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 suela

Inyección entresuela de eva moldeada

Sin costura aplicación de material

Flex paseo

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

Zapatillas B072MFRMF9 de 18554 running W630v5 para mujer New Alpha Balance Negro/ Alpha Pink/ Plata Metálica 16d5624Updated 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.

Zapatillas B072MFRMF9 de 18554 running W630v5 para mujer New Alpha Balance Negro/ Alpha Pink/ Plata Metálica 16d5624

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

Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / oscuro, pasas de rosa, mortal , Saucony Women s Ride 10 zapatillas para correr Blanco | , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Carbón / Blanco / Flash Coral Zapatillas de running NIKE RN 2017 para mujer gratis Wolf Gray / Off White-pure Platinum-black NIKE W Roshe One Womens 844994-601 Partícula rosa / vela negra , Zapatillas de running New Balance 1340v3 para mujer Gris oscuro Mujer adidas Cloudfoam Qt Racer Núcleo Negro, Cobre Met., Blanco , New Balance Women s w990v4 Running Shoe Gris / rosa En Running Womens Cloudflow Shoe Amanecer / Jade , NIKE WMNS Zoom Fly Women 897821-411 Blanco negro , adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Verde Ceniza Zapato de correr ideal para rejilla de mujer Saucony Blanco negro , Zapatillas de running definitivas Ryka para mujer Gris negro Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura , Zapatillas de running New Balance para mujer 490v4 Negro / Guayaba , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Blanco Zapatillas NIKE Kids Revolution 3 (GS) para correr Negro / Volt Noir / Blanco Saucony Women s Guide 9 Running Shoe Azul / Púrpura / Rosa NIKE Men s Roshe Two Running Shoe Wolf Gray / Dark Gray / Volt , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Phantom / Black / White Zapato de tenis ASICS Gel-Dedicate 5 para mujer Plata / Rojo Rojo / Blanco Zapatillas de running New Balance 1340v3 para mujer Gris oscuro , Zapatillas Nike Free TR Fit 3 para mujer 555158-402 Light Armory Blue / Armory Navy-armory Pizarra , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono , Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima ASICS Metrolyte II Shoe Women s Walking Blanco / Blanco / Blanco Zapatillas de running adidas Aerobounce Pr W para mujer Núcleo Negro, Plata Met., Ftwr Blanco Zapatillas de running Cavu para mujer HOKA ONE ONE Negro / Bluebird ,

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 Pantalón largo clásico NIKE Golf para mujer Tech Essentials Binario Azul / Gris Oscuro / Oscuro Obsidiana Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso NIKE Men s Roshe Two Running Shoe Midnight Navy / Black-sail-volt Zapatillas de running adidas Cloudfoam QT Racer para mujer Triple blanco Zapato casual bajo NIKE Lupinek Flyknit para hombre Cargo Khaki / Mica Green Vivien Flat de Bella Vita para mujeres Champagne cuero Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja , Calzado deportivo de punto Chelsea con memoria de Fila para mujer Abedul plateado / Abedul plateado / Blanco Chaco Mujer Zcloud 2 Sport Sandalia Eclipse cubierto , Birkenstock Arizona Sandalia de cuero unisex Cuero engrasado Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Negro / Negro / Carbono Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Plano extra ancho Feet Breeze Womens Zapatos para caminar Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer gris Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White , Walking Cradles Women s Bounce Flat Brown Lagart Patent Crocco Zapatillas de golf Skechers Women s Go Golf Birdie Malla natural / rosa Sandalia Merrell Terrano Ari Slide para mujer Dusty Olive Calvin Klein Dalana Slide Sandalia para mujer Negro / Peltre / Blanco 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1700- Trim oro / plata , Zapatilla con aplicación de perrito para mujer Haflinger Tierra Zapatillas de mujer Foamtreads Coddels Rosa claro , Zapatillas Spike Cross Country Kilkenny XC5 Spike de Country Saucony Aqua / Vizi Coral Bomba Lloret para mujer J.Renee Taupe / Gold Trotters para mujer Dea Slingback Pump Patente perlada multi acolchada / negra Cole Haan Women s 2.Zerogrand Laser Wing Oxford Simplemente Taupe Botín de arranque casual Lorca de Ted Baker para mujer Negro Negro Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso Pleaser Adore-1020 Clubbing de baile exótico Hot Sexy tobillo / pantorrilla sexy Bota de 7 \ Negro / Patente , Botín Colibri Nf63 para mujer El Naturalista Marrón / Hoja

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 Zapatillas B072MFRMF9 de 18554 running W630v5 para mujer New Alpha Balance Negro/ Alpha Pink/ Plata Metálica 16d5624 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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5454064

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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