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.

malla

Engineered malla proporciona transpirable soporte.

Tri-star patrón de suela mejora la flexibilidad.

Mínimo el diseño del talón proporciona un más natural en forma.

Co-moldeado entresuela usa una inyección más suave Núcleo de phylon (IP) espuma directamente debajo del pie y un poco más firme inyectado unidad única (IU) espuma transportadora a entrega comodidad y durabilidad para un viaje suave

Talón se redondea para una transición sin problemas en el toque inicial abajo.

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

Calzado de 2017 running NIKE Glow RN 2017 2017 para mujer de Negro/ Fuchsia Glow/ Hyper Magenta 011f384Updated 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.

Calzado de 2017 running NIKE Glow RN 2017 2017 para mujer de Negro/ Fuchsia Glow/ Hyper Magenta 011f384

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

Amortiguación 635v2 New Balance para mujer Negro , Brooks Womens Hyperion Negro / Dewberry / Aruba Blue , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Azul gris / Coral New Balance Women s w940v3 Running Shoe Plata Zapatillas de running NIKE Lunar Skyelux para mujer Gris oscuro / blanco / antracita / explosión rosa Brooks Womens Hyperion Peacoat / Purple Wine / Aruba Blue , Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura , Zapatillas de golf adidas W CC Ballerina II para mujer Correr Blanco / Plateado Metálico / Chambray , Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado Zapatillas de running NIKE Air Max Flair para hombre Blanco / puro platino Zapatillas NIKE Kids Revolution 3 (GS) para correr Gris oscuro / blanco / negro , Brooks Women s PureFlow 6 Bluebird / Peacoat / Lime Punch Zapatillas de running Comfort Ride 575v2 New Balance para mujer Púrpura , Zapatillas de running NIKE Air Max 90 para mujer Negro / Negro / Gran Gris Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running NIKE Air Huarache para hombre Gris oscuro / antracita Zapatillas para correr Skechers Women s Go MEB Razor Negro / Aqua Zapato de running ASICS GEL-Cumulus 20 para mujer Negro / Flash Coral , Zapatillas de running NIKE Air Huarache para hombre Stealth White 003 , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Negro / Onyx / Plata , Calzado de running Fila Memory Faction 2 de Fila Monumento / Sugarplum / Blanco , Nike Air Zoom Mariah Fk Racer para mujer Pale Gray / Dark Grey , Zapatillas de running Newton Gravity III para mujer Rosado Zapatillas de running ASICS Gel-Kayano 23 para mujer Cacatúa / Seguridad Amarillo / Lapis Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Phantom / imán , Zapatillas para correr ASICS Gel-DS Trainer 22 para mujer Acuario / Aqua Splash / Flash Coral Calzado de running NIKE RN 2017 2017 para mujer Cerúleo / Blanco / Azul Espacial / Espuma de Menta Zapatillas de running NIKE Women s Air Zoom Structure 20 Negro / Blanco-fresco gris-lobo gris ,

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 Reebok Women s Runner 2.0 MT Track Blanco / Poison Pink / Silver / Pink Ribbon Zapatillas de running Nike RN Distance Shield para hombre gratis Negro / Plata Metálico / Gris Oscuro / Sigilo Zapatillas de trail running Salomon Women s Sense Mantra 3 W Lotus rosa / blanco / verde abuelita Zapatillas deportivas de malla al aire libre de malla transpirable de Onemix para hombre Blanco negro NIKE Men s Flex 2017 RN Legion Blue / Black - Cloro Blue Zapatillas de running Skechers Performance para mujer Go Run 400 Action Negro / Aqua Prop & eacute; t Olivia Oxford para mujeres Hueso CLARKS Danelly Millie Sneaker para mujer Blush Pink Leather Skechers Women s Walk Walk Joy-15615 Wide Sneaker Rosa / Multi Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede , Sandalia para mujer Marc Fisher Negro , MBT Shoes GT 17 Athletic Athletic Leather / Mesh Lace-up Fucsia / menta Zapato de mujer WHITE MOUNTAIN Veronique Natural UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Negro / Naranja Calzado para caminar CUSH + WA365v1 New Balance para mujer Blanco negro , Zapatillas DC Mujeres Chelsea TX SE Sneaker Azul / azul / blanco , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Rosado ALMA NATURAL Sandalia Kiki Slide para mujer Negro , Birkenstock Arizona Sandalia de ante con suela suave Cuero de cobre , Sandalias AS-Mona Slide de mujer Ash Oro negro CAPEZIO Mujer cambre (1128W) -Petal Pink Pétalo rosa , CORRAL Cráter para mujer con bordado en hueso Vaquera Botín en punta - A1094 marrón CLARKS Womens Un Cruise Lace (Wave) Sneaker Combi de cuero blanco / arena nubuck J Slides Sandalia de mujer Naomi Slide Estaño Zapatillas de ciclismo Flynt Tri de mujer Giro Plata / blanco / azul lechoso , Mocasín Sabrina Moc II Lamo para mujer Raya azul , Zapatillas para mujer Del Rey TOMS Explosión de luz azul marino Sanuk Womens Bianca Prints Oxford Iris Sunrise , Botín de tobillo Mazin Cayto para mujer Hush Puppies Granito Gris Nubuck , D-Story Fashion Lace up Tall Punk Dancing Canvas Botas largas Sneakers Zapatos para mujeres Multicoloured14 ,

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 Calzado de 2017 running NIKE Glow RN 2017 2017 para mujer de Negro/ Fuchsia Glow/ Hyper Magenta 011f384 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