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Precalculus
TITLE : Unit 13: Conics SUGGESTED DURATION : 16 days

Unit Overview

This unit bundles student expectations that address determining, analyzing, and writing equations for the conic sections formed when a plane intersects a double-napped cone. Connections are made between the locus definitions of the conic sections and their equations in rectangular coordinates. Additionally, converting between parametric equations and rectangular relations for conic sections and using these representations to model and solve problems are also included. 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 Geometry Unit 07, students identified the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres. In Geometry Units 02 and 07, students derived and used the distance formula to verify geometric relationships. In Geometry Unit 06, students showed that the equation of a circle with center at the origin and radius r is x2 + y2 = r2 and wrote the equation for a circle centered at (h, k) with radius r using (xh)2 + (yk)2 = r2. In Algebra II Unit 05, students wrote the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening. In Precalculus Unit 11, students graphed parametric equations, used parametric equations to model and solve problem situations, and converted between parametric equations and rectangular relations for various linear and parabolic situations.

During this unit, students explore the cross-sections of a double-napped cone in order to determine the four conic sections formed when a plane intersects a double-napped cone. Additionally, students explore the special cases of the intersection of a plane and a double-napped cone yielding the degenerate conic sections. Students determine the general characteristics of the four types of conic sections (circles, parabolas, ellipses, and hyperbolas), including shape, cross-section conditions, key attributes (vertex/vertices, asymptotes, etc.), equations (for circles and parabolas), and real-world applications. Students explore the locus definition of a circle and make connections to its equation in rectangular coordinates. Specifically, students connect the distance formula to the equation of a circle. Students write equations of circles in both mathematical and real-world problem situations. Students explore the locus definition of an ellipse and make connections to its equation in rectangular coordinates. Specifically, students use the distance formula to determine the general equation of an ellipse. Students write rectangular equations of ellipses using their various characteristics, including major axis, semi-major axis, minor axis, semi-minor axis, focal length, coordinates for the center, coordinates for the foci, and coordinates for the vertices. Students write equations of ellipses in both mathematical and real-world problem situations. Students explore the locus definition of a hyperbola and make connections to its equation in rectangular coordinates. Specifically, students explore the conditions that give rise to a hyperbola and then use the distance formula to determine the general equation of a hyperbola. Students write rectangular equations of hyperbolas using their various characteristics, including focal length, coordinates for the center, coordinates for the foci, coordinates for the vertices, distance from the center to the vertices, slopes of the linear asymptotes, and equations for the linear asymptotes. Students write equations of hyperbolas in both mathematical and real-world problem situations. Students explore the locus definition of a parabola and make connections to its equation in rectangular coordinates. Specifically, students explore how the distance formula yields the equation of a parabola. Students write equations of parabolas in both mathematical and real-world problem situations. Students convert rectangular relations for circles, parabolas, ellipses, and hyperbolas into their parametric equations, then convert parametric equations for these conics into their rectangular relations. Students model real-world situations using rectangular and parametric equations of ellipses and hyperbolas and use these equations to solve problems.

After this unit, in subsequent mathematics courses, students will continue to apply conic sections as they arise in problem situations.

Algebraic and geometric reasoning serve an integral role in college readiness. Translating among multiple representations of equations and making connections between geometry and algebra are emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning B1, C1, D2; III. Geometric Reasoning C1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2000), students in grades 9-12 should “understand relations and functions and select, convert flexibly among, and use various representations for them” (p. 296). Additionally students should use visualization, spatial reasoning, and geometric modeling to solve problems including analyzing the cross-sections of three-dimensional objects (NCTM, 2000). Students should be provided with rich mathematical settings in which they can hone their visualization skills in order to use visualization as a problem-solving tool. Visualizing the shape of a conic section formed by the intersection of a plane and a cone can be facilitated by the use of physical models, drawings, and software capable of manipulating three-dimensional objects (NCTM, 2000). Herman (2012) describes how the modern study of conics is considered analytic geometry where the conic sections represent graphs of certain types of equations that are defined in terms of distance relationships between specific points and/or lines. Brown and Jones (2005) advocate for the use of two different coordinate systems which are well suited for graphing conic sections and exploring these distance relationships: a focus-directrix coordinate system and a focus-focus coordinate system. Using these alternative coordinate systems allows students to explore the definitions of the conic sections in a way that may be masked by using equations or technology to plot points alone (Brown & Jones, 2005). The study of conic sections continues into calculus, where students analyze conics and other implicitly defined curves using implicit differentiation (College Board, 2012).

