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 Instructional Focus DocumentGeometry
 TITLE : Unit 07: Relationships of Two- and Three-Dimensional Figures SUGGESTED DURATION : 13 days

#### Unit Overview

Introduction
This unit bundles student expectations that address exploring geometric relationships between sides, angles, and diagonals of various quadrilaterals to justify the type of quadrilateral; investigating patterns to make conjectures and generalizations about polygons; and identifying two-dimensional cross-sections of three-dimensional figures as well as identifying three-dimensional objects generated by rotations of two-dimensional shapes. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, 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.

Prior to this Unit
In Grade 3, students used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories. In Grade 5, students classified two-dimensional figures in a hierarchy of sets using graphic organizers based on their attributes and properties as well as investigated patterns to make conjectures about geometric relationships, including diagonals of quadrilaterals and interior and exterior angles of polygons. In Grade 7, students used patterns to analyze linear relationships. In Grade 8, students used patterns to analyze proportional and non-proportional linear relationships. In Algebra I Units 01 – 04, students studied linear functions and equations to represent data.

During this Unit
Students define types of quadrilaterals with a focus on identifying the characteristics (including sides, angles, and diagonal relationships) of parallelograms, rectangles, rhombi, and squares. Students use the characteristics of the quadrilateral to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares using coordinate geometry, two column proofs, paragraph proofs, and flow charts. Students apply quadrilateral relationships to solve real-world problems involving lengths of sides, angles, and midpoints. Students define and identify polygons, including regular polygons by number of sides. Students use tabular, graphical, and symbolic generalization to develop formulas for interior and exterior angles in terms of number of sides. Students extend and apply the properties of quadrilaterals and other polygons and interior and exterior angle theorems to determine lengths of sides, diagonals, midpoints, and all angle measures. Students identify shapes of two-dimensional cross sections of prisms, pyramids, cylinders, cones, and spheres. Students explore and identify three-dimensional objects generated by rotations of two-dimensional shapes.

After this Unit
In Units 08 – 09, students will use the concepts of polygons to investigate and solve problems involving properties and measurement of two- and three-dimensional figures. The concepts in this unit will be applied in subsequent mathematics courses.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1, C1, C2; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning A1, A2, C1, D3; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, C2, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to The National Council of Teachers of Mathematics (2000) this approach to studying quadrilaterals supports the need for all students to be able to analyze characteristics and properties of geometric shapes and develop mathematical arguments about geometric relationships. Students in grades 9 – 12 should specifically be able to investigate relationships among classes of two-dimensional geometric objects, make and test conjectures about them, and solve problems. At the conclusion of this unit, students are asked to create a graphic organizer to help classify quadrilaterals. The TxCCRS cites many skills related to the communication and representation of mathematical ideas. According to the National Council of Teachers of Mathematics (NCTM) in Focus in High School Mathematics: Reasoning and Sense Making (2009), the key elements of reasoning and sense making with geometry must include multiple representations of functions. In this unit, students gather data from geometric figures and organize this information into tables, graphs or diagrams. This leads to the development of symbolic expressions and verbal descriptions. A variety of representations helps make relationships more understandable to more students than working with symbolic representations alone. These approaches serve as the basis for this unit on polygons and circle. TxCCRS cites many skills related to the communication and representation of mathematical ideas. According to the National Council of Teachers of Mathematics (2012), using diagrams and constructions to interpret and communicate geometric relationships is essential in geometry. Using definitions of figures to characterize figures in terms of their properties is another essential in geometry. In geometry, the “proving process involves working with diagrams, variation and invariance, conjectures, and definitions.” (p. 92)

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. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2012). Developing essential understanding of Geometry for Teaching Mathematics in Grades 9 – 12. 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

