6.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


6.1A 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

6.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

6.1C 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

6.1D 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

6.1E 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

6.1F 
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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

6.1G 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

6.6 
Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to:


6.6C 
Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
Readiness Standard

Represent
A GIVEN SITUATION USING VERBAL DESCRIPTIONS, TABLES, GRAPHS, AND EQUATIONS IN THE FORM y = kx OR y = x + b
Including, but not limited to:
 Independent quantities are represented by the x coordinates or the input.
 Dependent quantities are represented by the y coordinates or the output.
 Various representations to describe algebraic relationships
 Verbal descriptions
 Tables
 Graphs
 Equations
 In the form y = kx, where k is the nonzero scale factor (constant of proportionality), from multiplicative problem situations
 Coefficient – a number that is multiplied by a variable(s)
 Integers
 Products of integers limited to an integer multiplied by an integer
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 In the form y = x + b, where b is the constant nonzero addend, from additive problem situations
 Constant – a fixed value that does not appear with a variable(s)
 Integers
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
Note(s):
 Grade Level(s):
 Grade 6 introduces representing a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
 Grade 7 will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Grade 8 will represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
 Grade 8 will represent linear nonproportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
 Grade 8 will write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of contexts
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

6.8 
Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to represent relationships and solve problems. The student is expected to:


6.8A 
Extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle.
Supporting Standard

Extend
PREVIOUS KNOWLEDGE OF TRIANGLES AND THEIR PROPERTIES TO INCLUDE THE SUM OF ANGLES OF A TRIANGLE AND THE RELATIONSHIP BETWEEN THE LENGTHS OF SIDES AND MEASURES OF ANGLES IN A TRIANGLE
Including, but not limited to:
 Properties of triangles – relationship of attributes within a triangle (e.g., an equilateral triangle has all sides and angles congruent, the sum of the lengths of two sides of a triangle is always greater than the length of the third side, the sum of the angles of a triangle is always 180°; etc.)
 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 angle – an angle that measures less than 90°
 Right angle – 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 angle – an angle that measures greater than 90° but less than 180°
 Congruent – of equal measure, having exactly the same size and same shape
 Angle congruency marks – angle marks indicating angles of the same measure
 Side congruency marks – side marks indicating side lengths of the same measure
 Equilateral – all side lengths of a polygon are congruent in measure
 Equiangular – all angles in a polygon are congruent in measure
 Triangle – a polygon with three sides and three vertices
 3 sides
 3 vertices
 Classification by angles
 Triangles are named based on their largest angle.
 Acute triangle
 3 sides
 3 vertices
 3 acute angles (less than 90°)
 Right triangle
 3 sides
 3 vertices
 2 acute angles (less than 90°)
 1 right angle (exactly 90°)
 Obtuse triangle
 3 sides
 3 vertices
 2 acute angles (less than 90°)
 1 obtuse angle (greater than 90° but less than 180°)
 Classification by length of sides
 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
 No congruent angles
 Right triangle with one 90° angle and two other angles each of different measures
 Obtuse triangle with one angle greater than 90° and two other angles each of different measures
 Acute triangle with all angles less than 90° and all angles of different measures
 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
 Obtuse triangle with one angle greater than 90° and two other angles each of different measures
 At least 2 congruent angles
 Right triangle with one 90° angle and two other angles each of the same measure
 Obtuse triangle with two angles of the same measure and one angle greater than 90°
 Acute triangle with all angles measuring less than 90° and at least two of the angles of the same measure
 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°
 Sum of the interior angles of a triangle is 180°
 Equations to determine a missing angle measure
 Relationship between lengths of sides and measure of angles in a triangle
 The shortest side length in a triangle is always opposite the smallest angle measure in a triangle.
 The longest side length in a triangle is always opposite the largest angle measure in a triangle.
 The sides opposite from angles of equal measure in a triangle are always congruent.
Note(s):
 Grade Level(s):
 Grade 4 applied knowledge of right angles to identify acute, right, and obtuse triangles.
 Grade 4 determined the measure of an unknown angle formed by two nonoverlapping adjacent angles given one or both angle measures.
 Grade 6 introduces extending previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle.
 Grade 7 will write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
 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:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.

6.8C 
Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.
Supporting Standard

Write
EQUATIONS THAT REPRESENT PROBLEMS RELATED TO THE AREA OF RECTANGLES, PARALLELOGRAMS, TRAPEZOIDS, AND TRIANGLES AND VOLUME OF RIGHT RECTANGULAR PRISMS WHERE DIMENSIONS ARE POSITIVE RATIONAL NUMBERS
Including, but not limited to:
 Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
 Various forms of positive rational numbers
 Counting (natural) numbers
 Decimals
 Fractions
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
 Types of polygons
 Triangle
 3 sides
 3 vertices
 No parallel sides
 Quadrilateral
 4 sides
 4 vertices
 Types of quadrilaterals
 Trapezoid
 Types of trapezoids
 Isosceles trapezoids
 4 sides
 4 vertices
 Exactly one pair of parallel sides
 At least 2 congruent sides, where 2 of the sides are opposite each other
 Right trapezoids
 4 sides
 4 vertices
 Exactly one pair of parallel sides
 2 pairs of perpendicular sides
 2 right angles
 Parallelogram
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 Opposite angles congruent
 Types of parallelograms
 Rectangle
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Rhombus
 4 sides
 4 vertices
 All sides congruent
 2 pairs of parallel sides
 Opposite angles congruent
 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
 Area – the measurement attribute that describes the number of square units a figure or region covers
 Area is a twodimensional square unit measure
 Positive rational number side lengths
 Formulas for area from STAAR Grade 6 Mathematics Reference Materials
 Rectangle or parallelogram
 A = bh, where b represents the length of the base of the rectangle or parallelogram and h represents the height of the rectangle or parallelogram
 Trapezoid
 A = (b_{1} + b_{2})h, where b_{1} represents the length of one of the parallel bases, b_{2} represents the length of the other parallel base, and h represents the height
 Triangle
 A = bh, where b represents the length of the base of the triangle and h represents the height of the triangle
 Threedimensional figure – a figure that has measurements including length, width (depth), and height
 Edge – where the sides of two faces meet on a threedimensional figure
 Vertex (vertices) in a threedimensional figure – the point (corner) where three or more edges of a threedimensional figure meet
 Face – a flat surface of a threedimensional figure
 Bases of a right rectangular prism – any two congruent, opposite, and parallel faces shaped like rectangles; exactly 3 possible sets
 Height of a right rectangular prism – the length of a side that is perpendicular to both bases
 Prism – a threedimensional figure containing two congruent and parallel faces that are polygons
 Right rectangular prisms and cubes
 6 rectangular faces (2 parallel rectangular faces [bases], 4 rectangular faces)
 12 edges
 8 vertices
 Volume – the measurement attribute of the amount of space occupied by matter
 One way to measure volume is a threedimensional cubic measure
 Positive rational number side lengths
 Formulas for volume from STAAR Grade 6 Mathematics Reference Materials
 Rectangular prism
 V = Bh, where B represents the base area and h represents the height of the prism, which is the number of times the base area is repeated or layered
 The base of a rectangular prism is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular prism may be found using V = Bh or V = (bh)h or V = (lw)h.
Note(s):
 Grade Level(s):
 Grade 5 represented and solved problems related perimeter and/or area and related to volume.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of contexts
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

6.8D 
Determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.
Readiness Standard

