Hello, Guest!
 TITLE : Unit 09: Geometry and Measurement SUGGESTED DURATION : 13 days

#### Unit Overview

Introduction
This unit bundles student expectations that address converting units of measure as well as modeling, writing, and solving equations to solve problems involving the area of triangles, rectangles, parallelograms, and trapezoids, and the volume of rectangular prisms. 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.”

Prior to this Unit
In Grade 3, students determined the area of rectangles by examining the number of rows times the number of unit squares in each row. They solved perimeter situations determining the perimeter of a polygon or a missing length. In Grade 4, students used models to determine the formula for the perimeter of a rectangle. They also represented and solved problems related to perimeter and area of rectangles. In Grade 5, students used concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism including the special forms for a cube. They represented and solved problems related to perimeter, area, and volume. In Grade 6, Unit 5, students were introduced to converting units within the same measurement system in relation to proportions and unit rates.

During this Unit
Students extend their knowledge of triangles and their properties to include the sum of the angles of a triangle, and how those angle measurements are related to the three side lengths of the triangle. Students examine and analyze the relationship between the three side lengths of a triangle and determine whether three side lengths will form a triangle using the Triangle Inequality Theorem. Students also decompose and rearrange parts of parallelograms (including rectangles), trapezoids, and triangles in order to model area formulas for each of the figures. Students write equations for and determine solutions to problems related to the area of rectangles, parallelograms, trapezoids, and triangles. Problems include situations where the equation represents the whole area of the shape or partial area of the shape. Writing equations and determining solutions is extended to the volume of right rectangular prisms. Positive rational numbers should be used in problem situations for this unit. Students expand previous knowledge of converting units within the same measurement system when determining solutions to problems involving length. Conversion processes for measurement extend beyond the use of proportions to now include dimensional analysis and conversions graphs.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 6

After this Unit
In Grade 7, students convert between measurement systems, including the use of proportions and the use of unit rates. Students will write and solve equations using geometry concepts, including the sum of the angles in a triangle and angle relationships. They will model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights, connect the relationship between a rectangular prism and a rectangular pyramid to the formulas, and solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Students will also extend concepts of length to include the circumference and area of circles.

In Grade 6, converting units within a measurement system, including the use of proportions and unit rates is STAAR Readiness Standard 6.4H and part of the Grade 6 Texas Response to Curriculum Focal Points (TxRCFP): Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships. 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 when three lengths form a triangle is STAAR Supporting Standard 6.8A and subsumed within the Grade 6 Focal Point: Grade Level Connections (TxRCFP).  Modeling area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes and writing 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 are identified as STAAR Supporting Standards 6.8B and 6.8C. Determining solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers is STAAR Readiness Standard 6.8D. These three standards are part of the Grade 6 Focal Point: Using expressions and equations to represent relationships in a variety of contexts (TxRCFP). All of the standards within this unit are part of the Grade 6 STAAR Reporting Category Geometry and Measurement. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1, C2; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning A1, A2, C1, D1, D3; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Principles and Standards for School Mathematics (2000) by the National Council of Teachers of Mathematics (NCTM), “Geometry not only provides a means for describing, analyzing, and understanding structures in the world around us but also introduces an experience of mathematics that complements and supports the study of other aspects of mathematics such as number and measurement” (2000, p. 2). NCTM (2005) also states, “whatever the context, measurement is indispensable to the study of number, geometry, statistics, and other branches of mathematics. It is the essential link between mathematics and science, art, social studies, and other disciplines, and it is pervasive in daily activities, from buying bananas or new carpet to charting heights of growing children…” (p. 1).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2005). Navigating through measurement in grades 6 – 8. Reston, VA: National Council of Teachers of Mathematics, Inc.
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

