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 Instructional Focus DocumentGeometry
 TITLE : Unit 06: Relationships of Circles, including Radian Measure and Equations of Circles SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address properties and attributes of circles, equations of circles, and angle and segment relationships within circles. 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 04, students investigated the properties and attributes of circles. In Grade 7, students defined pi, , as the ratio of the circumference to the diameter of a circle.

During this unit, students define new circle vocabulary, including special segments and angles of circles, using diagrams and definitions. Students use patterns, diagrams, and a variety of tools to investigate circles, chords, secants, tangents, and their angle relationships, including central and inscribed angles. Students use patterns, diagrams, and a variety of tools to investigate circles, chords, secants, tangents, and their segment length relationships. Students apply theorems about combined circle angle/segment length relationships, including central and inscribed angles, in non-contextual problems. Students describe and develop the concept of radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle. Students convert between degree and radian measures. Students develop the equation of a circle, x² + y² = r², using the coordinate grid and the Pythagorean Theorem, given the radius, r, and center at the origin. Students determine the equation of a circle, (xh)² + (yk)² = r², given the radius and a center of (h, k). Students represent real-world situations using equations of circles.

After this unit, in Geometry Unit 08, students will apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle and between the measure of the area of a sector of a circle and the area of the circle to solve problems. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving circles.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): III. Geometric Reasoning A1, A2, B2, C1, C3, D1, D1;. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

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. At the conclusion of this unit, students are asked to create graphic organizers. TxCCRS cites many skills related to the communication and representation of mathematical ideas. The National Council of Teachers of Mathematics (2000) said that all students in grades 9 – 12 should explore relationships in two-dimensional geometric figures, make and test conjectures about two-dimensional geometric figures, and solve problems involving two-dimensional geometric figures. 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/collegereadiness/crs.pdf

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Relations are algebraic models that describe how two quantities relate to one another. Functions are a subset of relations.

• What are types of relations?
• How can relations be represented?
• Why do some relations not define a function?

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 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 can constructions be used to validate conjectures about geometric figures?

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?

Attributes and properties of two-dimensional geometric shapes are foundational to developing geometric and measurement reasoning.

• Why is it important to compare and contrast attributes and properties of two-dimensional geometric shapes?
• How does analyzing the attributes and properties of two-dimensional geometric shapes develop geometric and measurement reasoning?

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.

Algebraic Reasoning

• Patterns/Rules
• Solve

Geometric Reasoning

• Conjectures
• Geometric Attributes/Properties
• Geometric Relationships
• Logical Arguments
• Theorems/Postulates/Axioms
• Two-Dimensional Figures

Measurement Reasoning

• Angle Measures
• Circumference
• Formulas
• Length

Associated Mathematical Processes

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

Geometric relationships exist between the angles, radii, intercepted arcs, and chords in a circle.

• What relationship occurs between a central angle and its intercepted arc?
• What relationship occurs between an inscribed angle and its intercepted arc?
• How is the measure of a central angle related to an inscribed angle with the same intersected arc?
• What geometric relationship is found between a tangent line drawn to a circle and its radius?
• What point of a circle is contained in the perpendicular bisector of a chord?
• What geometric relationship occurs if a radius of a circle bisects a chord of the circle that is not a diameter of the circle?
• What relationship can be found if a radius of a circle is perpendicular to a chord of the circle?
• What relationship can be found if two chords of a circle are congruent?
• When will two chords of a circle be congruent?
• What special geometric relationship can be found when an inscribed angle intercepts a semicircle?
• What relationship can be found if two angles of a circle intercept the same arc?

Geometric relationships exist with angles, arcs, and segments created by the intersection of chords, secants, and tangents of a circle.

• When two chords intersect in a circle, what conjecture(s) can be made?
• When two secants are drawn through a circle from the same exterior point, what conjecture(s) can be made?
• When a secant and a tangent are drawn through a circle from the same exterior point, what conjecture(s) can be made?
• When two tangents are drawn through a circle from the same exterior point, what conjecture(s) can be made?

Postulates and theorems pertaining to circles help develop an understanding of geometric relationships.

• What geometric relationships can be understood when investigating angles of circles?
• What geometric relationships can be understood when investigating special segments of circles?

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 the coordinate plane used to diagram two-dimensional figures?

Determining and comparing the measurements of the attributes of two-dimensional figures are foundational to analyzing the relationships in two-dimensional figures.

