Whistler Alley Mathematics

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.

Zapatillas de running Brooks running Womens Teal Glycerin 15 Neutral de Max Cushion Evening Blue/ Purple Cactus Flower/ Teal Victory 0253179




Malla
Importada
Caucho único
Dale a tu pies el impulso necesitan durante carreras cortas o largas con Brooks & reg; Glicerina 15. Con su plantilla acolchada y paseo constante, puedes ir desde marcador de milla 1-13 en comodidad cada paso del camino. Toma cada milla en zancada con ¡Brooks Running!
Predecesor: Glycerin 14.
Soporte Tipo: Neutro.
Amortiguación: Amortiguación máxima.
Superficie: La carretera. Malla de aire de 10 mm con estiramiento de 4 vías ofertas óptimas transpirabilidad durante carreras. superposición tiene una estrategia estirar y estructurar para un ajuste seguro Cordones centralizados el cierre entrega un seguro, ajustable ajuste. y collar proporciona comodidad contra pie. forro crea un gran desgaste dentro de zapato. Plantilla Soulhouse S-257 mantiene los pies en su lugar. Largometraje segmentado Crash Pad en conjunción con la felpa Zona de transición se jacta lisa
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


Zapatillas de running Brooks running Womens Teal Glycerin 15 Neutral de Max Cushion Evening Blue/ Purple Cactus Flower/ Teal Victory 0253179 - movie-store.online

Recent Changes

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 de running Brooks running Womens Teal Glycerin 15 Neutral de Max Cushion Evening Blue/ Purple Cactus Flower/ Teal Victory 0253179

Updated 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:

The Brachistochrone

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 de running Brooks running Womens Teal Glycerin 15 Neutral de Max Cushion Evening Blue/ Purple Cactus Flower/ Teal Victory 0253179

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

NIKE Men s Roshe Two Running Shoe Azul blanco , Zapatillas Nike Air Max Essential para hombre Stratus Blue / Gray / Black , Zapatillas de running Classic-M Vibram para hombre Negro , Zapatillas de running NIKE Lunar Skyelux para mujer Blanco / puro Platinum-off blanco , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Blanco / Negro Zapatillas de running Wstro para mujer New Balance Alpha Pink / Black / Lime Glow , Nike Men s Air Presto Essential Wolf Gray / Wolf Gray-white , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Gunmetal / Thunder Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Negro / Negro / Mtllc Slvr / Blanco , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Marrón, rojo, blanco , NIKE Women s WMNS Air Max 270, Blanco / Blanco-Blanco Blanco / Blanco-blanco Inov-8 Mujer F-Lite 260 (W) Cross Trainer Púrpura / negro Zapatillas de running NIKE Roshe Two para mujer Caqui / Negro / Negro / Blanco Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Negro / Fantasma , Zapatos ASICS para hombre Gel-Lyte-V Negro / negro / azul Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Plateado / Flash Coral , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer 010-black White Mtllc Silver Zapatillas NIKE Kids Revolution 3 (GS) para correr Universidad Rojo / Plata Metálica / Negro Zapatillas de running Reebok para mujer Acero / Gris plano / Orquídea exuberante / Resplandor cítrico / Blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Dark Grey Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Lobo gris / blanco puro Pure Platinum gris , Zapatillas para correr Mizuno Wave Inspire 12 para mujer Calypso Coral / Plata Brooks Womens Ghost 10 GTX Negro / Azul pavo real / Malvavisco , Zapatillas para correr NIKE Women s Air Zoom Structure 21 negro , Zapatillas de running W1540V2 New Balance para mujer Blanco azul , Zapatillas para correr Mizuno Women s Wave Creation 15 Púrpura / blanco / plateado , Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Onyx / Plata
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 ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Blanco negro Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul Zapatillas de running Adidas Performance Ultra s Boost M para hombre Negro / negro / amarillo solar , Columbia Women s Barraca & shy; Sunlight Athletic Sandal Fósil, Natural , Sandalia Merrell de mujer con gladiador de poste marrón , Sandalia de susurro de mujer Keen Licor de café / amarillo Zapatillas de running NIKE Air Max 2017 para mujer Negro / Blanco / Antracita Zapatillas de trail running Ryka Women s Intent XT 2 SR Negro / plata cromo Calzado deportivo New Balance Vazee Urge v1 para mujer Blanco negro , Zapatos Salomon Sense Pro Max Running Trail para mujer Jugo de uva, blanco, navegar por la web , Skechers Women-14960 Sneaker gris Sam Edelman Geren Sandalia para mujer Candy Red Zapatillas para correr Saucony Koa Tr para mujer Vio / Aqua Zapatillas de running retro Gel-Lyte III para mujer ASICS Azul clásico / azul clásico Drew Shoe Athena para mujeres Blanco , Zapatillas para caminar Brooks Women s Addiction Walker Negro , Sandalias de cuero de Viv Viv de TOMS Tribal negro Zapatillas de running NIKE Air Max Flair para hombre Negro / antracita-negro , Zapatillas para caminar Travelite para mujer Propet Blanco , Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro Zapatillas de deporte Skechers Sport para mujer Beathe Easy Our Song Fashion Sneaker Gris claro , Sandalia de diapositivas de ocio para mujer de Seychelles Azul , Sandalia de la diapositiva de junio de las mujeres de Propet Negro , Lacoste Women & rsquo; s Fraisier Slides Negro / Pnk sintético , Vince Camuto Sandalias Ellanna Slide para mujer Rojo cereza Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Blanco Blanco SOREL Women s Out N About Slippers Negro / Cantera , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Racer azul / negro Zapatillas de deporte de primera clase Microburst de Skechers Gris ,

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.

Zapatillas de running Brooks running Womens Teal Glycerin 15 Neutral de Max Cushion Evening Blue/ Purple Cactus Flower/ Teal Victory 0253179 - movie-store.online


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 de running Brooks running Womens Teal Glycerin 15 Neutral de Max Cushion Evening Blue/ Purple Cactus Flower/ Teal Victory 0253179 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3094246
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and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.

Back to Whistler Alley


Last update: March 16, 2016 ... Paul Kunkel whistling@whistleralley.com
For email to reach me, the word geometry must appear in the body of the message.