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within quadrilaterals?
• Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
• How is deductive reasoning used to understand, prove, and apply geometric conjectures about quadrilateral relationships?
• What structures and processes can be used to prove geometric relationships involving quadrilaterals?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent quadrilateral relationships?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate quadrilateral relationships?
• What tools and processes can be used to investigate …
• angle relationships
• side relationships
• diagonal relationships
• parallelism and perpendicularity
• relationships among various types
• How can the coordinate plane be used to make and validate conjectures about quadrilaterals?
• What conjectures can be made and validated by exploring the patterns and properties of quadrilateral relationships?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding quadrilateral relationships be applied when solving problem situations?
• How can measurable attributes related to …
• side lengths
• diagonals
• angle measures
… be distinguished and described in order to generalize geometric relationships of quadrilaterals?
• What processes can be used to determine the …
• length of sides and diagonals
• slope of sides and diagonals
• midpoint of sides and diagonals
• measures of angles
• Coordinate and Transformational Geometry
• Geometric Relationships
• Congruence
• Equivalence
• Parallelism
• Perpendicularity
• Geometric Representations
• Two-dimensional figures
• Two-Dimensional Coordinate Systems
• Distance
• Slope
• Midpoint
• Parallel and perpendicular lines
• Proof and Congruence
• Deductive Reasoning
• Definitions
• Conjectures
• Proofs
• Geometric Relationships
• Congruence
• Parallelism
• Perpendicularity
• Geometric Representations
• Angles
• Sides
• Diagonals
• Patterns, Operations, and Properties
• 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.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within polygons?
• How does examining variance and invariance lead to new conjectures and theorems about polygon side and angle relationships?
• Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
• How is deductive reasoning used to understand, verify, and apply geometric conjectures about polygon side and angle relationships?
• What structures and processes can be used to investigate patterns involving polygon side and angle relationships?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent polygon side and angle relationships?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate polygon side and angle relationships?
• What tools and processes can be used to investigate …
• angle relationships
• side relationships
• diagonal relationships
… of polygons?
• What conjectures can be made and validated by exploring the patterns and properties of polygons?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding polygon side and angle relationships be applied when solving problem situations?
• How can measurable attributes related to …
• side lengths
• diagonals
• angles
… be distinguished and described in order to generalize geometric relationships of polygons?
• What processes can be used to determine the …
• number of diagonals
• number of sides
• interior and exterior angle measures
… of geometric figures?
• Logical Arguments and Constructions
• Deductive Reasoning
• Definitions
• Conjectures
• Theorems
• Geometric Relationships
• Congruence
• Equivalence
• Side relationships
• Angle relationships
• Formulas
• Geometric Representations
• Angles
• Sides
• Diagonals
• Polygons
• Patterns, Operations, and Properties
• 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.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within and between two- and three-dimensional geometric figures?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent relationships between two- and three-dimensional geometric figures?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships within and between two- and three-dimensional geometric figures?
• What tools and processes can be used to investigate …
• cross-sections of three-dimensional figures?
• rotations of two-dimensional figures about an axis?
• What relationships can be identified by exploring the properties of …
• two-dimensional figures and their rotation about an axis?
• three-dimensional geometric figures and their cross sections?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding relationships between two- and three-dimensional geometric figures be applied when solving problem situations?
• What attributes can be used to describe the …
• cross-sections of three-dimensional figures?
• rotations of two-dimensional figures about an axis?
• Two-Dimensional and Three-Dimensional Figures
• Geometric Representations
• Two-dimensional figures
• Three-dimensional figures
• Cross sections of three-dimensional figures
• Rotations of two-dimensional shapes
• Two-Dimensional Coordinate Systems
• Patterns, Operations, and Properties
• 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 think the slopes of perpendicular lines are just negatives or opposites of each other or reciprocals of each other rather than being both opposite reciprocals of each other.
• Some students may incorrectly apply the distance formula even though they know the properties needed to classify quadrilaterals.
• Some students may confuse a statement and its converse. Students may understand that if a figure is a square, then it is also a rectangle; however, knowing a figure is a rectangle does not necessarily imply that it is a square.

#### Unit Vocabulary

• Congruent segments – line segments whose lengths are equal
• Conjecture – statement believed to be true but not yet proven
• Exterior angle of a polygon – angle on the outside of a polygon formed by the side of a polygon and an extension of its adjacent side
• Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
• Irregular polygon – polygon that is not equilateral or equiangular
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, my2 = my1.
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, my2 = • Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)

Related Vocabulary

 Concave polygon Cone Convex polygon Coordinate geometry proofs Cross sections of three-dimensional figures Cylinder Diagonals of quadrilaterals Distance formula of a line segment Kite Midpoint formula of a line segment Parallelogram Prism Polygon Pyramid Quadrilateral Rectangle Rhombus Rotations of two-dimensional figures Semi-circle Slope formula of a line segment Sphere Square Sum of the exterior angles of a polygon Sum of the interior angles of a polygon Three-dimensional figures Trapezium Trapezoid Two-dimensional figures
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

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

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

Texas Education Agency – 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 Geometry 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.
• 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
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
G.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

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.
• 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.
G.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.

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.
• 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.
G.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.

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.
• 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.
G.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

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.
• 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.
G.1E Create and use representations to organize, record, and communicate mathematical ideas.

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.
• 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.
G.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

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.
• 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.
G.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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.
• 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.
G.2 Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures. The student is expected to:
G.2B

Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.