Brown, E. M., & Jones, E. (2005). Understanding conic sections using alternate graph paper. Mathematics Teacher, 99(5), 322-327.
College Board.  (2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf.
Herman, M. (2012). Exploring conics: Why does b2 – 4ac matter? Mathematics Teacher, 105(7), 526-532.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

OVERARCHING UNDERSTANDINGS and QUESTIONS

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

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

 

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?
  • Why do some relations define a function?
  • Why can function models describe how two variable quantities change in relation to one another?

 

Equations can be used to model and solve problem situations.

  • Why are equations used to model problem situations?
  • How are equations used to model problem situations?
  • How are equations used to solve problem situations?
  • Why is it essential to solve equations using various methods?
  • How can solutions to equations be represented?
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

  • Equations
  • Multiple Representations
  • Relations

 

Geometric Reasoning

  • Conics

 

Associated Mathematical Processes

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

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

  • Why are models and diagrams necessary for visualizing the geometric relationships found in the problem situation?
  • How are models and diagrams used to organize information from the problem situation?
  • How do models and diagrams aid in calculations when solving problems?
  • Why is the coordinate plane used to diagram two-dimensional figures?

 

Conic sections have unique characteristics and can be represented in multiple ways, including as rectangular relations.

  • What are the conic sections?
  • How can conic sections be formed by the intersection of a plane with a double-napped cone?
  • What are the degenerate conic sections, and how are they formed by the intersection of a plane with a double-napped cone?
  • How can conic sections be represented using graphs?
  • How can conic sections be represented algebraically as rectangular relations?

 

A circle is one of the four basic forms of conic sections.

  • How is a circle defined as a locus of points? Explain.
  • How does the locus definition of a circle connect to the equation of a circle in rectangular coordinates?

 

A parabola is one of the four basic forms of conic sections.

  • How is a parabola defined as a locus of points? Explain.
  • How does the locus definition of a parabola connect to the equation of a parabola in rectangular coordinates?

 

An ellipse is one of the four basic forms of conic sections and can be represented by the equation   = 1.

  • How is an ellipse defined as a locus of points? Explain.
  • How does the locus definition of an ellipse connect to the equation of an ellipse in rectangular coordinates?
  • What are the characteristics of ellipses?
  • How can the characteristics of an ellipse be determined from the graph of the ellipse?
  • How can the characteristics of an ellipse be determined from the equation of the ellipse?
  • How can the characteristics of an ellipse be used to write the equation of an ellipse?
  • What situations can be modeled and solved by ellipses?

 

A hyperbola is one of the four basic forms of conic sections and can be represented by the equation  or  .

  • How is a hyperbola defined as a locus of points? Explain.
  • How does the locus definition of a hyperbola connect to the equation of a hyperbola in rectangular coordinates?
  • What are the characteristics of hyperbolas?
  • How can the characteristics of a hyperbola be determined from the graph of the hyperbola?
  • How can the characteristics of a hyperbola be determined from the equation of the hyperbola?
  • How can the characteristics of a hyperbola be used to write the equation of a hyperbola?
  • What situations can be modeled and solved by hyperbolas?

 

Parametric equations have unique characteristics and can be represented in multiple ways, including as rectangular relations.

  • What are the characteristics of parametric equations?
  • How do the characteristics of parametric equations and their corresponding rectangular relations compare?
  • How can parametric equations be used to represent functions and relations?
  • How can parametric equations be represented using tables?
  • How can parametric equations be represented using graphs?
  • How can parametric equations be represented algebraically?
  • How can parametric equations be converted into rectangular relations?
  • How can rectangular relations be converted into parametric equations?

 

Parametric equations can be used to model and solve mathematical real-world problem situations.

