 Hello, Guest!
 Instructional Focus DocumentGeometry
 TITLE : Unit 01: Introduction to Logic and Euclidean Geometry SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address distances in one-dimensional systems, constructions of congruent line segments and congruent angles, and the structure of a mathematical system, including undefined terms, definitions, postulates and conjectures. The unit also incorporates deductive reasoning to analyze conditional and related statements and to verify conjectures. 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 2, students were introduced to points on a number line. In Grade 4, students identified geometric terms (points, lines, line segments, rays, angles), determined distances from zero on a number line, measured angles, and drew angles to specified measures. In middle school, geometric vocabulary and applications were continued and expanded.

During this Unit
Students lay the foundation for geometry by developing an understanding of the structure of a geometric system through examination of the relationship between undefined terms (point, line, and plane), definitions, conjectures, and postulates. Students examine one-dimensional distance relationships in line segments, including fractional distances and midpoints, and make connections to the number line and segment addition. They also examine relationships in rays and angles making connections to the angle measure and angle addition. Constructions are used to explore and make conjectures about congruent geometric relationships in line segments and angles. They connect their understanding of definitions and postulates of lines, angles, and other geometric vocabulary to the context of the real world. Students also investigate logic statements and the conditions that make them true or false. Students explore conditional statements and their related statements (converse, inverse, and contrapositive) in both a real world and mathematical setting to develop an understanding of logic and the role it plays in geometry and the real world. Students are expected to recognize the connection between a biconditional statement and a true conditional statement with a true converse. Deductive reasoning and inductive reasoning are introduced and applied to make conjectures. Students verify that a conjecture is false using counterexamples.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Geometry

After this Unit
In Unit 02, students will extend their understanding of one-dimensional relationships to build an understanding of two-dimensional relationships. In Units 03 – 08, conjectures, postulates and theorems to investigate geometric relationships in parallel lines and transversals, polygons, and circles will be developed, and deductive reasoning will be used to verify conjectures and theorems. Connections will continue to be made between algebra and geometry. The concepts in this unit will also be applied in subsequent mathematics courses.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning A2, C1; 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 (NCTM), Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 9–12 (2012), Big Idea 3 identifies definitions and working with definitions as an important facet of geometry that is central to understanding geometric concepts. NCTM also identifies definitions as essential tools for conducting and analyzing geometric investigations. According to Geometry from Multiple Perspectives (1991) from the National Council of Teachers of Mathematics, many manufactured items are made of parts that are linear or circular in shape and are based on the geometry of Euclid, which is the geometry of the point set, of the straight line, and of the Euclidean tools of construction.

National Council of Teachers of Mathematics. (2012). Developing Essential Understanding of Functions, Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (1991). Curriculum and evaluation standards for school mathematics: Geometry from multiple perspectives. 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?
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 between length and distance on a number line?
• 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 length and distance relationships?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships on a number line?
• What tools and processes can be used to …
• represent fractional distances between two points on a number line?
• represent midpoint on a number line?
• construct congruent segments?
• construct congruent angles?
• How can constructions be used to make and validate conjectures about geometric relationships that occur with …
• congruent segments?
• congruent angles?
• How are undefined terms used to develop the concepts of distance (measure), congruence, and betweenness?
• What conjectures can be made and validated about one-dimensional distance relationships in line segments, the number line, and segment addition?
• What conjectures can be made and validated about rays and angles, angle measure, and angle addition?
• 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 the relationships of length and distance be applied when solving problem situations?
• How can measurable attributes related to …
• distance
• length
… be distinguished and described in order to generalize geometric relationships?
• What processes can be used to determine the coordinates of the …
• point that is a given fractional distance less than one from one end of a line segment?
• midpoint of a line segment?
• How are different processes that are used to determine coordinates on a number line related?
• Coordinate and Transformational Geometry
• Geometric Relationships
• Congruence
• Equidistance
• Equivalence
• Measure relationships
• Geometric Representations
• One-dimensional
• One-Dimensional Coordinate Systems
• Distance
• Midpoint
• Patterns, Operations, and Properties
• Logical Arguments and Constructions
• Constructions
• Congruent segments
• Congruent angles
• Deductive Reasoning
• Undefined terms
• Definitions
• Postulates
• Conjectures
• Geometric Relationships
• Congruence
• Geometries
• Euclidean
• 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.

 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)
• 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 are logical arguments applied in the study of geometric relationships and in real-world settings?
• How is deductive reasoning used to understand, prove, and apply geometric conjectures?
• How are logical arguments and deductive reasoning used to prove and disprove conditionals and their related statements?
• How can the validity of a conditional statement be determined?
• What relationships exist between a conditional statement and its converse, inverse, and contrapositive?
• How can symbolic and verbal representations be used to communicate logical equivalence?
• What relationships exist between a biconditional statement and a true conditional statement with a true converse?
• How can counterexamples be used to support or refute mathematical claims?
• Logical Arguments and Constructions
• Deductive Reasoning
• Definitions
• Conjectures
• Conditional statements
• Geometric Relationships
• Geometries
• Euclidean
• Patterns and Properties
• Associated Mathematical Processes
• Application
• 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 a conditional statement and its converse are logically equivalent, rather than testing to determine their truth value.

Underdeveloped Concepts:

• Although some students have an extensive visual mathematical vocabulary and may be able to connect geometric terms to pictures or examples, the students may have difficulty articulating formal verbal definitions of these terms.

