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Instructional Focus Document
Grade 6 Mathematics
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.

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.

Additional Notes
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, II. Algebraic Reasoning, III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands, IV. Measurement Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X. Connections.

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 & 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


  • 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 1 half.png 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 A over B.png, 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 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 6 Mathematics TEKS

 

Texas Instruments – Graphing Calculator Tutorials


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.
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:
    • X. Connections
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:
    • VIII. Problem Solving and Reasoning
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:
    • VIII. Problem Solving and Reasoning
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:
    • IX. Communication and Representation
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:
    • IX. Communication and Representation
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:
    • X. Connections
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:
    • IX. Communication and Representation
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.
Readiness Standard

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 Level(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. Numeric Reasoning
    • IV. Measurement Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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 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 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:
    • I. Numeric Reasoning
    • III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands
    • IV. Measurement Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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
        • Quadrilateral
          • 4 sides
          • 4 vertices
            • Types of quadrilaterals
              • 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 Level(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:
    • III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands
    • IV. Measurement Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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
        • 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 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 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:
    • III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands
    • IV. Measurement Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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
  • 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
        • 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 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 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. Numeric Reasoning
    • III.C. Geometric Reasoning – Connections between geometry and other mathematical content strands
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
Click here to collapse or expand this section.
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/01/2018
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