  • How are parametric equations used to model problem situations?
  • What type of problem situations can be modeled by parametric equations?
  • How can parametric equations be formulated to represent problem situations?
  • What methods can be used to solve problem situations involving parametric equations?
  • How is the reasonableness of solutions justified in problem situations?
  • Why must the reasonableness of solutions be justified in problem situations?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think that only four types of conic sections can be formed by the intersection of a plane with a double-napped cone: circles, parabolas, ellipses, and hyperbolas. Students may forget about the degenerate conic sections formed by a plane passing through the vertex of double-napped cones.
  • Students may confuse the Pythagorean relationships between a, b, and c for ellipses (c2 =|a2b2|) and for hyperbolas (c2 = a2b2).
  • Some students may believe that hyperbolas are made up of two symmetric parabolas. However, parabolas have different locus definitions than the branches of hyperbolas and therefore represent different geometric shapes.
  • Some students may mistakenly graph a hyperbola with an incorrect axis. For example, students may mistakenly graph the hyperbola  = 1 with a vertical axis rather than a horizontal axis.
  • Some students may incorrectly graph parametric equations if they use a calculator in the wrong angle mode.
  • When graphing parametric equations, some students may only graph the path of the parametric equations without noting specific t values along this path.

 

Underdeveloped Concepts:

  • 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.
  • Some students may fail to take the square root of the a2 or b2 portion of the equation of an ellipse or a hyperbola when determining the values of a and b.

Unit Vocabulary

Related Vocabulary:

  •   Asymptote
  •   Axis
  •   Axis of symmetry
  •   Center
  •   Circle
  •   Conic section
  •   Convert
  •   Coordinates
  •   Cross-section
  •   Degenerate conic section
  •   Directrix
  •   Distance
  •   Distance formula
  •   Double-napped cone
  •   Ellipse
  •   Equation
  •   Focal length
  •   Focus
  •   Horizontal
  •   Hyperbola
  •   Intersecting lines
  •   Intersection
  •   Line
  •   Locus
  •   Major axis
  •   Minor axis
  •   Parabola
  •   Parameter
  •   Parametric equations
  •   Parallel
  •   Path
  •   Perpendicular
  •   Point
  •   Radius
  •   Rectangular coordinates
  •   Rectangular relation
  •   Semi-major axis
  •   Semi-minor axis
  •   Slope
  •   Standard form
  •   Vertex
  •   Vertical
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 Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

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)
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.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
P.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
P.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
P.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
P.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
P.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
P.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
P.3 Relations and geometric reasoning. The student uses the process standards in mathematics to model and make connections between algebraic and geometric relations. The student is expected to:
P.3B Convert parametric equations into rectangular relations and convert rectangular relations into parametric equations.

Convert

PARAMETRIC EQUATIONS INTO RECTANGULAR RELATIONS

Including, but not limited to:

  • Algebraic methods
    • Solving one equation in a set of parametric equations (either x(t) or y(t)) for the parameter t
    • Substituting the expression for t into the other equation (either y(t) or x(t))
    • Applying trigonometric identities to eliminate the parameter t

Convert

RECTANGULAR RELATIONS INTO PARAMETRIC EQUATIONS

Including, but not limited to:

  • Rectangular relations into general parametric equations
    • Rectangular functions of the form y = f(x)
      • Letting x = t
      • Writing y as y = f(t)
    • Rectangular relations of the form x = f(y)
      • Letting y = t
      • Writing x as x = f(t)
    • Rectangular relations, given with specific information about x(t) or y(t)
      • Write an expression to describe x(t) or y(t)
      • Substitute the expression for x or y into the given rectangular relation
    • Rectangular relations of circles, ellipses, and hyperbolas
      • Circles
        • Applying the trigonometric identity sin2t + cos2t = 1 to parameterize a circle
      • Ellipses
        • Applying the trigonometric identity sin2t + cos2t = 1 to parameterize an ellipse
      • Hyperbolas
        • Applying the trigonometric identity sec2t – tan2t = 1 to parameterize a hyperbola

Note(s):

  • Grade Level(s):
    • Algebra I solved literal equations for a specified variable.
    • Algebra II used the composition of two functions.
    • Precalculus extends these skills to convert between parametric and rectangular equations.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • II. Algebraic Reasoning
      • C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
      • D2 – Translate among multiple representations of equations and relationships.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
P.3C Use parametric equations to model and solve mathematical and real-world problems.

