• Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths
• Multiply tenths by tenths (e.g., 0.3 × 0.7 = 0.21, 1.2 × 1.2 = 1.44, 14.3 × 1.3 = 18.59, etc.)
• Multiply tenths by hundredths or vice versa (e.g., 0.5 × 0.12 = 0.06, 1.4 × 0.15 = 0.21, 21.4 × 0.45 = 9.63, etc.)
• Multiply tenths by thousandths or vice versa (e.g., 0.4 × 0.125 = 0.05, 0.125 × 8.4 = 1.05, etc.)
• Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 × 1.3 = 3.9, 42 × 7.45 = 312.9, 7.02 × 78 = 547.56, 6 × 0.125 = 0.75, etc.)
• Products of fractions where factors are limited to a fraction and a whole number
• Mathematical and real-world numerical relationships
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Relationship between input-output tables and numerical patterns
• Input represented as x
• Output represented as y
• Numerical patterns from rules
• Replace the input (x) with a set of numbers to generate a related output (y).
• Input values must be sequential.
• List of output values creates numerical pattern
• Multiplicative rule in the form y = ax
• Additive rule in the form y = x + a

Graph

A NUMERICAL PATTERN WHEN GIVEN A RULE IN THE FORM y = ax OR y = x + a

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}
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• 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 (positive 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
• Sums of whole numbers
• Sums of decimals up to the thousandths
• Sums of fractions with equal and unequal denominators
• 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 three-digit factors by two-digit factors
• Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths
• Multiply tenths by tenths (e.g., 0.3 × 0.7 = 0.21, 1.2 × 1.2 = 1.44, 14.3 × 1.3 = 18.59, etc.)
• Multiply tenths by hundredths or vice versa (e.g., 0.5 × 0.12 = 0.06, 1.4 × 0.15 = 0.21, 21.4 × 0.45 = 9.63, etc.)
• Multiply tenths by thousandths or vice versa (e.g., 0.4 × 0.125 = 0.05, 0.125 × 8.4 = 1.05, etc.)
• Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 × 1.3 = 3.9, 42 × 7.45 = 312.9, 7.02 × 78 = 547.56, 6 × 0.125 = 0.75, etc.)
• Products of fractions where factors are limited to a fraction and a whole number
• Graphs of numerical patterns
• Limited to Quadrant I of the coordinate plane
• Horizontal axis represents an input (x)
• Vertical axis represents the related output (y)
• Ordered pairs written in the form (x, y) where x represents the input (x-coordinate) and y represents the output (y-coordinate)
• Numerical patterns from the rule y = ax create a graph of points that lie in a straight line and pass through the origin (0, 0).
• Numerical patterns from the rule y = x + a create a graph of points that lie in a straight line and do not pass through the origin (0, 0).
• Generating a set of ordered pairs from a rule using an input-output table
• Substitute values of x in the rule as the input to produce a related values of y as the output to create an ordered pair (x, y), including when x = 0.
• Multiplicative rule in the form y = ax
• Additive rule in the form y = x + a
• Process for graphing ordered pairs of numbers in the first quadrant
• To locate the x-coordinate, begin at the origin and move to the right along the x-axis the appropriate number of units according to the x-coordinate in the ordered pair.
• To locate the y-coordinate, begin at the origin and move up along the y-axis the appropriate number of units according to the y-coordinate in the ordered pair.
• The point of intersection of both the parallel movements on the x-axis and the y-axis is the location of the ordered pair.
• Mathematical and real-world numerical relationships
• Graphing ordered pairs from a numerical rule on a coordinate plane
• Multiplicative rule in the form y = ax
• Additive rule in the form y = x + a

Note(s):

• Grade 4 represented 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.
• Grade 6 will compare the two rules verbally, numerically, graphically, symbolically in the form y = ax or y = a + x in order to differentiate between additive and multiplicative relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and generating expressions and equations to solve problems
• TxCCRS:
• I. Numeric Reasoning
• II.D. Algebraic Reasoning – Representations
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
5.4D Recognize the difference between additive and multiplicative numerical patterns given in a table or graph.
Supporting Standard

