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 montaña de montaña de Esmeralda montaña Zapatillas Akyra La Sportiva para mujer Esmeralda/ menta 02951d1




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 montaña de montaña de Esmeralda montaña Zapatillas Akyra La Sportiva para mujer Esmeralda/ menta 02951d1 - 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 montaña de montaña de Esmeralda montaña Zapatillas Akyra La Sportiva para mujer Esmeralda/ menta 02951d1

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 montaña de montaña de Esmeralda montaña Zapatillas Akyra La Sportiva para mujer Esmeralda/ menta 02951d1

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 Zapatillas para correr Go Run Ride 5 de Skechers Performance para mujer Púrpura / Naranja Zapatillas de running adidas Aerobounce W para mujer Fácil Coral / Plata Metálica / Rojo Táctil , Zapatillas de running Hoka One One Vanquish 2 para mujer Azul corso / rojo amapola Zapato de running ASICS Alpine XT para mujer Gris medio / carbono / azufre primavera Zapatillas de running Gel-Moya para mujer de ASICS Durazno Pétalo / Blanco Zapato de running ASICS Gel-Kinsei 5 para mujer Azul / blanco / iridiscente , NIKE Women s Revolution 4 Running Shoe Pure Platinum / Sunset Pulse - Wolf Gray Nike Hombres Air Zoom Pegasus 33 Negro / blanco / antracita / gris fresco , Zapatillas de running ASICS Women s Roadhawk FF SP Cereza / flor / abedul Zapatillas de running New Balance Boracay V3 para mujer Supercell / Ozone Blue / Bleached Sunrise , Zapatillas de running NIKE Air Huarache para hombre Blanco / Negro / Gum Amarillo / Blanco , Zapatillas de entrenamiento NIKE Womens Metcon 3 Cross blancas, negras 849807 100 Blanco negro , Zapato de tenis ASICS Gel-Dedicate 5 para mujer Plata / Rojo Rojo / Blanco Zapatillas para correr Saucony Women s Cohesion 10 Gris / Coral , NIKE Womens Free RN Zapatillas para correr Dark Grey / Pink Blast 831509-006 Talla 9 - NIKE Men s Roshe Two Running Shoe Industrial Blue White 402 , En la ejecución de calzado de mujer en la nube Denim / Blanco , Nike Hombres Air Zoom Pegasus 33 Midnight Navy / Blanco-negro-puro Platino Zapatillas de running Reebok Print 2.0 para mujer Gable Gray / Fire Spark / Blanco / Pure Silver , Zapatillas para correr Pacev2 para mujer New Balance Sea Spray / Black Zapatillas para correr Mizuno Women s Wave Creation 15 Col / Bright Marigold / Columbia , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / antracita , Zapatillas de running adidas Ultraboost W para mujer Negro / Negro / Negro , Zapatillas de running NIKE para mujer Air Max Motion LW Soar / cloro azul / blanco Nike Hombres Air Zoom Pegasus 33 Vert Gorge / Noir / Platine / Black , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris , Zapatillas de running Mizuno Wave para mujer Inspire 10 Blanco ,
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-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul Sandalias KEEN Mujer Whisper-w Ágata gris / azul ópalo Zapatillas de agua Merrell para mujer All Out Blaze Aero Sport Beige / Caqui Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Verde , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede , Nike Hombres Air Zoom Pegasus 33 Racer Blue / Navy / Blue Glow Skechers Go Walk 4 en sintonía Womens Slip On Walking Sneakers Gris pardo , Sandalias EVA Gizeh de mujer Birkenstock Caqui Drew Shoe Women s Hope cuero terapéutico moda Oxfords Mezclilla Sandalia Estelle de Caja Amarilla para Mujer Blanco , Arche Women s Drick Flat Sand Nubuck , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul Zapato para caminar WW813 para mujer New Balance Negro Zapatillas de deporte WN Slip-on Farece para mujer de PUMA Pearl-puma White Converse CTAS Chuck Taylor All Star Mujeres Brookline Mid Negro / Negro / Blanco , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Armada Sandalia de diapositivas FIFI19SP de Donald J Pliner para mujer Adular , Vince Camuto Sanella deslizante de mujer Cool Blue Sandalia Janissah Slide de Nine West para mujer Cuero negro FRYE Botines Julia Bootie para mujer Forest Soft Nappa Lamb Dr. Comfort Vigor Mujer diabético terapéutico Extra profundidad senderismo botas de cuero de encaje castaña , Bomba Stillson para mujer de Sam Edelman Dusty Rose Suede Zapatilla clásica Dualyz Unisex Kush con suela extraíble Gris claro / rosa , Zapatilla de cuello de garganta 4640 de Jacques Levine para mujer Oro plata , Zapatillas de deporte Apollo Chukka de hombre nativas Azul , Nueve siete cuero de gamuza de las mujeres del dedo del pie redondo tacón grueso Comfort trabajo hecho a mano Botines Camello , CLARKS Women s CloudSteppers Sillian Jetay Flat Gris púrpura ,

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 montaña de montaña de Esmeralda montaña Zapatillas Akyra La Sportiva para mujer Esmeralda/ menta 02951d1 - 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 montaña de montaña de Esmeralda montaña Zapatillas Akyra La Sportiva para mujer Esmeralda/ menta 02951d1 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

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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.