• For which attributes of a two-dimensional figure can the measures be determined?
• How are the measures of the attributes determined?
• How can the measures of the attributes of two-dimensional figures be used to analyze relationships within the figure?
 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.

Algebraic Reasoning

• Equations
• Multiple Representations
• Ratios
• Relations

Geometric Reasoning

• Angles
• Geometric Attributes/Properties
• Geometric Relationships
• Two-Dimensional Figures

Measurement Reasoning

• Angle Measures
• Circumference
• Formulas
• Length
• Units of Measure

Associated Mathematical Processes

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

Radian measure of an angle is the ratio of the length of an arc intercepted by a central angle and the radius of the circle.

• What information about a central angle can be found by dividing the length of the arc of any central angle of a circle by the radius?
• How can the length of an arc intercepted by a central angle be found using the radian measure and the radius of the circle?
• How can the degree measure of an angle be given in radian measure?
• How can the radian measure of an angle be written in degrees?

A circle can be represented with the equation (xh)² + (yk)² = r² where r represents the radius and (h, k) represents the center.

• How is the equation of a circle related to the distance formula?
• How can the coordinates of the center of a circle and the length of the radius of the circle be used to find the equation for the circle?
• How can the general form of an equation for a circle be used to find the coordinates of the center of the circle and the length of the radius of the circle?
• How can the equation of a circle be written for a circle drawn on a coordinate plane?
• How can the equation of a circle be written given a point on the circle and the center of the circle?

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 the coordinate plane used to diagram two-dimensional figures?

Determining and comparing the measurements of the attributes of two-dimensional figures are foundational to analyzing the relationships in two-dimensional figures.

• For which attributes of a two-dimensional figure can the measures be determined?
• How are the measures of the attributes determined?
• How can the measures of the attributes of two-dimensional figures be used to analyze relationships within the figure?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think angles formed by intersecting chords are central angles even if the intersection is not at the center of the circle.
• Some students may think that a major arc can be named using only two points on the circle rather than using three points on the circle.
• Some students may think pi, , is just a number rather than a relationship between the circumference and radius of a circle.

Underdeveloped Concepts:

• Some students may think that  equals 180° instead of  radians equals 180° or approximately 3.14 radians equals 180°.
• Some students may think that if the vertex of an angle is on the circle the angle measure is the same as the measure of the intercepted arc rather than one-half the measure of the intercepted arc.
• Some students may determine the measure of an exterior angle created by the intersection of secants and/or tangents to be one-half the sum of the measures of the intercepted arcs rather than one-half the difference of the intercepted arcs.
• Some students may not recognize or use  as a variable.
• Some students may not realize that leaving  in an equation or formula represents an exact value.
• Some students may always use (0, 0) as the (h, k) when writing the equation of a circle.
• Some students may fail to take the square root of the r2 portion of the equation of a circle when determining the radius of the circle.

#### Unit Vocabulary

• Center of a circle – point equidistant from all points on the circle (origin or (h, k) as the center)
• Central angle – angle whose vertex is the center of the circle and whose sides are radii of the circle
• Chord of a circle – line segment that joins two points on the circle
• Circle – set of all points equidistant from a given point called the center of the circle
• Conjecture – statement believed to be true but not yet proven
• Diameter – line segment whose endpoints are on the circle and passes through the center of the circle
• Inscribed angle – angle whose vertex is on the circle and whose sides are chords of the circle
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, my2 = .
• Radian measure – ratio of the length of an arc intercepted by a central angle and the radius of the circle
• Radius – a line segment drawn from the center of a circle to any point on the circle and is half the length of the diameter of the circle
• Secant – a line, ray, or line segment that intersects the circle in exactly two points
• Tangent to a circle – line or line segment perpendicular to the radius and intersecting the circle at exactly one point called the point of tangency

Related Vocabulary:

 Arc Central angle Chord Circumference Diameter Distance Formula Equation of a circle Intercepted arc Major arc Minor arc Perpendicular   bisector Point of tangency Radian measure Radius Semicircle Tangent segment
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.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 FORMULA TO VERIFY GEOMETRIC RELATIONSHIPS

Including, but not limited to:

• Distance formula: d =
• Equation of a line
• Slope-intercept form, y = mx + b,
• Point-slope form, yy1 = m(xx1)
• Standard form, Ax + By = C
• Relationships of special segments and points in circles
• Center of circle
• Circle – set of all points equidistant from a given point called the center of the circle
• Center of a circle – point equidistant from all points on the circle
• Chord of a circle
• Chord of a circle – line segment that joins two points on the circle
• Diameter and radius of a circle
• Diameter – line segment whose endpoints are on the circle and passes through the center of the circle
• Radius – a line segment drawn from the center of a circle to any point on the circle and is half the length of the diamter of the circle
• Tangent to a circle
• Tangent to a circle – line or line segment perpendicular to the radius and intersecting the circle at exactly one point called the point of tangency

Note(s):

• Grade Level(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. 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.
• C1 – Make connections between geometry and algebra.
• 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.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 SPECIAL SEGMENTS AND ANGLES OF CIRCLES 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)
• Special segments of circles
• Intersecting chords
• Secant – a line, ray, or line segment that intersects the circle in exactly two points
• Secant-Secant
• Tangent to a circle – line or line segment perpendicular to the radius and intersecting the circle at exactly one point called the point of tangency
• Secant-Tangent
• Tangent-Tangent
• Special angles of circles
• Central angle – angle whose vertex is the center of the circle and whose sides are radii of the circle
• Angles formed by intersecting chords
• Inscribed angle – angle whose vertex is on the circle and whose sides are chords of the circle
• Tangent-Chord
• Tangent-Tangent
• Secant-Secant
• Secant-Tangent

Note(s):

• Grade Level(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. 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.12 Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:
G.12A Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems.

Apply

THEOREMS ABOUT CIRCLES, INCLUDING RELATIONSHIPS AMONG ANGLES, RADII, CHORDS, TANGENTS, AND SECANTS TO SOLVE NON-CONTEXTUAL PROBLEMS

Including, but not limited to:

• Geometric relationships
• Angles formed by intersecting radii, chords, tangents, and secants
• Central angles
• Inscribed angles
• Relationship between angles and intercepted arcs
• Radii
• Chords
• Tangents
• Secants
• Applications to non-contextual mathematical problem situations
• Use of appropriate units of measure

Note(s):

• Grade Level(s)
• Previous grade levels explored characteristics of circles.
• 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.
• D1 – Make and validate geometric conjectures
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.12D Describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.

Describe

RADIAN MEASURE OF AN ANGLE AS THE RATIO OF THE LENGTH OF AN ARC INTERCEPTED BY A CENTRAL ANGLE AND THE RADIUS OF THE CIRCLE

Including, but not limited to:

• Radian measure – ratio of the length of an arc intercepted by a central angle and the radius of the circle
• Comparison of radian measure of a circle and degree measure of a circle
• Generalization of the common conversion factors between degree and radian measure
• 180° = π
• Conversions of degrees into radians and radians into degree measures (values in radians can be left in terms of π)
• Radian measure can be described as , where  is the radian measure of the central angle, ℓ is the length of the arc intercepted by the central angle, and r is the length of the radius of the circle
• Applications of radian measure

Note(s):

• Grade Level(s)
• Geometry lays the foundation for development of radian measurement in Precalculus.
• 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.
• C3 – Make connections between geometry and measurement.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
G.12E Show that the equation of a circle with center at the origin and radius r is x2 + y2 = r2 and determine the equation for the graph of a circle with radius r and center (h, k), (x - h)2 + (y - k)2 =r2.

Show

THAT THE EQUATION OF A CIRCLE WITH CENTER AT THE ORIGIN AND RADIUS r IS x2 + y2 = r2

Including, but not limited to:

• Derivation of equation of a circle using the distance formula

Determine

THE EQUATION FOR THE GRAPH OF A CIRCLE WITH RADIUS r AND CENTER (h, k), (x - h)2 + (y - k)2 = r2

Including, but not limited to:

• General equation for a circle with center (h, k) and radius of length r: (x - h)2 + (y - k)2 = r2
• Graphs of circles from their equations
• Equations for circles given their graphs

Note(s):

• Grade Level(s)
• Geometry further develops the foundation of conic sections measurement in Precalculus.
• 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.
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills. The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum. School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency. Choose appropriate ELPS to support instruction. Click here to collapse or expand this section. ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.1A use prior knowledge and experiences to understand meanings in English ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment ELPS.c.1F use accessible language and learn new and essential language in the process ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations. ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs. ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency ELPS.c.3E share information in cooperative learning interactions ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired ELPS.c.3I adapt spoken language appropriately for formal and informal purposes ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment. ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to: ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs ELPS.c.4H read silently with increasing ease and comprehension for longer periods ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs. ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to: ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as: ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/11/2016

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