Determine
SOLUTIONS FOR PROBLEMS INVOLVING THE AREA OF RECTANGLES, PARALLELOGRAMS, TRAPEZOIDS, AND TRIANGLES AND VOLUME OF RIGHT RECTANGULAR PRISMS WHERE DIMENSIONS ARE POSITIVE RATIONAL NUMBERS
Including, but not limited to:
 Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
 Various forms of positive rational numbers
 Counting (natural) numbers
 Decimals
 Fractions
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
 Types of polygons
 Triangle
 Quadrilateral
 Types of quadrilaterals
 Trapezoid
 Types of trapezoids
 Isosceles trapezoid
 Right trapezoid
 Parallelogram
 Types of parallelograms
 Rectangle
 Rhombus
 Square (a special type of rectangle and a special type of rhombus)
 Area – the measurement attribute that describes the number of square units a figure or region covers
 Area is a twodimensional square unit measure
 Positive rational number side lengths
 Formulas for area from STAAR Grade 6 Mathematics Reference Materials
 Rectangle or parallelogram
 A = bh, where b represents the length of the base of the rectangle or parallelogram and h represents the height of the rectangle or parallelogram
 Trapezoid
 A = (b_{1} + b_{2})h, where b_{1} represents the length of one of the parallel bases, b_{2} represents the length of the other parallel base, and h represents the height of the trapezoid
 Triangle
 A = bh, where b represents the length of the base of the triangle and h represents the height of the triangle
 Threedimensional figure – a figure that has measurements including length, width (depth), and height
 Prism – a threedimensional figure containing two congruent and parallel faces that are polygons
 Right rectangular prisms and cubes
 Volume – the measurement attribute of the amount of space occupied by matter
 One way to measure volume is a threedimensional cubic measure
 Positive rational number side lengths
 Formulas for volume from STAAR Grade 6 Mathematics Reference Materials
 Rectangular prism
 V = Bh, where B represents the base area and h represents the height of the prism, which is the number of times the base area is repeated or layered
 The base of a rectangular prism is a rectangle whose area may be found with the formula, A = bh or A = lw, meaning the base area, B, may be found with the formula B = bh or B = lw; therefore, the volume of a rectangular prism may be found using V = Bh or V =(bh)h or V = (lw)h.
 Problem situations could involve using a ruler to determine side lengths when solving problem situations.
Note(s):
 Grade Level(s):
 Grade 5 represented and solved problems related to perimeter and/or area and related to volume.
 Grade 7 will solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids.
 Grade 7 will determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles.
 Grade 7 will solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of contexts
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.1. Recognize characteristics and dimensional changes of two and threedimensional figures.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

6.10 
Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to solve problems. The student is expected to:


6.10A 
Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts.
Readiness Standard

Model, Solve
ONEVARIABLE, ONESTEP EQUATIONS THAT REPRESENT PROBLEMS, INCLUDING GEOMETRIC CONCEPTS
Including, but not limited to:
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Variable – a letter or symbol that represents a number
 One variable on one side of the equation
 Coefficient – a number that is multiplied by a variable(s)
 Integers
 Products of integers limited to an integer multiplied by an integer
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 Constant – a fixed value that does not appear with a variable(s)
 Integers
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 Onestep equations
 A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
 Solution set – a set of all values of the variable(s) that satisfy the equation
 Constraints or conditions
 Distinguishing between equations and inequalities
 Characteristics of equations
 Equates two expressions
 Equality of the variable
 One solution
 Equality words and symbols
 Relationship of order of operations within an equation
 Order of operations – the rules of which calculations are performed first when simplifying an expression
 Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
 Exponents: rewrite in standard numerical form and simplify from left to right
 Limited to positive whole number exponents
 Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
 Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
 Models and solutions to onevariable, onestep equations from problem situations (concrete, pictorial, algebraic)
 Solutions to onevariable, onestep equations from geometric concepts
 Sum of the angles in a triangle, complementary angles, supplementary angles, sum of angles in a quadrilateral, etc.
 Supplementary angles – two angles whose degree measures have a sum of 180°
 Complementary angles – two angles whose degree measures have a sum of 90°
Note(s):
 Grade Level(s):
 Grade 4 introduced geometric concepts such as geometric attributes, parallel and perpendicular lines, and angle measures including complementary and supplementary angles.
 Grade 5 represented and solved multistep problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
 Grade 7 will model and solve onevariable, twostep equations and inequalities.
 Grade 7 will write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of contexts
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.1. Describe and interpret solution sets of equalities and inequalities.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
