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Common Core High School Geometry Teaching and Learning Resources

Domain: GPE: Expressing Geometric Properties with Equations

Common Core Math

Select the cluster for resources on this page:

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



Technology-enhanced investigations: 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:

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

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

Math Open Reference:

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:

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:



Algebra Lab:

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.

Performance tasks:

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:


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



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: 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.

Performance tasks:

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:

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.


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