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.

Sintético

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

Importado

Adidas

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

Adidas Adizero B01LXQOANW Shotput Running Shoe Núcleo Adizero Running Negro, Ftwr Blanco, Naranja bf65718Updated 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.

Adidas Adizero B01LXQOANW Shotput Running Shoe Núcleo Adizero Running Negro, Ftwr Blanco, Naranja bf65718

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

Under Armour Women s Micro G Assert 6 SA Negro / Gris Glaciar Zapatillas de running Saucony Women s Triumph ISO 4 Negro / Aqua / Violeta Calzado de running FuelCore Rushv3 para mujer New Balance Oro blanco , I-RUNNER Elite Terapia atlética atlética extra Profundo Cuero / Velcro de malla Negro Zapatillas de running NIKE Juvenate para mujer Blanco negro , Zapatillas de running Ultra Kalani 3.0 de Zoot para mujer Negro / Rosa Brillo / Azul atómico Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial azul marino / colegiado azul marino / azul marino nocturno , NIKE Women s Tanjun Negro / Metálico Rojo Bronce , Zapatillas Nike Superflyte para mujer Blanco negro , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Blanco / Blanco / Negro , Zapatillas para correr A7200WX085 de Apex para mujeres Rosado NIKE Men s Roshe Run Obsidiana , NIKE Mujeres Air Zoom Pegasus 33 Pure Platinum / blanco-negro-voltio , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Negro , NIKE Women s Air Force 1 Mid 07 Le , Zapatillas deportivas 574S Sport New Balance para hombre Driftwood / White , Converse Chuck Taylor All Star Seasonal Color Hola Teal atómico oscuro Zapatillas de running NIKE Unisex Flyknit Racer Negro / azul / rosa , Zapatillas deportivas 574S Sport New Balance para hombre Blanco , Sandalia deportiva NIKE Kawa Slide para hombre Gris oscuro / blanco / gris lobo , NIKE Women s Tanjun Negro / Metálico Rojo Bronce NIKE Women s WMNS Zoom Fly, negro / blanco Equator Blue / White , Sandalias OLKAKAI Ohana - Hombre Sombra negra / oscura NIKE Men s Lunar Fingertrap Cross Trainer Binario Azul Blanco Negro Adidas Adizero Rush Womensg48876 Shagre / Corene / Metsil Zapato de running ASICS GEL-Venture 5 para mujer Frost Grey / Grey / Silver / Magenta Zapatillas de running New Balance Coast V3 para mujer Phantom / Overcast , Brooks Womens mazama 2 Rosa / ciruela / azul marino , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Volt / Black ,

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 Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Volt / Blue Lagoon / Blanco , Zapatillas de running adidas Women s Edge Lux 2 Blanco / Blanco / Plata Metálica Chaco Women s Z2 Classic Athletic Sandal Eclipse estático , Brooks Womens PureFlow 7 Gris / negro / verde , Converse Chuck Taylor All Star Seasonal Color Hola Piel de tiburon , Zapatillas para correr Saucony Women s Cohesion 10 Gris / verde azulado / cidra Zapatillas de running Adidas Performance Ultra s Boost M para hombre Negro / Negro / Negro Camiseta adidas Swift Run W para mujer Traza púrpura / Traza púrpura / Traza púrpura Zapato de running ASICS GT-2000 5 para mujer Regatta Blue / Flash Coral / Indigo Blue Hush Puppies Mujeres Power Walker Ii Loafer Cuero Taupe Zapatillas de running New Balance Boracay V3 para mujer Plata Metálica / Trueno , Sam Edelman Sandalia Dariel para mujer Avena de Gamuza , Zapatillas para caminar Ryka Women s Devo Plus 2 Azul marino / verde azulado Sandalia de cuero Camilia de TOMS para mujer Ante negro , WHITE MOUNTAIN Sandalia Gwinett plana para mujer Negro , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Negro , Entrenador cruzado 611v1 de New Balance para mujeres Artic Fox / Metallic , Sandalias de mujer Ros Hommerson Mellow Negro , Calzado de senderismo Go Joy 15603 de Skechers para mujer Armada nativo AP Mercury Liteknit Regatta Blue / Shell White / Natural Rubber Zapatillas Casual NIKE WMNS Lunar Force 1 Duckboot para mujer Sidra / Sidra-negro-gris pálido Mujer NIKE SF AIR FORCE 1 MID Verde al aire libre / Verde al aire libre , Zapato de golf NBGW1006 para mujer New Balance Azul , Sam Edelman - Sandalia de mujer Liliana Slide Denim Blue , Bernardo Women s Bridget Slide Sandal Guante de cemento , Calzado de senderismo Nine West Women s Palyla de cuero Dark Natural Multi , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro , Ray Rose Breeze 410 - Zapatos de baile de enseñanza de cuero y ante unisex (tacón de 1,5 pulgadas) Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 6027- Tan Satin & Flesh Mesh SOREL Women s Out N Acerca de las diapositivas Cantera / Natural

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 Adidas Adizero B01LXQOANW Shotput Running Shoe Núcleo Adizero Running Negro, Ftwr Blanco, Naranja bf65718 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
2197527

5431008

7238831

7954398

12749973

14646255

16527962

16994932

17044558

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