Use

PARAMETRIC EQUATIONS

Model, Solve

MATHEMATICAL AND REAL-WORLD PROBLEMS USING PARAMETRIC EQUATIONS

Including, but not limited to:

  • Conics
    • Circles
      • General form: 
      • Variables and constants
        • r (radius)
        • (hk) (center of circle)
    • Ellipses
      • General form: 
      • Variables and constants
        • a (horizontal semi-axis)
        • b (vertical semi-axis)
        • (hk) (center of ellipse)
    • Hyperbolas
      • Hyperbola with a horizontal axis
        • General form: 
        • Variables and constants
          • a (distance from center to vertices)
          • b (where ± represents the slopes of the linear asymptotes)
          • (hk) (center of hyperbola)
      • Hyperbola with a vertical axis
        • General form: 
        • Variables and constants
          • a (where ± represents the slopes of the linear asymptotes)
          • b (distance from center to vertices)
          • (hk) (center of hyperbola)
    • Applications

Note(s):

  • Grade Level(s):
    • Geometry applied the trigonometric ratios (sine, cosine) to determine side lengths and angle measures in triangles.
    • Algebra I analyzed the features of linear and quadratic functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • II. Algebraic Reasoning
      • D2 – Translate among multiple representations of equations and relationships.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections 
P.3F Determine the conic section formed when a plane intersects a double-napped cone.

Determine

THE CONIC SECTION FORMED WHEN A PLANE INTERSECTS A DOUBLE-NAPPED CONE

Including, but not limited to:

  • Circle
    • Figure: a closed curve
    • Intersection: The plane is perpendicular to the symmetry axis of the double-napped cone.
    • Nappes: The plane intersects only one nappe of the double-napped cone.
  • Parabola
    • Figure: an unbounded curve
    • Intersection: The plane is parallel to an edge of the double-napped cone.
    • Nappes: The plane intersects only one nappe of the double-napped cone.
  • Ellipse
    • Figure: a closed curve
    • Intersection: The plane is neither parallel to an edge, nor perpendicular to the symmetry axis, of the double-napped cone.
    • Nappes: The plane intersects only one nappe of the double-napped cone.
  • Hyperbola
    • Figure: two unbounded curves
    • Intersection: The plane is parallel to the symmetry axis of the double-napped cone.
    • Nappes: The plane intersects both nappes of the double-napped cone.
  • Degenerate conic sections
    • A single point
      • Intersection: The plane passes through the apex at an angle greater than the angle made by the symmetry axis and the edge of the cone.
    • A single line
      • Intersection: The plane passes through the apex and is tangent to the edges of the double-napped cone.
    • A pair of lines
      • Intersection: The plane passes through the apex at an angle less than the angle made by the symmetry axis and the edge of the cone.
  • Applications
    • Determine if objects in art and architecture can be formed using a cross-section of a double-napped cone.

Note(s):

  • Grade Level(s):
    • Geometry identified the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, spheres, and cones.
    • Precalculus extends this knowledge to the double-napped cone and includes shapes such as the ellipse, hyperbola and parabola.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections 
P.3G Make connections between the locus definition of conic sections and their equations in rectangular coordinates.

Make

CONNECTIONS BETWEEN THE LOCUS DEFINITION OF CONIC SECTIONS AND THEIR EQUATIONS IN RECTANGULAR COORDINATES

Including, but not limited to:

  • Circles
    • Locus definition of a circle
      • A circle is the set of points (xy) in a plane such that (xy) is a distance of r units from a center point (hk).
    • Application of the distance formula and its transformation to the general rectangular equation for a circle
  • Ellipses
    • Locus definition of an ellipse
      • For two points in a plane (a focus at F1 and another focus at F2), an ellipse is the set of points P at (xy) such that the sum of the distances from P to each focus is constant (or PF1 + PF2 = 2a, where a is a constant).
    • Application of the distance formula and its transformation to the general rectangular equation for an ellipse
  • Parabolas
    • Locus definition of a parabola
      • For a point (the focus, F) and a line (called the directrix), a parabola is the set of all points (xy) such that the distance from F to (xy) is equivalent to the distance from (xy) to the directrix.
    • Application of the distance formula and its transformation to the general rectangular equation for a parabola
  • Hyperbolas
    • Locus definition of a hyperbola
      • For two points (a focus at F1 and another focus at F2), a hyperbola is the set of points P at (xy) such that the difference of the distances from P to each focus is constant (or |PF1 - PF2| = 2a, where a is a constant).
    • Application of the distance formula and its transformation to the general rectangular equation for a hyperbola

Note(s):

  • Grade Level(s):
    • Geometry used the distance formula to verify geometric relationships.
    • Geometry developed and used the equation of a circle, (x – h)2 + (y – k)2 = r2, centered at (hk) with radius r.
    • Algebra II developed the equation for a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
    • Precalculus extends this knowledge to include the other conic sections (ellipses and hyperbolas).
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • II. Algebraic Reasoning
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections 
P.3H Use the characteristics of an ellipse to write the equation of an ellipse with center (h, k).

