Common Core is written to require that students examine conics from two perspectives: geometric descriptions (graphical representations) and equations. If students only cover the required material, teachers will not allow their pupils the opportunity to explore more than the bare minimum. Even when some teachers do include the origin of conic sections as being derived from the intersection of a plane and a cone, little is done to develop connections between three-dimensional representations, graphs in a Cartesian plane, and their equations.
The Common Core Standards associated with conics appear in high school mathematics as:
Expressing Geometric Properties with Equations (G.GPE)
Translate between the geometric description and the equation for a conic section
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.
2. Derive the equation of a parabola given a focus and directrix.
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.
Although it does not appear in the same set of standards, the section on Geometric Measurement and Dimension can be utilized in introductory lessons to conic sections. This standard is as follows:
Geometric Measurement and Dimension (G.GMD)
Visualize relationships between two‐dimensional and three‐dimensional objects
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
To incorporate a richer introduction of conic sections, instructors may begin by reviewing two-dimensional cross-sections of three-dimensional figures. This concept first appears in the middle school curriculum; therefore, teachers should feel confident that students have already mastered it. After a re-introduction to the two-dimensional cross-sections of more basic three-dimensional figures (e.g. cubes, prisms, cylinders, and pyramids), the class can focus its attention on cones (eventually double-napped cones).
The intersection of a plane with a double-napped cone can be represented using the graphing calculator function on a computer, images and animations through web searches, physical manipulatives, and a flashlight.
Teachers may also want to provide a basic history of conic discoveries by Ancient Greek mathematicians, such as Menaechmus, Euclid, Aristaeus, Archimedes, and of course Apollonius. From there, instructors can tie in astronomy by discussing Kepler discovery of elliptical orbits of the planets. This historical perspective may be used as a hook to gather student interest.
The standards require that students master how to derive the equations of circles, parabolas, ellipses, and hyperbolas given specific guidelines. It is important that students also are able to derive the equation of each conic section by other means than just what is given in the Common Core standards in order to make connections and build greater conceptual understanding.
Instructors should ensure that students can graph each figure, label the appropriate parts of that graph, and make connections between the equations and the graphs. With the availability of applets and other technology, it is a waste for students to only graph each conic section using paper and pencil. Desmos, graphing calculators, and Geometer's Sketchpad are all exciting media for investigating how to construct, manipulate, and explore conics in ways that were unavailable decades ago.
In the classroom, the standards should be viewed as the floor of knowledge. This means that instructors should teach their students no less that what is stated in the standards, but should always attempt to provide more tools and deeper knowledge. The wider variety of material a student experiences, the greater the chance of an individual realizing her true passion.
For additional resources and potential lesson ideas, please visit:
Conic Flyer
Conic Section Explorer
Conic Section Lesson Plan-Math and Science Center
Cutting Conics
Flashlights and Conic Sections
Human Conics
Instructional Unit: Conic Sections
References
High School: Geometry » Expressing Geometric Properties with Equations. Home. Retrieved May 21, 2014, from http://www.corestandards.org/Math/Content/HSG/GPE/
High School: Geometry » Geometric Measurement & Dimension. Home. Retrieved May 21, 2014, from http://www.corestandards.org/Math/Content/HSG/GMD/
The Common Core Standards associated with conics appear in high school mathematics as:
Expressing Geometric Properties with Equations (G.GPE)
Translate between the geometric description and the equation for a conic section
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.
2. Derive the equation of a parabola given a focus and directrix.
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.
Although it does not appear in the same set of standards, the section on Geometric Measurement and Dimension can be utilized in introductory lessons to conic sections. This standard is as follows:
Geometric Measurement and Dimension (G.GMD)
Visualize relationships between two‐dimensional and three‐dimensional objects
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
To incorporate a richer introduction of conic sections, instructors may begin by reviewing two-dimensional cross-sections of three-dimensional figures. This concept first appears in the middle school curriculum; therefore, teachers should feel confident that students have already mastered it. After a re-introduction to the two-dimensional cross-sections of more basic three-dimensional figures (e.g. cubes, prisms, cylinders, and pyramids), the class can focus its attention on cones (eventually double-napped cones).
The intersection of a plane with a double-napped cone can be represented using the graphing calculator function on a computer, images and animations through web searches, physical manipulatives, and a flashlight.
Teachers may also want to provide a basic history of conic discoveries by Ancient Greek mathematicians, such as Menaechmus, Euclid, Aristaeus, Archimedes, and of course Apollonius. From there, instructors can tie in astronomy by discussing Kepler discovery of elliptical orbits of the planets. This historical perspective may be used as a hook to gather student interest.
The standards require that students master how to derive the equations of circles, parabolas, ellipses, and hyperbolas given specific guidelines. It is important that students also are able to derive the equation of each conic section by other means than just what is given in the Common Core standards in order to make connections and build greater conceptual understanding.
Instructors should ensure that students can graph each figure, label the appropriate parts of that graph, and make connections between the equations and the graphs. With the availability of applets and other technology, it is a waste for students to only graph each conic section using paper and pencil. Desmos, graphing calculators, and Geometer's Sketchpad are all exciting media for investigating how to construct, manipulate, and explore conics in ways that were unavailable decades ago.
In the classroom, the standards should be viewed as the floor of knowledge. This means that instructors should teach their students no less that what is stated in the standards, but should always attempt to provide more tools and deeper knowledge. The wider variety of material a student experiences, the greater the chance of an individual realizing her true passion.
For additional resources and potential lesson ideas, please visit:
Conic Flyer
Conic Section Explorer
Conic Section Lesson Plan-Math and Science Center
Cutting Conics
Flashlights and Conic Sections
Human Conics
Instructional Unit: Conic Sections
References
High School: Geometry » Expressing Geometric Properties with Equations. Home. Retrieved May 21, 2014, from http://www.corestandards.org/Math/Content/HSG/GPE/
High School: Geometry » Geometric Measurement & Dimension. Home. Retrieved May 21, 2014, from http://www.corestandards.org/Math/Content/HSG/GMD/