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities can vary together and be reasoned up and down in situations involving invariant (constant) relationships builds flexible proportional reasoning.
• Ratios and rates are modeled and described to develop an understanding of proportional relationships and these relationships are applied to represent equivalence and solve problem situations involving ratios and rates.
• How are …
• scale factors
• proportions
• unit rates
• conversion graphs
… used to convert within a measurement system?
• Equations can be modeled, written, and solved using various methods to gain insight into the context of the situation and make critical judgments about algebraic relationships and efficient strategies.
• How can an equation be used to represent the sum of the angles of a triangle?
• How can an equation be used to solve for a missing angle in a triangle?
• What is the process to determine if three side lengths can form a triangle?
• What generalization can be made about the lengths of the sides to form a triangle?
• How are constraints or conditions within a problem situation represented in an equation or a formula?
• What is the process for writing an equation and determining the solution for the area of a …
• rectangle?
• parallelogram?
• trapezoid?
• triangle?
• What is the process for writing an equation and determining the solution for the volume of a right rectangular prism?
• How can the height of a right rectangular prism be determined when given the area of the base and its volume?
• How can the area of the base of a right rectangular prism be determined when given the height and its volume?
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing attributes of geometric figures with quantifiable measures and equations in order to generalize geometric relationships and solve problems.
• How can decomposition and composition of figures simplify the measurement process?
• How can decomposition and composition of a …
• parallelogram
• triangle
• trapezoid
… be used to model the area formula of the figure?
• How can the formula for the area of a rectangle be used to determine the formula for the area of a …
• parallelogram?
• triangle?
• trapezoid?
• Proportionality
• Ratios and Rates
• Unit rates
• Scale factors
• Relationships and Generalizations
• Proportions
• Systems of Measurement
• Customary
• Metric
• Representations
• Solution Strategies
• Expressions, Equations, and Relationships
• Composition and Decomposition of Figures and Angles
• Geometric Representations
• Two-dimensional figures
• Geometric Relationships
• Formulas
• Area
• Volume
• Measure relationships
• Geometric properties
• Operations
• Properties of operations
• Order of operations
• Numeric and Algebraic Representations
• Expressions
• Equations
• Equivalence
• Representations
• 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 multiply only the base and height to find the area of a triangle and forget to multiply by or divide by 2.
• Some students may multiply by a side length that they believe represents the height of a trapezoid, triangle, or parallelogram rather than using the actual height of the figure.
• Some students may not realize that a parallelogram can always be formed from two congruent trapezoids or two congruent triangles.

Underdeveloped Concepts:

• Some students may not realize that all rectangles are parallelograms.

#### Unit Vocabulary

• Acute angle – an angle that measures less than 90°
• Angle – two rays with a common endpoint (the vertex)
• Angle congruency marks – angle marks indicating angles of the same measure
• Area – the measurement attribute that describes the number of square units a figure or region covers
• Bases of a right rectangular prism – any two congruent, opposite, and parallel faces shaped like rectangles; exactly 3 possible sets
• Congruent – of equal measure, having exactly the same size and same shape
• Edge – where the sides of two faces meet on a three-dimensional figure
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Equiangular – all angles in a polygon are congruent in measure
• Equilateral – all side lengths of a polygon are congruent in measure
• Face – a flat surface of a three-dimensional figure
• Height of a right rectangular prism – the length of a side that is perpendicular to both bases
• Obtuse angle – an angle that measures greater than 90° but less than 180°
• Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Prism – a three-dimensional figure containing two congruent and parallel faces that are polygons
• 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.)
• Right angle – an angle (formed by perpendicular lines) that measures exactly 90°
• Side congruency marks – side marks indicating side lengths of the same measure
• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Triangle – a polygon with three sides and three vertices
• Triangle Inequality Theorem – the sum of the lengths of any two sides in a triangle must be greater than the length of the third side of the triangle
• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Unit rate – a ratio between two different units where one of the terms is 1
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Volume – the measurement attribute of the amount of space occupied by matter

Related Vocabulary:

 Acute triangle Base Cubic unit Customary unit Degree Dimension Dimensional analysis Equilateral triangle Height Interior angle Isosceles triangle Length Metric unit Obtuse triangle Opposite Parallel Parallelogram Perpendicular Proportion Quadrilateral Ratio Rectangle Rectangular prism Rhombus Right triangle Scale factor Scalene triangle Side Square Square unit Trapezoid Unit Width
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 Center 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 6 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE 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.