Use

THE CHARACTERISTICS OF AN ELLIPSE

Including, but not limited to:

  • Characteristics of an ellipse
    • Major axis
    • Semi-major axis
    • Minor axis
    • Semi-minor axis
    • Focal length
    • Coordinates for the center
    • Coordinates for the foci
    • Coordinates for the vertices

Write

THE EQUATION OF AN ELLIPSE WITH CENTER (hk)

Including, but not limited to:

  • Using the general equation for an ellipse
    • Ellipse with a horizontal major axis:  = 1, with a > b
    • Ellipse with a vertical major axis:  = 1, with b > a
  • Given a combination of two or more characteristics of an ellipse
    • Major axis, 2(horizontal) or 2b (vertical)
    • Semi-major axis, (horizontal) or b (vertical)
    • Minor axis, 2(horizontal) or 2b (vertical)
    • Semi-minor axis, (horizontal) or b (vertical)
    • Focal length, c
    • Equation relating focal length and the length of each semi-axis, |a2 – b2| = c2
    • Coordinates for the center, (hk)
    • Coordinates for the foci
      • Ellipse with a horizontal major axis: (h ± ck)
      • Ellipse with a vertical major axis: (hk ± c)
    • Coordinates for the vertices
      • On the horizontal axis: (h ± ak)
      • On the vertical axis: (hk ± b)
    • Description of effects of changes in characteristics of an ellipse on the represetnative equation
  • Use conic sections to solve problems involving modeling.

Note(s):

  • Grade Level(s):
    • Geometry wrote the equation for a circle centered at (hk) with radius r using (x – h)2 + (y – k)2 = r2.
    • Algebra II wrote the equation for a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
    • Precalculus extends this knowledge to write equations for the other conic sections (ellipses and hyperbolas).
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • II. Algebraic Reasoning
      • D2 – Translate among multiple representations of equations and relationships.
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections 
P.3I Use the characteristics of a hyperbola to write the equation of a hyperbola with center (h, k).

Use

THE CHARACTERISTICS OF A HYPERBOLA

Including, but not limited to:

  • Characteristics of a hyperbola
    • Coordinates for the center
    • Distance from the center to the vertices
    • Coordinates for the vertices
    • Slopes of the linear asymptotes
    • Equations for the linear asymptotes
    • Focal length
    • Coordinates for the foci

To Write

THE EQUATION OF A HYPERBOLA WITH CENTER (hk)

Including, but not limited to:

  • Using the general equation for a hyperbola
    • Hyperbola with a horizontal axis:  = 1
    • Hyperbola with a vertical axis: = 1
  • Given a combination of two or more characteristics of a hyperbola
    • Coordinates for the center, (hk)
    • Distance from the center to the vertices, a (horizontal) or b (vertical)
    • Coordinates for the vertices
      • Hyperbola with a horizontal axis: (h ± ak)
      • Hyperbola with a vertical axis: (hk ± b)
    • Slopes of the linear asymptotes, ±
    • Equations for the linear asymptotes, y – k = ± (– h)
    • Focal length, c
    • Equation relating focal length to a and b, a2 + b2 = c2
    • Coordinates for the foci
      • Hyperbolas with a horizontal axis: (h ± ck)
      • Hyperbolas with a vertical axis: (hk ± c)
    • Description of effects of changes in characteristics of a hyperbola on the representative equation
  • Use conic sections to solve problems involving modeling.

Note(s):

  • Grade Level(s):
    • Geometry wrote equations for circles centered at (hk) with radius r using (x – h)2 + (y – k)2r2.
    • Algebra II wrote the equation for a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.
    • Precalculus extends this knowledge to write equations for the other conic sections (ellipses and hyperbolas).
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • II. Algebraic Reasoning
      • D2 – Translate among multiple representations of equations and relationships.
    • III. Geometric Reasoning
      • 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 08/24/2016
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