# Common Core High School Geometry Teaching and Learning Resources

## Domain: GPE: Expressing Geometric Properties with Equations

Select the cluster for resources on this page:

• GPE-A: Translate between the geometric description and the equation for a conic section

• GPE-B: Use coordinates to prove simple geometric theorems algebraically

### GPE-A: Translate between the geometric description and the equation for a conic section

Standards:

• GPE-A.1. Derive the equation of a circle of given center and radius using the Pythagorean. Theorem; complete the square to find the center and radius of a circle given by an equation.
• GPE-A.2. Derive the equation of a parabola given a focus and directrix.
• GPE-A.3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with GPE-A and GPE-B:

 Conic Sections Ellipse Directrices of Hyperbola Point of Division Formula Circle Foci of Ellipse Asymptote Midpoint formula Parabola Directrices of Ellipse Analytic (Coordinate) Geometry Perpendicular Lines Focus of Parabola Hyperbola Coordinate Parallel Lines Directrix of Parabola Foci of Hyperbola Distance formula Slope of a Line

CK-12: Geometry: Circles in the Coordinate Plane: Concepts for writing the equation of the circle, video, and practice problems for learners to complete.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

• Lesson set: Understand and use equations for circles: Three video lessons: How to derive the equation of a circle using the Pythagorean Theorem, find the center and radius of a circle, and how to change an equation of a circle into its standard form using the completing the square method.  Aligns with GPE-A.1.
• Lesson set: Derive the equation of a circle using the Pythagorean Theorem: Four video lessons: Define a circle, derive the equation of the circle centered at the origin or (h,k), translate the equation of a circle from general form to standard form.  Aligns with GPE-A.1.
• Lesson set: Derive the equation of a circle, and complete the square to find the center and radius: Six video lessons: Find distances between points using the distance formula, relate points on a circle to the center, understand the standard form equation for circles, complete the square, solve quadratic equations by completing the square, find the center and radius of a circle by completing the square.  Aligns with GPE-A.1.

Thinking Mathematics: The Equation of a Circle: A video on YouTube.

A. Dendane: Analyze Math: Select the following, which align with GPE-A:

• Equation of a Circle: HTML5 Applet to explore the equation of a circle in standard center-radius form and the properties of the circle.
• Tutorial on Equation of a Circle: Tutorials with detailed solutions to examples and matched exercises on finding equation of a circle, radius and center. Detailed explanations are also provided. Aligns with GPE-A.1.
• Construct a Parabola: This applet explores the use of the definition of the parabola to create its graph from a directrix and focus.
• Equation of a Parabola: This is an applet to explore the equation of a parabola and its properties. The equation used is the standard equation that has the form (y - k)2= 4a(x - h).  Related applications of the parabola are included.
• Equation of Hyperbola: The equation and properties of a hyperbola are explored interactively using an applet.  Learners can investigate the hyperbola definition, foci, vertices, and asymptotes. The equation used has the form x2/a2 - y2/b2 = 1

Math Open Reference:

• Ellipse: This applet visually shows the generation of the ellipse from its definition.  Content includes major properties of the ellipse.
• General Equation of the Ellipse: This applet shows changes in the equation of an ellipse by manipulating the center, and lengths of major and minor axes.

Math Warehouse:

PhET Interactive Simulations: Graphing Quadratics. Explore including standard form, vertex form, and focus/directrix.  Aligns with IF-C.7, IF-C.7.a, IF-C.8, IF-C.8.a, and GPE-A.2.

Purple Math: These lessons focus on finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conic equations to solve typical word problems:

Shodor Interactivate:

• Conic Flyer Equations: This is a lesson using the geometric interpretations of the various conic sections to explain their equations.  Aligns with GPE-A.
• Conic Flyer is a virtual manipulative for learners to "manipulate different types of conic section equations on a coordinate plane using slider bars. Learn how each constant and coefficient affects the resulting graph. Choose from vertical or horizontal parabola, circle, ellipse, and vertical or horizontal hyperbola."  Aligns with GPE-A.
• Cross Sections: This lesson utilizes the concepts of cross-sections of three-dimensional figures to demonstrate the derivation of two-dimensional shapes.  Among objectives are that students learn the difference between ellipses, parabolas, hyperbolas, and circles as they relate to conic sections.  Aligns with GPE-A.
• Cross Section Flyer: Use this virtual manipulative to explore cross sections of different geometric solids: cone, double cone, cylinder, pyramid, and prism. Manipulate the cross section with slider bars, and see how the graphical representation changes.  Aligns with GPE-A.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 1 manipulative addressing GPE-A.1, 7 manipulatives addressing GPE-A.2 and 11 manipulatives addressing GPE-A.3.  Among those:

Multiple Choice:

Constructed-response:

Algebra Lab:

• Conic Sections: Circles: Lesson on deriving the equations of circles and relating those to their graphs.  Learners see how a circle is formed by passing a plane through a cone.  Worked examples are presented, followed by problems for learners to complete.  Aligns with GPE-A.1.
• Writing the Equation of a Circle: 10 practice problems for learners to complete given the center and radius. Aligns with GPE-A.1.
• Focus and Directrix: Lesson in which learners see a worked example for deriving the equation of the parabola from its definition using a focus and directrix and worked examples involving the equation of the parabola and finding the vertex, focus, and directrix.  They then complete associated problems. Aligns with GPE-A.2.
• Conic Sections: Ellipses: Lesson in which learners see an ellipse formed by a plane passing through a cone and a specific derivation of the equation of an ellipse with given foci and constant sum of distances from a point on the ellipse to the foci.  This helps learners better relate to a general derivation of the equation of the ellipse from its definition.  Several worked examples are provided for writing equations of the ellipse and identifying its properties.  These are followed by problems for learners to complete.  Aligns with GPE-A.3.
• Conic Sections: Hyperbolas: Lesson in which learners see how the hyperbola is formed by a plane passing through a cone.  The general derivation of equation of the hyperbola is presented, along with worked examples of writing equations and identifying properties.  Learners then complete related problems.  Aligns with GPE-A.3.