Specificity 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.
TEKS# SE# TEKS SPECIFICITY
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 – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 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:
• 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 problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving 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 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:
• 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 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:
• 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.
6.4 Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:
6.4H Convert units within a measurement system, including the use of proportions and unit rates.

Convert

UNITS WITHIN A MEASUREMENT SYSTEM, INCLUDING THE USE OF PROPORTIONS AND UNIT RATES

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
• Unit conversions within systems
• Customary
• Metric
• Unit rate – a ratio between two different units where one of the terms is 1
• Multiple solution strategies
• Dimensional analysis using unit rates
• Scale factor between ratios
• Proportion method
• Conversion graph

Note(s):

• Grade 4 converted measurements within the same measurement system, customary or metric, from a smaller unit into a large unit or a large unit into a smaller unit when given other equivalent measures represented in a table.
• Grade 6 introduces using proportions and unit rates to convert units within a measurement system.
• Grade 7 will convert between measurement systems, including the use of proportions and the use of unit rates.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• I.C. Numeric Reasoning – Systems of measurement
• I.C.2. Convert units within and between systems of measurement.
• 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.
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, THE RELATIONSHIP BETWEEN THE LENGTHS OF SIDES AND MEASURES OF ANGLES IN A TRIANGLE, AND DETERMINING WHEN THREE LENGTHS FORM 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 mearsure
• 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 of different measures
• Acute triangle with all angles less than 90° and all angles of different measure
• 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
• 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.
• Triangle Inequality Theorem – the sum of the lengths of any two sides in a triangle must be greater than the length of the third side in the triangle

Note(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 non-overlapping 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 three-dimensional 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.
6.8B Model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes.
Supporting Standard

Model

AREA FORMULAS FOR PARALLELOGRAMS, TRAPEZOIDS, AND TRIANGLES BY DECOMPOSING AND REARRANGING PARTS OF THESE SHAPES

Including, but not limited to:

• Two-dimensional 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
• 4 sides
• 4 vertices
• Trapezoid
• Types of trapezoids
• Isosceles trapezoid
• 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 trapezoid
• 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 two-dimensional square unit measure
• 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 represents the height of the rectangle or parallelogram
• A parallelogram can be decomposed and rearranged to form a rectangle.
• Trapezoid
• A = (b1 + b2)h, where b1 represents the length of one of the parallel bases, b2 represents the length of the other parallel base, and represents the height
• Two congruent trapezoids can be arranged to form a parallelogram.
• A trapezoid can be decomposed and rearranged to form a parallelogram.
• A trapezoid can be decomposed to form two triangles.
• Triangle
• A = bh, where b represents the length of the base of the triangle and h represents the height of the triangle
• Two congruent triangles can be arranged to form a parallelogram.

Note(s):

• Grade 5 used concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism including the special forms for a cube (V = l × w × h, V = s × s × s, and V = Bh).
• Grade 7 will model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas.
• 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.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• 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.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
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
• Two-dimensional 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
• 4 sides
• 4 vertices
• 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 two-dimensional 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 = (b1 + b2)h, where b1 represents the length of one of the parallel bases, b2 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
• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Edge – where the sides of two faces meet on a three-dimensional figure
• Vertex (vertices) in a three-dimensional figure – the point (corner) where three or more edges of a three-dimensional figure meet
• Face – a flat surface of a three-dimensional 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 three-dimensional 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 three-dimensional 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 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.

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
• Two-dimensional 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
• 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 two-dimensional 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 = (b1 + b2)h, where b1 represents the length of one of the parallel bases, b2 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
• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Prism – a three-dimensional 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 three-dimensional 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 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 three-dimensional figures.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.1. Find the perimeter and area of two-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.