 Hello, Guest!
 Instructional Focus DocumentGrade 6 Mathematics
 TITLE : Unit 13: Essential Understanding of Equations SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address modeling, writing, and solving equations as well as solving problems involving area and volume. 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 Unit 06, students modeled and solved one-variable, one-step equations that represent problems, including geometric concepts. In Unit 08, students used concrete models, pictorial models, and algebraic representations to model and solve one-variable, one-step equations that represent problems, including geometric concepts. In Unit 09, students extended their knowledge of triangles and their properties to include the sum of the angles of the triangle and the relationship between angle measures and the three side lengths of the triangle. Students wrote equations and determined solutions to problems involving the area of rectangles, parallelograms, trapezoids, and triangles, as well as the volume of right rectangular prisms.

During this Unit
Students revisit and solidify essential understandings of equations. Students represent two-variable algebraic relationships, including additive and multiplicative relationships, in the form of verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b, and model and solve one-variable, one-step equations that represent problems, including geometric concepts. Although certain models, such as algebra tiles, may limit a student’s ability to model solving equations with whole number or integer constants or coefficients, students should also solve equations with positive rational number constants or coefficients with an algebraic model. Students apply their knowledge of triangles and their properties to include the sum of the angles of the triangle and how those angle measurements are related to the three side lengths of the triangle. Students write equations and determine solutions to problems related to 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. Students extend concepts of equations to determining the volume of right rectangular prisms.

After this Unit
In Grade 7, students will model, write, solve, and represent solutions for one-variable, two-step equations and inequalities. In Algebra I, students work with exponents will be extended as students add, subtract, and multiply polynomials of degree one and degree two. Students will also rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property. Also 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.

Research
According to Van de Walle, Karp, and Bay-Williams (2010), “students must understand the connections between context, tables, graphs, equations, and verbal descriptions; it is not enough just to teach each one separately (Hackbarth & Wilsman, 2008)” (p. 252). Additionally, in Principles and Standards for School Mathematics (2000), the National Council of Teachers of Mathematics, states that “Students should also become flexible in recognizing equivalent forms of linear equations and expressions, This flexibility can emerge as students gain experience with multiple ways of representing a contextualized problem” (p. 282).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. 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
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.

 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)
• 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.
• What are the similarities and differences between representing a problem situation using a table, graph, or equation in the form y = kx or y = x + b?
• 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?
• How can a(n) …
• concrete model
• pictorial model
• algebraic representation
... be used to represent and solve an equation?
• What models effectively and efficiently represent how to solve equations?
• What is the process for solving an equation, and how can the process be …
• described verbally?
• represented algebraically?
• When considering equations, …
• why is the variable isolated in order to solve?
• how are negative values represented in concrete and pictorial models?
• why must the solution be justified in terms of the problem situation?
• why does equivalence play an important role in the solving process?
• Why is it important to understand when and how to use standard algorithms?
• How does knowing more than one solution strategy build mathematical flexibility?
• 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 problem situations involving …
• complementary angles
• supplementary angles
• the sum of the angles in a triangle
• the sum of the angles in a quadrilateral
… be represented and solved using an equation?
• What model(s) can be used to represent …
• complementary angles
• supplementary angles
• the sum of the angles in a triangle
• the sum of the angles in a quadrilateral
…, and how can the model lead to a generalization that can be represented with an equation?
• When angles are complementary, why does the sum always equal 90°?
• When angles are supplementary, why does the sum always equal 180°?
• When finding the sum of the angles in a triangle, why does the sum always equal 180°?
• When finding the sum of the angles in a quadrilateral, why does the sum always equal 360°?
• Expressions, Equations, and Relationships
• Geometric Representations
• Two-dimensional figures
• Geometric Attributes/Properties
• Formulas
• Area
• Volume
• Geometric properties
• Measure relationships
• 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

Underdeveloped Concepts:

• Some students may confuse the independent and dependent quantities in tables and graphs.
• Students may write an equation that expresses x in terms of y instead of y in terms of x.
• Some students may graph an ordered pair incorrectly by going up or down first and then moving to the left or right.
• Some students may think they can use either number in an ordered pair to graph a point on coordinate plane rather than always associating the first number in an ordered pair to the x-axis and the second number to the y-axis.
• Some students may think the equals sign means “solve this” or “the answer is” rather than understanding that equal sign represents a quantitative and balanced relationship.
• 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.

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
• Coefficient – a number that is multiplied by a variable(s)
• Complementary angles – two angles whose degree measures have a sum of 90°
• Congruent – of equal measure, having exactly the same size and same shape
• Constant – a fixed value that does not appear with a variable(s)
• Edge – where the sides of two faces meet on a three-dimensional figure
• Equiangular – all angles in a polygon are congruent in measure
• Equilateral – all side lengths of a polygon are congruent in measure
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• 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
• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Obtuse angle – an angle that measures greater than 90° but less than 180°
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• 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
• Solution set – a set of all values of the variable(s) that satisfy the equation or inequality
• Supplementary angles – two angles whose degree measures have a sum of 180°
• Three-dimensional figure – a figure that has measurements including length, width (depth), and height
• Triangle – a polygon with three sides and three vertices
• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Variable – a letter or symbol that represents a number
• 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:

 Cubic unit Customary unit Degree Dependent Dimension Equivalent Evaluate Graph Height Horizontal Independent Interior angle Input Length Metric unit Ordered pair Output Quadrant I Quadrant II Quadrant III Quadrant IV Scale Scale factor Simplify Solution Solve Square unit Unit Vertical x-axis x coordinate x-value y-axis y coordinate y-value
Unit Assessment Items System Resources Other Resources

Show this message:

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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
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 PAPER AND PENCIL AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• 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
6.6 Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to:
6.6C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.

