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

#### Unit Overview

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.

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): III. Geometric Reasoning A1, A2, B1, B3, D1, D2; VIII. Problem Solving and Reasoning B1, B2; IX. Communication and Representation A3, C1.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

• Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
• Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?
• How do geometric and spatial reasoning allow for the understanding of different geometric systems as models for the world?

Geometric systems are axiomatic systems built on undefined terms, defined terms, postulates, and theorems which are fundamental in verifying conjectures through logical arguments.

• What roles do undefined terms, defined terms, postulates, and theorems serve in an axiomatic system?
• How does the investigation of geometric patterns lead to the development of conjectures and postulates?
• How are one- and two-dimensional coordinate systems and algebra used to investigate and verify geometric relationships?
• How are logical arguments applied in the study of geometric relationships and their application in real-world settings?
• How is deductive reasoning used to understand, prove, and apply geometric conjectures and theorems pertaining to geometric relationships?
• How are logical arguments and deductive reasoning used to prove and disprove conditionals and their related statements?
• How can constructions be used to validate conjectures about geometric figures?

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Numerical Understanding

• Fractions
• Number Line

Geometric Reasoning

• Conjectures
• Constructions
• Definitions
• Geometric Attributes/Properties
• Geometric Relationships
• Geometric Systems
• Logical Arguments
• One-Dimensional Figures
• Postulates
• Undefined Terms

Measurement Reasoning

• Angle Measures
• Distance/Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

In Euclidean geometry, undefined terms (point, line, and plane), definitions and postulates are used to describe and represent many aspects of our world.

• What are the undefined terms in Euclidean geometry?
• What are some real world examples of undefined terms?
• What is the relationship between undefined terms, definitions, postulates, and conjectures?
• How are undefined terms used to develop the concepts of distance (measure), congruence, and “betweenness” (segment and angle addition postulates)?

The distance and midpoint between two points on a number line can be calculated to verify geometric relationships.

• How is the coordinate of a point that is a given fractional distance less than one from one end of a line segment to the other determined?
• How is the coordinate of the midpoint of a line segment determined?
• How is the concept of betweenness applied when finding the point a given fractional distance less than one from one end of a line segment or the midpoint of a line segment?

Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

• Why are diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are diagrams used to organize information from the problem situation?
• How do diagrams aid in calculations when solving problems?
• Why is a number line used to diagram one-dimensional figures?

Construction of congruent line segments and congruent angles can be used to verify geometric congruence relationships.

• What tools can be used to construct congruent segments and congruent angles?
• How is a construction of congruent line segments used to verify the concept of equidistance?
• How can a counterexample be used to prove a conjecture to be false?
• Under what condition can the converse, inverse, or contrapositive of a conditional be used to support a statement in a deductive proof?
 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 Reasoning

• Conjectures
• Definitions
• Geometric Attributes/Properties
• Logical Arguments
• Postulates
• Undefined Terms

Associated Mathematical Processes

• Application
• Communication
• Representations
• Relationships
• Justification

Deductive reasoning can be used to determine the validity of a conditional statement and its related statements (converse, inverse, contrapositive, and biconditional).

• What is a conditional statement?
• How can you determine the validity of a conditional statement?
• What is the relationship between a conditional statement its converse, inverse, and contrapositive?
• What is the relationship between a biconditional statement and a true conditional statement with a true converse?

Deductive reasoning is used to justify conjectures about geometric relationships.

• How do conjectures relate to conditional statements?
• How is deductive reasoning related to the verification of conjectures?
• How can a counterexample be used to prove a conjecture to be false?
• Under what condition can the converse, inverse, or contrapositive of a conditional be used to support a statement in a deductive proof?

#### 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) – true statements not requiring proof
• Undefined Terms – terms not formally defined and used to define other terms/concepts in a mathematical system

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 Postulate Ray Right angle Segment addition Space Straight angle Straight edge Supplementary angles Vertex Vertical angles
Unit Assessment Items System Resources Other Resources

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

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

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

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

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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:
• X. Connections
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:
• IX. Communication and Representation
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 = |x1 – x2| or |x2 – x1|
• 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 expressionx1 + k(|x2 x1|).

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
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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
• III. Geometric Reasoning
• D2 – Understand that Euclidean geometry is an axiomatic system.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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
• 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
• p → q 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. Geometric Reasoning
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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, 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
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B2 – Identify the symmetries in a plane figure.
• D1 – Make and validate geometric conjectures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections