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 mujer montaña de montaña Akyra de Akyra La Sportiva para mujer Esmeralda/ menta c64ab77




tejido y sintético
IR LA DISTANCIA - Una estructura segura y estructurada zapato ideal para sendero de larga distancia corredores; Rocker2 suela promueve exterior natural talón a la punta del pie soporte para que puede ir más lejos sin fatiga
TRANSPIRABLE - Pares de AirMesh con un innovador Esqueleto de TPU a permitir un mayor flujo de aire, reduciendo la humedad y acumulación de calor
TOTAL CONTROL - Agresivo asas combinadas con Caucho FriXion y Trail Bite talón proporcionar todo terreno tracción y agarre, incluso en superficies mojadas; Impact Brake System para paradas rápidas y control de descenso
COMPLETO PROTECCIÓN - FlexGuard inserciones en el superior y choque absorbente MEMlex EVA y 2.4 mm LaSpEVA protect el pie y el tobillo; Amortiguación de la media suela suministros agregados comodidad
ESPECIFICACIONES - La montaña Akyra Zapato para correr una altura de pila de 25 mm en el tacón y 16 mm en la punta (9 mm soltar); oz (278 g)
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 mujer montaña de montaña Akyra de Akyra La Sportiva para mujer Esmeralda/ menta c64ab77 - 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 mujer montaña de montaña Akyra de Akyra La Sportiva para mujer Esmeralda/ menta c64ab77

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 mujer montaña de montaña Akyra de Akyra La Sportiva para mujer Esmeralda/ menta c64ab77

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 NIKE RN 2017 para mujer gratis Glaciar Azul / Plata Metálica Hoka ONE ONE Zapato Clayton 2 Speed ​​para mujer Frost / Snap , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Ágata Púrpura / Negro / Tierra Púrpura / Blanco Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Blanco Calzado deportivo New Balance Vazee Urge v1 para mujer Blanco negro , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Racer Pink / Vastgrey Zapatillas de running Mizuno Wave para mujer Inspire 14 Folkstone Gray / Pink Glo , ASICS Metrolyte II Shoe Women s Walking Hielo medio gris / gris medio / albaricoque Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple , Zapatillas de running NIKE Zooms Winflo 4 para mujer Atmósfera rosa , Zapatillas de running Topo Athletic Ultrafly - Mujer Púrpura / Lila NIKE Women s Tanjun Marrón nocturno , Zapatillas Shimano SH-TR5W Tri para mujer Blanco Zapatillas de running Altra Women s Intuition 4 Ocean / Teal , Zapatillas de running ASICS Gel-Nimbus 18 para mujer Melocotón Melba / rosa fuerte / guayaba Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta Short NIKE Tempo para mujer Deporte Rojo / Negro , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Dark Mulberry / Black Rose Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Dragon Fruit / Sea Salt / Phantom Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Lobo gris / blanco-antracita-cool gris , NIKE Men s Benassi Just Do It Athletic Sandal Negro / Gimnasio Rojo Zapatillas de running ASICS Gel-Kayano 23 para mujer Blanco / Nieve / Plata Adicción a Brooks 12 Antracita / Alto riesgo Rojo / Plata , Zapatillas de running Nike Air Huarache Run Ultra para hombre Negro / Negro / Negro Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Supercell / Sunrise Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Mora Zapatillas de running Saucony para mujer Stretch N Go Breeze Negro Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower Zapatillas adidas Response TR Shoe para mujer Gris medio / Calzado blanco / gris oscuro
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 Gris blanco Zapatillas de running adidas Women s Element V (8.5, Grey Three) - Wolky Comfort Jewel 70820 canales de mezclilla Zapatillas de deporte casual Gazelle de adidas para hombre Blanco Chaco Mujer ZX1 Classic Sport Sandalia Adobe Clan Wolky Comfort Jewel 12120 Offwhite Nubuck Saucony Women s Guide 9 Running Shoe Gris / Azul / Citron , Zapatillas de deporte Sneak Name de FLEXX para mujer Corda Elba Intreccio Sandalias Rainbow 301ALTS Single Layer Premier Leather Marron oscuro , Zapato de running 840v4 New Balance para mujer Gris oscuro , Zapatillas NIKE Kids Revolution 3 (GS) para correr Pure Platinum / Pink Blast / Wolf Gray / White , Sandalia de 2 piezas Cobb Hill Women s Hollywood Cuero blanco , Sandalias de cuero de Viv Viv de TOMS Yute metálico natural , Sandalia Skechers Mujer Reggae-Haystack Toe Ring Fucsia / Amp / Natural KEEN Women s Uneek 8MM Sandal Zinfandel / Vino morado , Zapatillas de deporte Marilena de Spring Step para mujer Armada Reebok Crossfit Nano 7.0 Track Shoe para mujer Blanco / Calavera Gris / Negro Kenneth Cole REACTION Womens Cake Shop 2 Equipaje sintético Lakai Limited Footwear Mens Griffin Textil negro , Sandalia de la diapositiva de Stella de las mujeres de Rachel Zoe Negro , Sandalia Heather Slide de mujer Vía Spiga Midnight Suede , Zapatilla Moxie Bootie de mujer de piel de oveja Acorn castaña , Bota Cobb Hill para mujer Penfield Cuero de Burdeos Felicity | Cuero negro Botín de mujer Kike Fixed Kitten de The Fix para mujer Silver Metallic , Botín Jillian para tobillo de mujer Paul Green Cilantro de gamuza Bomba antideslizante a medida Nolan para mujer Franco Sarto Negro Zapatilla de deporte Stella Nd58 de El Naturalista para mujer Negro , Chanclas Kate Spade Nova, estampado de pimienta de chiles negros brillantes - Brunswick Aura Ladies Bowling Shoes Azul marino / Bebé

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 mujer montaña de montaña Akyra de Akyra La Sportiva para mujer Esmeralda/ menta c64ab77 - 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 mujer montaña de montaña Akyra de Akyra La Sportiva para mujer Esmeralda/ menta c64ab77 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 1147953
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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.