Khan Academy: HSG-G-GPE includes HSG-GPE.A.1 through HSG-GPE.A.3 Common Core aligned problems.

< p>Also see the Conic Sections unit with practice questions with videos.

Georgia Standards of Excellence Framework: Pre-Calculus Unit 6: Conics.  This unit contains eight tasks: Our Only Focus: Circles and Parabolas Review, The Focus is the Foci: Ellipses and Hyperbolas, Deriving the General Equation of a Parabola, Parabolas in Other Directions, Writing the Equations of Parabolas, The Intersection of a Line and a Quadratic, A Conic Application, and Culminating Task: Dr. Cone's New House.  Tasks align with GPE-A.1, GPE-A.2, and GPE-A.3, and algebra standard REI-C.7.

Illustrative Mathematics: Geometry:

Mathematics Assessment Project: High School: Geometry: Task 406: Equations of Circles 1

Mathematics Vision Project, Secondary 2 Student Edition:

### GPE-B: Use coordinates to prove simple geometric theorems algebraically

Standards:

• GPE-B.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
• GPE-B.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
• GPE-B.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
• GPE-B.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Technology-enhanced investigations:

Imagine Learning Classroom (formerly LearnZillion) subscription needed: Lesson set: Apply the coordinate plane to geometry: Two video lessons: How to prove whether a figure is a rectangle in the coordinate plane and whether a point is on a circle.  Aligns with GPE-B.4.

Math Open Reference: Coordinate Geometry includes the introduction to coordinate geometry, and addresses multiple topics in coordinate geometry related to points, lines, planes, polygons, circles, and ellipses.

Math Warehouse: Coordinate Geometry: Using the coordinate plane in proofs: This is an interactive demonstrating how to do coordinate proofs.  Do the proofs (perpendicular segments, trapezoid is not isosceles, and rectangle, parallelogram) and see the answers.  Aligns with GPE-B.4.

National Math and Science Initiative: Lesson: Using Linear Equations to Define Geometric Solids: Use this lesson for geometry within a unit on volume applications.  Students write the equations of lines and graph linear equations that define bounded regions, calculate the areas and perimeters of the regions, revolve the planar regions about horizontal and vertical lines to create solids, and calculate the volumes of the resulting solids; they use technology (virtual manipulatives) to create and revolve planar figures about horizontal and vertical lines and model the conceptual understanding using a real world situation.  Targeted standards include GMD-A.3, GMD-B.4, GPE-B.7 and MG-A.1.  Common Core Math Practice standards 1-7 are also addressed.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 2 manipulatives addressing GPE-B.4, 1 manipulative addressing GPE-B.5, and 4 manipulatives addressing GPE-B.7.  Among those:

Multiple Choice:

Khan Academy: Practice questions with videos.

Constructed-response:

AlgebraLab.org: Algebra I: Distance and Midpoint Formulas.  Study aids with examples and practice problems (short answer and T/F).  Given two ordered pairs, determine length of a segment and midpoint.  Given three ordered pairs, determine if the vertices form a right triangle.

Khan Academy: HSG-G-GPE includes HSG-GPE.B.4 through HSG-GPE.B.7 Common Core aligned problems.

Also see the following practice questions with videos.

L. Spector: The Math Page: Skill in Algebra.  Students see worked examples of concepts and following the presentation within each section, students are presented problems in constructed response form.  They would solve using paper-pencil and then by using their mouse to roll over a box, they see the solution to problems presented.

• Lesson 31: Rectangular Coordinates: Use with standard GPE-B.4.
• Lesson 32: Pythagorean Distance Formula: Distance of a point from the origin, distance between points, and proof of Pythagorean theorem.  Use with standard GPE-B.7.
• Lesson 34: The slope of a straight line: Includes the slope intercept form of the equation of a straight line, the general form; parallel and perpendicular lines; the point-slope formula and the two-point formula.  Use with standard GPE-B.5.

Illustrative Mathematics: Geometry:

Mathematics Assessment Project: High School: Geometry: Task 226: Finding Equations of Parallel and Perpendicular Lines

Mathematics Vision Project, Secondary 1 Student Edition:

• Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks.  Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5.  Task 2: Is It Right?; and Task 5: Leap Year align with GPE-B.5.
• Module 7: Connecting Algebra and Geometry: This module contains six classroom tasks.  Three tasks in Module 6 each address the function standards BF-A.1, BF-B.3, and IF-C.9.  The other three tasks address geometry standards GPE-B.4, GPE-B.5, and GPE-B.7.  Task 1: Go the Distance aligns with GPE-B.7.  Task 2: Slippery Slopes align with GPE-B.5.  Task 3: Prove It! aligns with GPE-B.4.

NCTM's Reasoning and Sense Making Task Library: As the Crow Flies "is designed to help students develop an understanding of the meaning of the [distance] formula. It would be appropriate as an introduction (or review) of the distance formula for students who are familiar with the Pythagorean theorem and coordinate systems."  Learners "compute the distance between two locations in a city with the streets laid out on an evenly spaced square grid." Aligns with GPE-B.4, GPE-B.7, and math practice standards MP-2, MP-4, and MP-7.