Hello, Guest!
 TITLE : Unit 12: Essential Understandings of Geometry SUGGESTED DURATION : 12 days

#### Unit Overview

Introduction
This unit bundles student expectations that address angle relationships and measurements of circles, composite figures, rectangular and triangular prisms, and pyramids. 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 6, students determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, triangles, and the volume of right rectangular prisms where the dimensions were positive rational numbers.

During this Unit
Students revisit and solidify essential understandings of geometry. Students use the formulas for circumference and area of a circle to solve problems. Students extend previous knowledge of the area of rectangles, parallelograms, trapezoids, and triangles along with the area of circles in determining the area of composite figures consisting of rectangles, triangles, parallelograms, squares, quarter circles, semicircles, and trapezoids. Students also solve problems involving the volume of rectangular and triangular prisms and pyramids. Students extend their algebraic understandings of writing equations to represent geometric concepts, including the sum of the angles in a triangle, and other angle relationships. Students describe angle relationships such as adjacent angles, vertical angles, complementary angles, supplementary angles, and straight angles.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 7

After this Unit
In Grade 8, students students will use informal arguments to establish facts about the sum of angles and measures of exterior angles of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Additionally, in Grade 8, students will extend their knowledge of circles to finding the volume and surface area of cylinders, cones, and spheres. Students will describe the volume formula of a cylinder, V = Bh, in terms of its base area and its height and will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights as well as connect that relationship to their respective formulas. Students will solve problems involving the volume of cylinders, cones, and spheres and will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area to determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders.

In Grade 7, solving problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids, determining the circumference or circles, area of circles, and the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles are identified as STAAR Readiness Standard 7.9A, 7.9B and 7.9C. Writing and solving equations using geometry concepts is identified as STAAR Supporting Standard 7.11C. These four standards are listed under the Grade 7 STAAR Reporting Category: Geometry and Measurement. These standards are part of the Grade 7 Focal Point: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning C1, D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Curriculum and Evaluation Standards for School Mathematics (1989) by the National Council of Teachers of Mathematics (NCTM), “Geometric models provide a perspective from which students can analyze and solve problems, and geometric interpretations can help make an abstract (symbolic) representation more easily understood. Many real-world objects can be viewed geometrically. For example, the use of area models provides the interpretation for much of the arithmetic of decimals, fractions, ratios, proportions, and percents. At the middle school level, geometry should focus on investigating and using geometric ideas and relationships rather than on memorizing definitions and formulas” (p. 112). As students develop concepts of length and area, it should be noted that “Students need to become fluent in the use of procedures or formulas to solve problems; however, they also need to learn those skills with understanding rather than just through memorization. Algorithms and formulas have the potential to simplify calculation and clarify topics, but without understanding they can become an impediment to further learning” (NCTM, 2009, p. 7). In regards to the algorithm for calculating volume, “students who understand where formulas come from, do not seem as mysterious, tend to remember them, and reinforce the idea that mathematics makes sense. Rote use of formulas from a book offers none of these advantages” (Van de Walle, Karp, & Bay-Williams, 2010, p. 391).

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in Grades 6 – 8: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
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)
• 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 a parallelogram that is created by rearranging equal-sized pieces of a circle be used to find the …
• area
• circumference
… of the circle?
• How is the formula for the area of a parallelogram related to the formula for the area of a circle?
• What is the process to determine the circumference of a circle if the …
• diameter is known?
• What is the process to determine the …
• diameter
… if the circumference is known?
• How can geometric, spatial, and measurement reasoning affect how one visualizes composite figures?
• How can composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles be decomposed and/or composed to simplify the measurement process?
• What is the process to determine the area of the composite figure, and how can the area be represented with an equation and/or formula?
• Expressions, Equations, & Relationships
• Composition and Decomposition of Figures
• Geometric Representations
• Two-dimensional figures
• Composite figures
• Geometric Relationships
• Formulas
• Circumference
• Area
• Measure relationships
• Geometric measures
• Operations
• Properties of operations
• Order of operations
• 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.

