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.

Textil

Importado

Sintético único

Moda zapatilla de deporte con malla superior que ofrece superposiciones sintéticas con cordones Frente

Memoria-espuma plantilla

Non-marking suela

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

Skechers Sport Run Skech negra Air Run High fuerte Fashion Sneaker Adornos de malla negra/ rosa fuerte a3cd956Updated 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.

Skechers Sport Run Skech negra Air Run High fuerte Fashion Sneaker Adornos de malla negra/ rosa fuerte a3cd956

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

Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , NIKE Men s Roshe Two Running Shoe University Red / White / Track Red , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel Zapato de running ASICS Alpine XT para mujer Gris Pizarra / Gris Pizarra Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul , Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver Skechers Performance Women s Go Run 400 Obstruct Azul marino / Aqua , Zapatillas de running adidas Ultraboost W para mujer Misterio Rojo / Misterio Rojo / Rosa Táctil Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento , NIKE Women s Tanjun Iced Jade Aurora verde blanco Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo , NIKE Men s Benassi Just Do It Athletic Sandal Midnight Navy / Windchill , Nike Men s Air Presto Essential Universidad / Rojo / Negro / Negro Calzado de carretera Inov-8 Road-X-Treme 250 Gris / Negro / Rojo Pantalón largo clásico NIKE Golf para mujer Tech Essentials Binario Azul / Blanco-azul Luna Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Negro , NIKE Women s Revolution 4 Running Shoe Gunsmoke / Ocean Bliss / Dark Grey , Adicción a Brooks 12 Ombre Azul / Obsidiana / Púrpura Fucsia Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Peacoat Zapatillas de running adidas Ultraboost W para mujer Aero Blue S, Aero Blue S, Noble Indigo S , Zapatillas de running NIKE para mujer Epic React Flyknit Perla rosa / rosa perla rosa apenas rosa Zapato de running ASICS GEL-Cumulus 20 para mujer Gris piedra / negro , Nike Hombres Air Zoom Pegasus 33 Wolf Gray / Blanco-cool gris , Zapatillas de running New Balance 990v4 para mujer Verde militar / verde militar Zapatillas para correr ASICS Gel-Venture 6 para mujer Phantom / Coral / Lima , Zapatillas de running New Balance 990V3 para mujer Blanco azul Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-corredor Rosa-gamma Azul

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 ASICS Gel-Cumulus 19 para hombre Azul índigo / negro / amarillo de seguridad , Brooks Womens Adrenaline GTS 18 Blanco / morado / gris Zapato deportivo ASICS Gel-Kayano 24 para mujer Prune / Pink Glow / White Sandalias Columbia Sorrento Flip Athletic para mujer Iceberg / Zing , Zapatillas de trail running Arc teryx Norvan VT GTX - Mujer Liberty / Arcturus Zapatillas de running NIKE Air Max 2017 para mujer Blanco / polvo Zapatillas deportivas NIKE Lunar Apparent para hombre Jade claro de carbono / negro claro , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris blanco , CLARKS Sandalia plana Brizo Cady para mujer Puntos blancos fucsia Adrianna Papell Mujeres Natalia Ballet Flat Silver Beehive Mesh Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma White-puma Silver , Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Taos Taupe / Blanco Under Armour Womens UA Remix Negro / Blanco / Trópico Rosa , Salomon Womens Speedcross 3 GTX Scuba Blue / Cosmic Purple / Black , Vans Unisex Old Skool Zapatillas clásicas de skate Blanc De Blanc Sandalia para mujer Merrell, 1SIX8 Linna AC + Slide Azalea , Sandalia KAANAS Women s Recife Bow Fashion Slide Desnudo , Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse Sandalia UGG Royale Slide para mujer Blanco Sandalias Birkenstock de mujer con aplicaciones suaves de Granada Cuero engrasado , Birkenstock Mujer Mayari Birko-Flor Sandal Negro , Sandalia para mujer Cadiz Dolce Vita Slide Gray Multi Stella 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- Black Pu y Black Mesh , 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 1620- Black Pu Zapatillas ajustables de ancho extra extra ancho para mujer - Calzado para diabéticos y edemas Vino , Puño de hierro - Bota de patente de murciélago para mujer rojo Belleville Hot Weather Desierto Botas y vehículos de combate (cisterna) Botas, Hombres, Zapatos Broncearse , CLARKS Womens Un Haven Lace Sand Nubuck , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Marrón-azul , Sam Edelman Loraine Loaine para mujer Roccia Snake ,

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 Skechers Sport Run Skech negra Air Run High fuerte Fashion Sneaker Adornos de malla negra/ rosa fuerte a3cd956 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
38237

347630

4209833

6873063

7035822

9221377

10499732

12542598

15332713

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