Use

THE DISTANCE, SLOPE, AND MIDPOINT FORMULAS TO VERIFY GEOMETRIC RELATIONSHIPS, INCLUDING CONGRUENCE OF SEGMENTS AND PARALLELISM OR PERPENDICULARITY OF PAIRS OF LINES

Including, but not limited to:

• Distance formula: d = • Formula for slope of a line: m = • Formula for midpoint of a line segment on a coordinate plane: • Congruent segments – line segments whose lengths are equal
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, my2my1.
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, my2 = – .
• Equation of a line
• Slope-intercept form, y = mx + b
• Point-slope form, yy1 = m(xx1)
• Standard form, Ax + By = C
• Coordinate geometry proofs
• Identification of polygons
• Trapezium
• Kite
• Trapezoid
• Parallelogram
• Rectangle
• Rhombus
• Square

Note(s):

• Grade 8 introduced and applied the Pythagorean Theorem to determine the distance between two points on a coordinate plane.
• Grade 8 introduced slope as or .
• Algebra I addressed determining equations of lines using point-slope form, slope intercept form, and standard form.
• Algebra I wrote equations of lines that contain a given point and are parallel or perpendicular to a given line.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• VII.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.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
G.5 Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:
G.5A

Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.

Investigate

PATTERNS TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS, INCLUDING DIAGONALS OF QUADRILATERALS AND INTERIOR AND EXTERIOR ANGLES OF POLYGONS CHOOSING FROM A VARIETY OF TOOLS

Including, but not limited to:

• Conjecture – statement believed to be true but not yet proven
• Investigations should include good sample design, valid conjecture, and inductive/deductive reasoning.
• Patterns include numeric and geometric properties.
• Utilization of a variety of tools in the investigations (e.g., compass and straightedge, paper folding, manipulatives, dynamic geometry software, technology)
• Isosceles trapezoid: Diagonals are congruent.
• Parallelogram: Diagonals bisect each other.
• Rectangle: Diagonals are congruent and exhibit all properties of the diagonals of a parallelogram.
• Rhombus: Diagonals bisect angles, are perpendicular to each other, and exhibit all properties of the diagonals of a parallelogram.
• Square: Exhibits all properties of a parallelogram, rectangle, and rhombus.
• Interior and exterior angles of a polygon
• Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
• Exterior angle of a polygon – angle on the outside of a polygon formed by the side of a polygon and an extension of its adjacent side
• Relationship between interior and exterior angles (one pair per vertex)
• Concave and convex polygons
• Sum of the interior angles of a polygon: sum = 180(n – 2) degrees, where n is the number of sides of the polygon
• Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)
• Measure of each interior angle: ∠ measure = degrees, where n is the number of sides of the polygon
• Irregular polygon – polygon that is not equilateral or equiangular
• Sum of the exterior angles of a polygon (one angle per vertex): sum = 360°
• Regular polygon
• Measure of each exterior angle: ∠ measure = degrees, where n is the number of sides of the polygon

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Grade 8 used 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.
• Geometry introduces analyzing patterns in geometric relationships and making conjectures about geometric relationships which may or may not be represented using algebraic expressions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• 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.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
G.6 Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:
G.6E Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

Prove

A QUADRILATERAL IS A PARALLELOGRAM, RECTANGLE, SQUARE, OR RHOMBUS USING OPPOSITE SIDES, OPPOSITE ANGLES, OR DIAGONALS

Including, but not limited to:

• Identification of type of quadrilateral
• Parallelogram
• Rectangle
• Square
• Rhombus
• Comparisons by coordinate proofs
• Opposite sides
• Opposite angles
• Diagonals

Apply

RELATIONSHIPS IN QUADRILATERALS TO SOLVE PROBLEMS

Including, but not limited to:

• Mathematical problem situations involving quadrilaterals
• Real-world problem situations involving quadrilaterals
• Parallelogram
• Rectangle
• Square
• Rhombus

Note(s):

• Geometry introduces coordinate proofs of conjectures about figures.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• III.A. Geometric and Spatial Reasoning – Figures and their properties
• III.A.2. Form and validate conjectures about one-, two-, and three-dimensional figures and their properties.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.3. Determine a 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.
• VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
• 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.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.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.
G.10 Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures. The student is expected to:
G.10A Identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations of two-dimensional shapes.

Identify

THE SHAPES OF TWO-DIMENSIONAL CROSS-SECTIONS OF PRISMS, PYRAMIDS, CYLINDERS, CONES, AND SPHERES

Including, but not limited to:

• Three-dimensional figures
• Prisms
• Cylinders
• Pyramids
• Cones
• Spheres
• Description of cross sectional intersections
• Verbal
• Pictorial
• Representations of cross sections
• Point
• Line segment
• Region of a plane (polygon or circle)

Identify

THREE-DIMENSIONAL OBJECTS GENERATED BY ROTATIONS OF TWO-DIMENSIONAL SHAPES

Including, but not limited to:

• Rectangle
• Triangle
• Semi-circle
• Trapezoid
• Rotation about x-axis and y-axis

Note(s): 