 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)
• 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.
• What is the process for determining the volume of a …
• rectangular prism?
• rectangular pyramid?
• triangular prism?
• triangular pyramid?
• How can the height of a …
• rectangular prism
• rectangular pyramid
• triangular prism
• triangular pyramid
… be determined when given the area of the base and its volume?
• How can the area of the base of a …
• rectangular prism
• rectangular pyramid
• triangular prism
• triangular pyramid
… be determined when given the height and its volume?
• What relationship exists between the volume of a rectangular prism and the volume of a rectangular pyramid?
• What relationship exists between the volume of a triangular prism and the volume of a triangular pyramid?
• How can problem situations involving …
• complementary angles
• supplementary angles
• the sum of the angles in a triangle
• the Triangle Inequality Theorem
… be represented and solved using an equation or inequality?
• 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°?
• What relationship exists among vertical angles?
• What relationship exists among supplementary angles and straight angles?
• Why is the sum of two side lengths of a triangle always greater than the third side length?
• What model(s) can be used to represent …
• vertical angles
• complementary angles
• supplementary angles
• straight angles
• the sum of the angles in a triangle
• the Triangle Inequality Theorem
…, and how can the model lead to a generalization that can be represented with an equation and/or inequality?
• Expressions, Equations, & Relationships
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Geometric Relationships
• Formulas
• Area
• Volume
• Measure relationships
• Geometric properties
• Numeric and Algebraic Representations
• Expressions
• Equations
• Inequalities
• Equivalence
• Operations
• Properties of operations
• Order of operations
• 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 think the area of a composite figure changes rather than remaining the same when the figure is disassembled.
• Some students may confuse the formulas for the circumference and area of a circle.
• Some students may multiply the radius by 2 rather than squaring it when determining the area of a circle.
• Some students may think that “B” is synonymous with “b”, the length of the base, instead of “B”, which represents the area of the base of a three-dimensional figure.
• Some students may think that a constant term can be combined with a variable term (e.g., 2x + 5 = 7x).

#### Unit Vocabulary

• Adjacent angles – two non-overlapping angles that share a common vertex and exactly one ray
• Area – the measurement attribute that describes the number of square units a figure or region covers
• Circle – a two-dimensional figure formed by a closed curve with all points equal distance from the center
• Circumference – a linear measurement of the distance around a circle
• Coefficient – a number that is multiplied by a variable(s)
• Complementary angles – two angles whose degree measures have a sum of 90°
• Composite figure – a figure that is composed of two or more two-dimensional figures
• Congruent angles – angles whose angle measurements are equal
• Constant – a fixed value that does not appear with a variable(s)
• Diameter – a line segment whose endpoints are on the circle and passes through the center of the circle
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Pi (π) – the ratio of the circumference to the diameter of a circle
• 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
• Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex
• Quarter circle – one-fourth of a circle
• Radius – a line segment drawn from the center of a circle to any point on the circle and is half the length of diameter of the circle
• Semicircle – half of a circle
• Solution set – a set of all values of the variable(s) that satisfy the equation
• Straight angle – an angle with rays extending in opposite directions and whose degree measure is 180°
• 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
• Two-dimensional figure – a figure with two basic units of measure, usually length and width
• Variable – a letter or symbol that represents a number
• Vertical angles – a pair of non-adjacent, non-overlapping angles formed by two intersecting lines creating angles that are opposite and congruent to each other
• Volume – the measurement attribute of the amount of space occupied by matter

Related Vocabulary:

 Angle measure Base of a rectangular pyramid Base of a triangular pyramid Bases of a rectangular prism Bases of a triangular prism Edge Equal to Face Height of a rectangular prism Height of a rectangular pyramid Height of a triangular prism Height of a triangular pyramid Length Line segment Parallel Parallelogram Polygon Quadrilateral Perpendicular Rectangle Rectangular prism Rectangular pyramid Solution Solve Square Trapezoid Triangle Triangular prism Triangular pyramid Vertex Width
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 Center if your district has granted access to that tool.

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

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 7 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

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

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
7.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
7.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
7.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
7.1C

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

Process Standard

Select

TOOLS, INCLUDING PAPER AND PENCIL AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
7.1D

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, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
7.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
7.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
7.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII. A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII. C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
7.9 Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems. The student is expected to:
7.9A Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids.

Solve

PROBLEMS INVOLVING THE VOLUME OF RECTANGULAR PRISMS, TRIANGULAR PRISMS, RECTANGULAR PYRAMIDS, AND TRIANGULAR PYRAMIDS

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
• 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
• Rectangular prism, including a cube or square prism
• Triangular prism
• Pyramid – a three-dimensional figure containing a base that is a polygon and the faces are triangles that share a common vertex, also known as an apex
• Rectangular pyramid, including a square pyramid
• Triangular pyramid
• 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
• Recognition of volume embedded in mathematical and real-world problem situations
• Formulas for volume from STAAR Grade 7 Mathematics Reference Materials
• 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)
• Rectangular prism
• 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.
• Triangular prism
• The base of a triangular prism is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular prism may be found using V = Bh or V = .
• Pyramid
• V = Bh, where B represents the base area and h represents the height of the pyramid
• Rectangular pyramid
• The base of a rectangular pyramid 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 pyramid may be found using V = Bh or V = (bh)h or V = (lw)h.
• Triangular pyramid
• The base of a triangular pyramid is a triangle whose area may be found with the formula, A = bh, meaning the base area, B, may be found using B = bh; therefore, the volume of a triangular pyramid may be found using V = Bh or V = or V = .