Represent

A GIVEN SITUATION USING VERBAL DESCRIPTIONS, TABLES, GRAPHS, AND EQUATIONS IN THE FORM y = kx OR y = x + b

Including, but not limited to:

• Independent quantities are represented by the x coordinates or the input.
• Dependent quantities are represented by the y coordinates or the output.
• Various representations to describe algebraic relationships
• Verbal descriptions
• Tables
• Graphs
• Equations
• In the form y = kx, where k is the non-zero scale factor (constant of proportionality), from multiplicative problem situations
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• In the form y = x + b, where b is the constant non-zero addend, from additive problem situations
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values

Note(s):

• Grade 6 introduces representing a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
• Grade 7 will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
• Grade 8 will represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
• Grade 8 will represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
• Grade 8 will write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• I. Numeric Reasoning
• II. Algebraic 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 AND THE RELATIONSHIP BETWEEN THE LENGTHS OF SIDES AND MEASURES OF ANGLES IN A TRIANGLE

Including, but not limited to:

• Properties of triangles – relationship of attributes within a triangle (e.g., an equilateral triangle has all sides and angles congruent, the sum of the lengths of two sides of a triangle is always greater than the length of the third side, the sum of the angles of a triangle is always 180°; etc.)
• Angle – two rays with a common endpoint (the vertex)
• Various angle types/names
• Right angle, 90°, used as a benchmark to identify and name angles
• Acute angle – an angle that measures less than 90°
• Right angle – an angle (formed by perpendicular lines) that measures exactly 90°
• Notation is given as a box in the angle corner to represent a 90° angle.
• Obtuse angle – an angle that measures greater than 90° but less than 180°
• Congruent – of equal measure, having exactly the same size and same shape
• Angle congruency marks – angle marks indicating angles of the same measure
• Side congruency marks – side marks indicating side lengths of the same measure
• Equilateral – all side lengths of a polygon are congruent in measure
• Equiangular – all angles in a polygon are congruent in measure
• Triangle – a polygon with three sides and three vertices
• 3 sides
• 3 vertices
• Classification by angles
• Triangles are named based on their largest angle.
• Acute triangle
• 3 sides
• 3 vertices
• 3 acute angles (less than 90°)
• Right triangle
• 3 sides
• 3 vertices
• 2 acute angles (less than 90°)
• 1 right angle (exactly 90°)
• Obtuse triangle
• 3 sides
• 3 vertices
• 2 acute angles (less than 90°)
• 1 obtuse angle (greater than 90° but less than 180°)
• Classification by length of sides
• Scalene triangle
• 3 sides
• 3 vertices
• No congruent sides
• No parallel sides
• Up to one possible pair of perpendicular sides
• Right triangle with two sides that are perpendicular to form a right angle and three different side lengths
• No congruent angles
• Right triangle with one 90° angle and two other angles each of different measures
• Obtuse triangle with one angle greater than 90° and two other angles each of different measures
• Acute triangle with all angles less than 90° and all angles of different measures
• Isosceles triangle
• 3 sides
• 3 vertices
• At least 2 congruent sides
• No parallel sides
• Up to one possible pair of perpendicular sides
• Right triangle with two sides that are perpendicular to form a right angle and are each of the same length
• Obtuse triangle with one angle greater than 90° and two other angles each of different measures
• At least 2 congruent angles
• Right triangle with one 90° angle and two other angles each of the same measure
• Obtuse triangle with two angles of the same measure and one angle greater than 90°
• Acute triangle with all angles measuring less than 90° and at least two of the angles of the same measure
• Equilateral triangle/Equiangular triangle (a special type of isosceles triangle)
• 3 sides
• 3 vertices
• All sides congruent
• No parallel or perpendicular sides
• All angles congruent
• Acute triangle with all angles measuring 60°
• Sum of the interior angles of a triangle is 180°
• Equations to determine a missing angle measure
• Relationship between lengths of sides and measure of angles in a triangle
• The shortest side length in a triangle is always opposite the smallest angle measure in a triangle.
• The longest side length in a triangle is always opposite the largest angle measure in a triangle.
• The sides opposite from angles of equal measure in a triangle are always congruent.

Note(s):

• Grade 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
6.8C 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
• 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 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.

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

Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.

Model, Solve

ONE-VARIABLE, ONE-STEP EQUATIONS THAT REPRESENT PROBLEMS, INCLUDING GEOMETRIC CONCEPTS

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Variable – a letter or symbol that represents a number
• One variable on one side of the equation
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• One-step equations
• A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the equation
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Equality words and symbols
• Equal to, =
• Relationship of order of operations within an equation
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Limited to positive whole number exponents
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Models and solutions to one-variable, one-step equations from problem situations (concrete, pictorial, algebraic)
• Solutions to one-variable, one-step equations from geometric concepts
• Sum of the angles in a triangle, complementary angles, supplementary angles, sum of angles in a quadrilateral, etc.
• Supplementary angles – two angles whose degree measures have a sum of 180°
• Complementary angles – two angles whose degree measures have a sum of 90°

Note(s):

• Grade 4 introduced geometric concepts such as geometric attributes, parallel and perpendicular lines, and angle measures including complementary and supplementary angles.
• Grade 5 represented and solved multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
• Grade 7 will model and solve one-variable, two-step equations and inequalities.
• Grade 7 will write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• 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 