#### Unit Vocabulary

• Biconditional statement – statement for which both the conditional statement and its converse are true
• Conditional statement – statement composed of a hypothesis (if) and a conclusion (then)
• Congruent angles – angles whose angle measurement is equal
• Congruent segments – line segments whose lengths are equal
• Conjecture – statement believed to be true but not yet proven
• Contrapositive of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged and negated
• Converse of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged
• Counterexample – example used to disprove a statement
• Definition – words or terms used to describe new terms/concepts
• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Inverse of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are negated
• Line segment – part of a line between two points on the line, called endpoints of the segment
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• One-dimensional coordinate system – numbers used as locations of points on a number line
• Postulates (axioms) – statements accepted as true without requiring proof
• Undefined terms – terms not formally defined and used to define other terms/concepts in a mathematical system
• Validity – determination of statement as either true or false

Related Vocabulary:

 Acute angle Adjacent angles Angle Angle addition Angle mesasure “Betweenness” Collinear Compass Complementary angles Coplanar Defined terms Degree measure Distance Exterior of an angle Interior of an angle Intersection Length Line Linear pair of angles Non-collinear Non-coplanar Number line Obtuse angle Point Ray Right angle Segment addition Space Straight angle Straight edge Supplementary angles Vertex Vertical angles
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 – 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.2A

Determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint.

Determine

THE COORDINATES OF A POINT IN A ONE-DIMENSIONAL COORDINATE SYSTEM THAT IS A GIVEN FRACTIONAL DISTANCE LESS THAN ONE FROM ONE END OF A LINE SEGMENT TO THE OTHER, INCLUDING FINDING THE MIDPOINT

Including, but not limited to:

• A point in a one-dimensional coordinate system that is a given fractional distance less than one from one end of a line segment to the other
• One-dimensional coordinate system – numbers used as locations of points on a number line
• Line segment – part of a line between two points on the line, called endpoints of the segment
• Distance formula: d = |x1x2| or d|x2x1|
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• For any k (where 0 < k < 1), the location on a number line that is the fraction k of the distance from any point, x1 to any other point, x2 and can be found using the expression x1 + k(|x2x1|).

Note(s):

• Prior grade levels addressed points and distance on a number line.
• Prior grade levels addressed points and lines on a coordinate plane.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.
• 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.
• 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.4 Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to:
G.4A

Distinguish between undefined terms, definitions, postulates, conjectures, and theorems.

Distinguish

BETWEEN UNDEFINED TERMS, DEFINITIONS, POSTULATES, AND CONJECTURES

Including, but not limited to:

• Undefined terms – terms not formally defined and used to define other terms/concepts in a mathematical system
• Undefined terms in Euclidean geometry
• Point
• Line
• Plane
• Definitions – words or terms used to describe new terms/concepts
• Postulates (axioms) – statements accepted as true without requiring proof
• Conjecture – statement believed to be true but not yet proven
• Representation and notation for undefined and defined terms
• Connections between undefined terms and defined terms
• Connections between definitions, postulates, and conjectures
• Conjectures are suspected of being true but can be disproved with a counterexample or proven with a logical argument at which point a conjecture becomes a theorem.
• Definitions and postulates can be used to support a logical argument that a conjecture is true.

Note(s):

• Geometry introduces the vocabulary of undefined terms, definitions, postulates, conjectures, and theorems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts 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.
G.4B Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse.

Identify, Determine

THE VALIDITY OF THE CONVERSE, INVERSE, AND CONTRAPOSITIVE OF A CONDITIONAL STATEMENT

Including, but not limited to:

• Conditional statement – statement composed of a hypothesis (if) and a conclusion (then)
• If p then q
• Converse of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged
• If q then p
• Inverse of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are negated
• If not p then not q
• Contrapositive of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged and negated
• If not q then not p
• Validity – determination of statement as either true or false
• Connections between conditional, converse, inverse, and contrapositive statements
• Conditional statements in a variety of forms
• If p then q
• q if p
• p implies q
• Symbolic representations
• pq for if p then q
• ~p → ~q for if not p then not q
• Logical equivalence of a statement and its contrapositive
• Mathematical and non-mathematical conditional statements
• Examples and counterexamples

Recognize

THE CONNECTION BETWEEN A BICONDITIONAL STATEMENT AND A TRUE CONDITIONAL STATEMENT WITH A TRUE CONVERSE

Including, but not limited to:

• Connection between a conditional statement and a biconditional statement
• Biconditional statement – statement for which both the conditional statement and its converse are true
• Truth value of a biconditional or related statement is determined by analysis of its hypothesis and conclusion (truth table, Venn diagram)
• A test for determining if a definition is a good definition is to test its converse, meaning a good definition is a biconditional statement.

Note(s):

• Geometry introduces conditional statements, including converse, inverse, contrapositive, and biconditional statements.
• 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.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.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.
• 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.4C Verify that a conjecture is false using a counterexample.

Verify

A CONJECTURE IS FALSE USING A COUNTEREXAMPLE

Including, but not limited to:

• Counterexample – example used to disprove a statement
• Mathematical and non-mathematical examples and counterexamples

Note(s):

• 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.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.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.
• 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.5B

Construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge.

Construct

CONGRUENT SEGMENTS AND CONGRUENT ANGLES USING A COMPASS AND A STRAIGHTEDGE

Including, but not limited to:

• Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Congruent segments – line segments whose lengths are equal
• Congruent angles – angles whose angle measurements are equal

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Geometry introduces constructions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• 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.
G.5C

Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships.

Use

THE CONSTRUCTIONS OF CONGRUENT SEGMENTS AND CONGRUENT ANGLES TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS

Including, but not limited to:

• Geometric construction – creation of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
• Use of various tools
• Compass and straightedge
• Dynamic geometric software
• Patty paper
• Constructions
• Congruent segments
• Congruent angles
• Conjectures about attributes of figures related to the constructions
• Number line and segment addition
• Angle measure and angle addition

Note(s):

• Previous grade levels investigated attributes of geometric figures.
• Geometry introduces the use of constructions to make conjectures about geometric relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• 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.
• 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. 