Note(s):

• Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.
• Grade 8 will solve problems involving the volume of cylinders, cones, and spheres.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.2. Determine the surface area and volume of three-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
7.9B Determine the circumference and area of circles.

Determine

THE CIRCUMFERENCE AND AREA OF CIRCLES

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
• Circle – a two-dimensional figure formed by a closed curve with all points equal distance from the center
• Diameter – a line segment whose endpoints are on the circle and passes through the center of the circle
• Radius – a line segment drawn from the center of a circle to any point on the circle and is half the length of the diameter of the circle
• Semicircle – half of a circle
• Quarter circle – one-fourth of a circle
• Circumference – a linear measurement of the distance around a circle
• Pi (π) – the ratio of the circumference to the diameter of a circle
• π ≈ 3.14 or
• Formulas for circumference from STAAR Grade 7 Mathematics Reference Materials
• Circumference using the radius of a circle
• C = 2πr, where C represents the circumference of the circle and r represents the radius of the circle, and π is approximately 3.14 or
• Circumference using the diameter of a circle
• C = πd, where C represents the circumference of the circle, d represents the diameter of the circle, and π is approximately 3.14 or
• 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 dimensions
• Formula for area of a circle from STAAR Grade 7 Mathematics Reference Materials
• Area
• Aπr2, where A represents the area of the circle, r represents the radius of the circle, and π is approximately 3.14 or

Note(s):

• Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.
• Grade 8 will solve problems involving the volume of cylinders, cones, and spheres.
• Grade 8 will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.1. Find the perimeter and area of two-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
7.9C Determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles.

Determine

THE AREA OF COMPOSITE FIGURES CONTAINING COMBINATIONS OF RECTANGLES, SQUARES, PARALLELOGRAMS, TRAPEZOIDS, TRIANGLES, SEMICIRCLES, AND QUARTER CIRCLES

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
• Composite figure – a figure that is composed of two or more two-dimensional figures
• Rectangles
• Squares
• Parallelograms
• Trapezoids
• Triangles
• Circles
• Semicircles
• Quarter circles
• Any combination of these figures
• 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 7 Mathematics Reference Materials
• Triangle
• A = bh, where b represents the length of the base of the triangle and h represents the height of the triangle
• 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
• Circle
• A = &pir2, where A represents the area of the circle, r represents the radius of the circle, and π is approximately 3.14 or
• Problem situations could involve using a ruler to determine side lengths when solving problem situations.

Note(s):

• Grade 6 determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.1. Find the perimeter and area of two-dimensional figures.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
7.11 Expressions, equations, and relationships. The student applies mathematical process standards to solve one-variable equations and inequalities. The student is expected to:
7.11C Write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
Supporting Standard

Write, Solve

EQUATIONS USING GEOMETRY CONCEPTS, INCLUDING THE SUM OF THE ANGLES IN A TRIANGLE, AND ANGLE RELATIONSHIPS

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
• Decimals
• Fractions
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Solution set – a set of all values of the variable(s) that satisfy the equation
• Equations from geometry concepts
• Angle measures as numeric and/or algebraic expressions
• Sum of the angles in a triangle
• Other angle relationships
• Adjacent angles – two non-overlapping angles that share a common vertex and exactly one ray
• Complementary angles – two angles whose degree measures have a sum of 90°
• Supplementary angles – two angles whose degree measures have a sum of 180°
• Straight angle – an angle with rays extending in opposite directions and whose degree measure is 180°
• Congruent angles – angles whose angle measurements are equal
• Arc(s) on angles are usually used to indicate congruency (one set of congruent angles would have 1 arc, another set of congruent angles would have 2 arcs, etc.).
• Arcs and tick marks on angles can be used to indicate congruency (one set of congruent angles would have 1 arc with 1 tick mark, another set of congruent angles would have 1 arc with 2 tick marks, etc.)
• Vertical angles – a pair of non-adjacent, non-overlapping angles formed by two intersecting lines creating angles that are opposite and congruent to each other
• Real-life situation involving angle measures

Note(s):

• Grade 4 determined the measure of an unknown angle formed by two non-overlapping adjacent angles given one or both angle measures.
• Grade 6 extended 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 6 modeled and solved one-variable, one-step equations and inequalities that represent problems, including geometric concepts.
• Grade 8 will use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.