These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.

Cuero

Hecho en EE. UU. o importados

Caucho único

Con cada par de zapatos que compra, TOMS & reg; dará un nuevo par de zapatos para un niño necesitado One for One & reg;.

El TOMS & reg; Alpargata sin cordones zapato tiene un relajado, estilo fácil que está listo para todos los días desgaste.

Superior hecho de gamuza, chambray, lavado lienzo o lino.

Material se establece en el nombre del color.

Suave, lengua flexible y vuelta en el talón para una entrada fácil Plantilla textil suave proporciona luz amortiguación, soporte, y transpirabilidad. Flexible, de una sola pieza, caucho mezclado suela.Importado.Medidas: Peso: 6 oz Mediciones del producto fueron tomadas usando tamaño 7.5, ancho B - Medio. tenga en cuenta que las mediciones puede variar por tamaño

Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015

Mocasines casuales Metálico Alpargata TOMS Deconstructed de TOMS para mujer para Peltre Metálico Cuero 3169ceeUpdated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.

The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.

This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.

Includes GSP-3 files

Includes GSP-4 files

Includes GSP-5 files
Contents:

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.

Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.

Mocasines casuales Metálico Alpargata TOMS Deconstructed de TOMS para mujer para Peltre Metálico Cuero 3169cee

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

PUMA zapatillas de deporte Fenty x Bow Trinomic para mujer Sweet Lavender / Sweet Lavender , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Black Ditzy Daisy , Calzado de senderismo Nine West Women s Palyla de cuero Dark Natural Multi , Sansha Motion Dance Sneaker Negro rojo , Sansha Motion Dance Sneaker Negro rojo VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Negro Negro Sansha Motion Dance Sneaker Negro , CLARKS Mujer McKella Morris Sneaker Negro Zapatilla de deporte Blair Fashion de Naturalizer para mujer Negro CLARKS Womens Un Cruise Lace (Wave) Sneaker Cuero negro VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Pewter / Black NIKE Women s Free 5.0 Tr Fit 4 Print Negro / gris frío / gris lobo / blanco , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Negro / Blanco Gunsmoke Oil Grey , Zapatillas de deporte CVT-Hemp de mujer Vibram Negro , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Black Heritage Canvas , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Rose Gold Metallic Leather Mocasines casuales Alpargata Deconstructed de TOMS para mujer Hojas de flor CLARKS Womens Un Cruise Lace (Wave) Sneaker Combi de cuero blanco / arena nubuck , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Brillante Crimson / Ice Blue-sail , Sansha Tutto Nero Dance Sneaker Negro , NIKE Women s Free 5.0 Tr Fit 4 Print Fucsia Force / Light Magnet Gray / Hyper Pink / White Mocasines casuales Alpargata Deconstructed de TOMS para mujer Mustard Slub Chambray , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Desierto Taupe Gamuza , CLARKS Womens Un Cruise Lace (Wave) Sneaker Combi de cuero blanco / arena nubuck Mocasines casuales Alpargata Deconstructed de TOMS para mujer Sarga lavada azul pizarra , Zapatilla de deporte Blair Fashion de Naturalizer para mujer Grano de café , PUMA zapatillas de deporte Fenty x Bow Trinomic para mujer Plata Rosa / Plata Rosa / Plata Rosa , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Birch Retro Floral Sailor Moon Chibi Usa Cosplay Shoes Boots por encargo

trigonometry, and probability. Fair warning: The discussion revolves about
a figure that appears only when the sketch is animated. Without Geometer’s
Sketchpad, this one will make little sense at all.
Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.

The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.

Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.

Inversion Geometry New Balance Women s w990v4 Running Shoe Gris / rosa Sandalia Drew Angela para mujer Dusty Multi Metallic Sanda Verra Teva para mujer Surf Taupe Multi Calzado de trail running In-X-Talon 212 (S) para hombres Inov-8 Negro / Rojo Zapatillas de running New Zealand Kaymin Trail v1 Fresh Foam Trail para mujer Vintage Indigo , CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Faded Denim Sandalias Joanie II de mujer SOREL Harina de avena , Chaco Mujer Zcloud Sport Sandal Penny Black Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral Zapatillas para caminar RYKA Women s Devotion Plus Azul marino BareTraps Women s Dasie Platform Sandal Negro , Vionic Rest Farra - Sandalias de apoyo para mujeres Black-patent , Zapatillas de senderismo Merrell Moab 2 GTX para mujer Beluga , Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño Jambu Women s Spain Flat rojo , PUMA Mujer Ignite Flash Evoknit WN Sneaker Puma Black-asphalt-puma White , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Llovizna Gum Grey , Zapatillas para mujer Del Rey TOMS Gray Diamond Melange Vans Auténtico Brite Neon Verde / Azul , Vans Old Skool Unisex Adults Low-Top Trainers Control primario negro / blanco Converse Chuck Taylor All Star Core Hola Camo Green Sandalia Mason para mujer Naturalizer Slide Negro , Zapatilla unisex Veitsch Lodge de Giesswein rojo Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7017EB Cómodo-Muy Fino 3 \ de 5] Negro Bomba de Angel II para mujeres estilo suave Negro Cósmico Coach Womens Rhoda lona alpargata naranja Zapatillas Nike Women s Flex Trainer 5 Baya de té / plata metalizada / Burdeos , Hush Puppies Womens Chowchow WT Oxford Ante negro Bed | Stu Botines para mujer estridentes Tinte Di Moro Dip Dye , Mephisto Sabatina Tuxedo Oxford para mujer Patente negra

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.

Linkages

Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.

On the Square

The challenge: Given four arbitrary, unique points, *A*, *B*,
*C*, and *D*, construct one line through each point such that the four lines form a square.

Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.

The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.

Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.

Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.

Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.

The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.

Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.

Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.

Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured Mocasines casuales Metálico Alpargata TOMS Deconstructed de TOMS para mujer para Peltre Metálico Cuero 3169cee it out, and I got a chance to try out this new graphics software.

Resources

The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge
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and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

For email to reach me, the word