Hello, Guest!

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

Unit Overview

Introduction
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, π, 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 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.

Additional Notes
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.

Research
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/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9


  • Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life.
    • Why is developing geometric, spatial, and measurement reasoning essential?
    • How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
  • Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments.
    • Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life?
    • What elements of logical reasoning influence the truth of a statement?
    • How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
  • Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
    • What invariant (unchanging) and variant (changing) relationships exist within circles?
    • How does examining variance and invariance lead to new conjectures and theorems about circle relationships?
  • Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
    • How is deductive reasoning used to understand, prove, and apply geometric conjectures about circles?
    • What conjectures can be made about circles and their …
      • angle measure relationships?
      • special circle segment length relationships?
      • combined measure relationships?
  • Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
    • What types of problem situations represent circle relationships?
    • How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships about circle relationships?
    • What tools and processes can be used to investigate …
      • circles?
      • the relationships between arc angle measures and arc length measures?
      • special segments of circles?
      • special angles of circles?
    • How can constructions be used to make and validate conjectures about geometric relationships in circles?
    • What conjectures can be made about …
      • relationships in angles of circles?
      • relationships in special circle segments?
  • 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
    • How can understanding circle relationships be applied when solving problem situations?
    • How can measurable attributes related to …
      • angle measures, including central and inscribed angles
      • arc lengths and arc angles
      • special circle segments lengths, including radii, chords, tangents, and secants
      … be distinguished and described in order to generalize geometric relationships of circles?
    • What processes can be used to determine the …
      • angle measure of a circle?
      • length of a segment of a circle?
      • combined angle measures of a circle?
      • combined segment lengths of a circle?
  • Circles
    • Circles
      • Theorems
      • Circle relationships
      • Measure relationships
    • Geometric Relationships
      • Congruence
      • Equidistance
      • Equivalence
      • Perpendicularity
      • Formulas
    • Geometric Representations
      • Two-dimensional figures
    • Triangle Relationships
      • Right triangles
  • Logical Arguments and Constructions
    • Constructions
    • Deductive Reasoning
      • Definitions
      • Conjectures
      • Theorems
    • Geometric Relationships
      • Angle relationships
    • Geometric Representations
      • Angles
      • Segments
      • Circles
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships
    • Justification
Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

  • Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life.
    • Why is developing geometric, spatial, and measurement reasoning essential?
    • How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
  • Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments.
    • Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life?
    • What elements of logical reasoning influence the truth of a statement?
    • How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
  • Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
    • What invariant (unchanging) and variant (changing) relationships exist within circles?
    • How does examining variance and invariance lead to new conjectures about circle relationships?
  • Deductive reasoning can be used to determine the validity of a conditional statement and its related statements and conjectures about geometric relationships in order to support or refute mathematical claims through the process of proving.
    • What conjectures can be made about circles and their …
      • angle measure relationships?
      • segment length relationships?
  • Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
    • What types of problem situations represent circle relationships?
    • How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships about circle relationships?
    • What tools and processes can be used to investigate …
      • circles?
      • arcs and radii of circles?
      • central angles of circles?
      • degree and radian measure of angles of circles?
      • equations of circles?
    • How can the coordinate plane be used to make and validate conjectures about geometric relationships in circles?
    • What conjectures can be made about …
      • relationships in angles of circles?
      • relationships in special circle segments?
      • the relationship of the equation of a circle and the Pythagorean Theorem?
  • 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 circle relationships be applied when solving problem situations?
    • How can measurable attributes related to …
      • distance
      • special segments of circles, including arcs and radii
      • Pythagorean Theorem
      • degree and radian measure of angles of circles
      … be distinguished and described in order to generalize geometric relationships of circles?
    • What processes can be used to determine the …
      • degree or radian measure of an angle of a circle?
      • equation of a circle?
  • Coordinate and Transformational Geometry; Circles
    • Circles
      • Circle relationships
      • Measure relationships
      • Equation of a circle
    • Geometric Relationships
      • Equidistance
      • Equivalence
      • Proportionality
      • Formulas
    • Geometric Representations
      • Two-dimensional figures
    • Two-Dimensional Coordinate Systems
      • Distance
  • Logical Arguments and Constructions
    • Deductive Reasoning
      • Definitions
      • Conjectures
    • Geometric Relationships
      • Angle relationships
    • Geometric Representations
      • Angles
      • Segments
      • Circles
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Application
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships
    • Justification
Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think 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
  • 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 – a 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
  • Radian measure – ratio of the length of an arc intercepted by a central angle and the radius of the circle
  • Radius – 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 to a circle – line, ray, or line segment that intersects the circle in exactly two points
  • Tangent to a circle – line, ray, or line segment perpendicular to the radius and intersecting the circle at exactly one point, the point of tangency

Related Vocabulary:

  • Arc
  • Circumference
  • Distance formula
  • Equation of a circle
  • Intercepted arc
  • Major arc
  • Minor arc
  • Perpendicular bisector
  • Point of tangency
  • Semicircle
  • Tangent segment
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:
    • 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 – a line segment whose endpoints are on the circle and passes through the center of the circle
      • Radius – 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, ray, or line segment perpendicular to the radius and intersecting the circle at exactly one point, 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 and angles of circles
    • Central angle – angle whose vertex is the center of the circle and whose sides are radii of the circle

      • Measure of a central angle is equal to the measure of the intercepted arc.
    • Inscribed angle – angle whose vertex is on the circle and whose sides are chords of the circle

      • Measure of an inscribed angle is the measure of the intercepted arc.
        • Inscribed angle with semicircle intercepted arc is a right angle.
      • Measure of an inscribed angle is the measure of the central angle that shares the same intercepted arc.
    • Chord of a circle – line segment that joins two points on the circle
      • Two chords are congruent if and only if their corresponding intercepting arcs are congruent.
      • Diameter of a circle bisects a chord if and only if the diameter and chord are perpendicular.
      • Products of the lengths of the segments of intersecting chords are equal.
      • Vertical angles formed by intersecting chords are equal in measure.
      • Adjacent angles formed by intersecting chords are supplementary.
      • Measure of the angle formed by intersecting chords is the sum of the measures of the intercepted arcs.
    • Secant to a circle – line, ray, or line segment that intersects the circle in exactly two points
      • Secant-Secant
        • Product of the length of the secant segment and its external segment is equal to the product of the other secant segment and its external segment from the same point outside the circle.
        • Measure of the angle formed by two secants intersecting at a point outside the circle is the difference of the measures of the intercepted arcs.
    • Tangent to a circle – line, ray, or line segment perpendicular to the radius and intersecting the circle in exactly one point, the point of tangency
      • Tangent-Tangent
        • Intersecting tangent segments are equal in length from the same point outside the circle.
        • Measure of angle formed by the intersection of two tangents at a point outside the circle is the difference of the measures of the intercepted arcs.
    • Radii
      • Tangent line is perpendicular to a radius of the circle at the point of tangency.
    • Combinations of chords, secants, and tangents
      • Chord-Tangent
        • Measure of the angle formed by the intersection of a tangent and chord is the measure of the intercepted arc.
      • Secant-Tangent
        • Product of the length of a secant segment and its external part is equal to the product of the square of the length of a tangent segment intersecting the secant segment at a point outside the circle.
        • Measure of the angle formed by a secant and a tangent intersecting at the same point outside the circle is the difference of the measures of the intercepted arcs.

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 among angles, radii, chords, tangents, and secants
    • Measure of the arc is denoted as m.
    • Central angle – angle whose vertex is the center of the circle and whose sides are radii of the circle
      • Measure of a central angle is equal to the measure of the intercepted arc.
    • Inscribed angle – angle whose vertex is on the circle and whose sides are chords of the circle

      • Measure of an inscribed angle is the measure of the intercepted arc.
        • Inscribed angle with semicircle intercepted arc is a right angle.
      • Measure of an inscribed angle is the measure of the central angle that shares the same intercepted arc.
    • Radii
      • Radius of a circle is perpendicular to a tangent line at the point of tangency.
    • Chord of a circle – line segment that joins two points on the circle
      • Two chords are congruent if and only if their corresponding intercepting arcs are congruent.
      • Diameter of a circle bisects a chord if and only if the diameter and chord are perpendicular.
      • Products of the lengths of the segments of intersecting chords are equal.
      • Vertical angles formed by intersecting chords are equal in measure.
      • Adjacent angles formed by intersecting chords are supplementary.
      • Measure of an angle formed by intersecting chords is the sum of the measures of the intercepted arcs.
    • Tangent to a circle – line, ray, or line segment perpendicular to the radius and intersecting the circle in exactly one point, the point of tangency
      • Intersecting tangent segments are equal in length from the same point outside the circle.
      • Measure of angle formed by the intersection of two tangents outside the circle is the difference of the measures of the intercepted arcs.
    • Secant to a circle – line, ray, or line segment that intersects the circle in exactly two points
      • Product of the length of the secant segment and its external segment is equal to the product of the other secant segment and its external segment from the same point outside the circle.
      • Measure of angle formed by two secants intersecting at a point outside the circle is  the difference of the measures of the intercepted arcs.
    • Combinations of chords, secants, and tangents
      • Chord-Tangent
        • Measure of the angle formed by the intersection of a tangent and chord is the measure of the intercepted arc.
      • Secant-Tangent
        • Product of the length of a secant segment and its external part is equal to the product of the square of the length of a tangent segment intersecting the secant segment at a point outside the circle.
        • Measure of the angle formed by a secant and a tangent intersecting at the same point outside the circle is the difference of the measures of the intercepted arcs.
  • 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 θ = , where θ 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: 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 + (yk)2 = r2

Including, but not limited to:

  • General equation for a circle with center (h, k) and radius of length r: (x h)2 + (yk)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
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.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
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 11/15/2018
Loading
Data is Loading...