Recognize

THE DIFFERENCE BETWEEN ADDITIVE AND MULTIPLICATIVE NUMERICAL PATTERNS GIVEN IN A TABLE OR GRAPH

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}
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• 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 (positive 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
• Sums of whole numbers
• Sums of decimals up to the thousandths
• Sums of fractions with equal and unequal denominators
• 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 three-digit factors by two-digit factors
• Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths
• Multiply tenths by tenths (e.g., 0.3 × 0.7 = 0.21, 1.2 × 1.2 = 1.44, 14.3 × 1.3 = 18.59, etc.)
• Multiply tenths by hundredths or vice versa (e.g., 0.5 × 0.12 = 0.06, 1.4 × 0.15 = 0.21, 21.4 × 0.45 = 9.63, etc.)
• Multiply tenths by thousandths or vice versa (e.g., 0.4 × 0.125 = 0.05, 0.125 × 8.4 = 1.05, etc.)
• Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 × 1.3 = 3.9, 42 × 7.45 = 312.9, 7.02 × 78 = 547.56, 6 × 0.125 = 0.75, etc.)
• Products of fractions where factors are limited to a fraction and a whole number
• Additive numerical pattern – a pattern that occurs when a constant non-zero value is added to an input value to determine the output value (y = x + a)
• Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value (y = ax)
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Relationship between input-output tables and tables of numerical patterns
• x is the input.
• y is the output.
• Additive numerical patterns exist in a table when a constant non-zero value is added to each input value to result in a respective output value.
• Multiplicative numerical patterns exist in a table when a constant non-zero value is multiplied by each input value to result in a respective output value.
• Graphs of numerical patterns
• Limited to Quadrant I of the coordinate plane
• Horizontal axis represents an input (x)
• Vertical axis represents the related output (y)
• Ordered pairs written in the form (x, y) where x represents the input (x-coordinate) and y represents the output (y-coordinate)
• Additive numerical patterns exist in a graph when the points lie in a straight line that does not pass through the origin (0, 0).
• Multiplicative numerical patterns exist in a graph when the points lie in a straight line that passes through the origin (0, 0).
• Relationship between numerical patterns in tables and graphs
• An additive numerical pattern exists when each value of x is added to a constant non-zero value of a to result in a set of respective values of y and will result in a set of ordered pairs which, when graphed, lie on a straight line that does not pass through the origin (0, 0).
• A multiplicative numerical pattern exists when each value of x is multiplied by a constant non-zero value of a to result in a set of respective values of y and will result in a set of ordered pairs which, when graphed, lie on a straight line that passes through the origin (0, 0).

Note(s):

• Grade 4 represented 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.
• Grade 6 will compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and generating expressions and equations to solve problems
• TxCCRS:
• I. Numeric Reasoning
• II.D. Algebraic Reasoning – Representations
• IX. Communication and Representation
5.8 Geometry and measurement. The student applies mathematical process standards to identify locations on a coordinate plane. The student is expected to:
5.8A Describe the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and the given point (0, 0); the x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number, indicates movement parallel to the y-axis starting at the origin.
Supporting Standard

Describe

THE KEY ATTRIBUTES OF THE COORDINATE PLANE, INCLUDING PERPENDICULAR NUMBER LINES (AXES) WHERE THE INTERSECTION (ORIGIN) OF THE TWO LINES COINCIDES WITH ZERO ON EACH NUMBER LINE AND THE GIVEN POINT (0, 0); THE X-COORDINATE, THE FIRST NUMBER IN AN ORDERED PAIR, INDICATES MOVEMENT PARALLEL TO THE X-AXIS STARTING AT THE ORIGIN; AND THE Y-COORDINATE, THE SECOND NUMBER, INDICATES MOVEMENT PARALLEL TO THE Y-AXIS STARTING AT THE ORIGIN

Including, but not limited to:

• Coordinate plane (coordinate grid) – a two-dimensional plane on which to plot points, lines, and curves
• Perpendicular lines – lines that intersect at right angles to each other to form square corners
• Axes – the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane
• Intersecting lines – lines that meet or cross at a point
• Origin – the starting point in locating points on a coordinate plane
• Quadrants – any of the four areas created by dividing a plane with an x-axis and y-axis
• Attributes of the coordinate plane
• Two number lines intersect perpendicularly to form the axes, which are used to locate points on the plane.
• The horizontal number line is called the x-axis.
• The vertical number line is called the y-axis.
• The x-axis and the y-axis cross at 0 on both number lines and that intersection is called the origin.
• The ordered pair of numbers corresponding to the origin is (0, 0).
• Four quadrants are formed by the intersection of the x- and y-axes and are labeled counterclockwise with Roman numerals beginning with Quadrant I that includes the positive x- and y-values.
• The first quadrant plots positive rational numbers.
• Positive numbers on the x-axis are located to the right of the origin.
• Positive numbers on the y-axis are located above the origin.
• Iterated units are labeled and shown on both axes to show scale.
• Intervals may or may not be increments of one.
• Intervals may or may not include decimal or fractional amounts.
• Relationship between ordered pairs and attributes of the coordinate plane
• A pair of ordered numbers names the location of a point on a coordinate plane.
• Ordered pairs of numbers are indicated within parentheses and separated by a comma (x, y).
• When graphing in Quadrant I, the first number in the ordered pair represents the parallel movement on the x-axis, starting at the origin and moving right.
• When graphing in Quadrant I, the second number in the ordered pair represents the parallel movement on the y-axis, starting at the origin and moving up.

Note(s):

• Grade 3 represented fractions of halves, fourths, and eighths as distances from zero on a number line.
• Grade 6 will graph points in all four quadrants using ordered pairs of rational numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.8B Describe the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane.
Supporting Standard

Describe

THE PROCESS FOR GRAPHING ORDERED PAIRS OF NUMBERS IN THE FIRST QUADRANT OF THE COORDINATE PLANE

Including, but not limited to:

• Coordinate plane (coordinate grid) – a two-dimensional plane on which to plot points, lines, and curves
• Axes – the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane
• Intersecting lines – lines that meet or cross at a point
• Origin – the starting point in locating points on a coordinate plane
• Quadrants – any of the four areas created by dividing a plane with an x-axis and y-axis
• Attributes of the coordinate plane
• Two number lines intersect perpendicularly to form the axes, which are used to locate points on the plane.
• The horizontal number line is called the x-axis.
• The vertical number line is called the y-axis.
• The x-axis and the y-axis cross at 0 on both number lines and that intersection is called the origin.
• The ordered pair of numbers corresponding to the origin is (0, 0).
• Four quadrants are formed by the intersection of the x- and y-axes and are labeled counterclockwise with Roman numerals beginning with Quadrant I that includes the positive x- and y-values.
• Relationship between ordered pairs and attributes of the coordinate plane
• A pair of ordered numbers names the location of a point on a coordinate plane.
• Ordered pairs of numbers are indicated within parentheses and separated by a comma (x, y).
• When graphing in Quadrant I, the first number in the ordered pair represents the parallel movement on the x-axis, starting at the origin and moving right.
• When graphing in Quadrant I, the second number in the ordered pair represents the parallel movement on the y-axis, starting at the origin and moving up.
• Limited to the first quadrant for graphing ordered pairs of positive rational numbers
• Various forms of positive rational numbers as ordered pairs
• Counting (natural) numbers
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Fractions (positive proper, improper, or mixed numbers with equal or unequal denominators)
• Process for graphing ordered pairs of numbers in the first quadrant
• To locate the x-coordinate, begin at the origin and move to the right along the x-axis the appropriate number of units according to the x-coordinate in the ordered pair.
• To locate the y-coordinate, begin at the origin and move up along the y-axis the appropriate number of units according to the y-coordinate in the ordered pair.
• The point of intersection of both the parallel movements on the x-axis and the y-axis is the location of the ordered pair.
• Multiple ordered pairs may be graphed on the same coordinate plane.

Note(s):

• Grade 3 represented fractions of halves, fourths, and eighths as distances from zero on a number line.
• Grade 6 will graph points in all four quadrants using ordered pairs of rational numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.8C Graph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table.

Graph

IN THE FIRST QUADRANT OF THE COORDINATE PLANE ORDERED PAIRS OF NUMBERS ARISING FROM MATHEMATICAL AND REAL-WORLD PROBLEMS, INCLUDING THOSE GENERATED BY NUMBER PATTERNS OR FOUND IN AN INPUT-OUTPUT TABLE

Including, but not limited to:

• Coordinate plane (coordinate grid) – a two-dimensional plane on which to plot points, lines, and curves
• Axes – the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane
• Intersecting lines – lines that meet or cross at a point
• Origin – the starting point in locating points on a coordinate plane
• Quadrants – any of the four areas created by dividing a plane with an x-axis and y-axis
• Attributes of the coordinate plane
• Two number lines intersect perpendicularly to form the axes, which are used to locate points on the plane.
• The horizontal number line is called the x-axis.
• The vertical number line is called the y-axis.
• The x-axis and the y-axis cross at 0 on both number lines and that intersection is called the origin.
• The coordinate pair of numbers corresponding to the origin is (0, 0).
• Four quadrants are formed by the intersection of the x- and y-axes and are labeled counterclockwise with Roman numerals beginning with Quadrant I that includes the positive x- and y-values.
• Relationship between ordered pairs and attributes of the coordinate plane
• A pair of ordered numbers names the location of a point on a coordinate plane.
• Ordered pairs of numbers are indicated within parentheses and separated by a comma (x, y).
• When graphing in Quadrant I, the first number in the ordered pair represents the parallel movement on the x-axis, starting at the origin and moving right.
• When graphing in Quadrant I, the second number in the ordered pair represents the parallel movement on the y-axis, starting at the origin and moving up.
• Limited to the first quadrant for graphing ordered pairs of positive rational numbers
• Various forms of positive rational numbers as ordered pairs
• Counting (natural) numbers
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Fractions (positive proper, improper, or mixed numbers with equal or unequal denominators)
• Process for graphing ordered pairs of numbers in the first quadrant
• To locate the x-coordinate, begin at the origin and move to the right along the x-axis the appropriate number of units according to the x-coordinate in the ordered pair.
• To locate the y-coordinate, begin at the origin and move up along the y-axis the appropriate number of units according to the y-coordinate in the ordered pair.
• The point of intersection of both the parallel movements on the x-axis and the y-axis is the location of the ordered pair.
• Multiple ordered pairs may be graphed on the same coordinate plane.
• Ordered pairs in mathematical and real-world problem situations
• Ordered pairs generated from number patterns or those found in an input-output table

Note(s):