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 New Balance para mujer 940v3 Azul mujer running para negro 23cfc45




Textil y sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
ABZORB entresuela
T-BEAM tecnología
Medial Enviar
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 New Balance para mujer 940v3 Azul mujer running para negro 23cfc45 - 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 New Balance para mujer 940v3 Azul mujer running para negro 23cfc45

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 New Balance para mujer 940v3 Azul mujer running para negro 23cfc45

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

Calzado deportivo NIKE CK Racer para mujer negro , NIKE New Air Max 90 Ultra 2.0 Running Sneaker para mujer Blanco Zapatillas de running Skechers Performance para mujer Go Run 400 Action Azul marino / blanco Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco / Bahía / Plata Zapatillas de running Nike Kids Air Max Zero Essential GS Blanco / lobo blanco gris , Zapatillas de running Reebok Print Smooth Ultk para mujer Mar gris / niebla / blanco / carbón Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Vintage Indigo / Pigment / White Munsell , Short NIKE Tempo para mujer Blanco / Hyper Pink , Zapato deportivo para acolchado 635v2 New Balance para mujer Silver Mink Zapatillas de running NIKE Air Max 90 para mujer Blanco negro , Zapatillas de running ASICS Gel-Contend 4 para mujer Diva Azul / Plata / Orquídea , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino Blanco / Blanco-puro Zapatillas de running NIKE Air Huarache para hombre Hueso Ligero / Vela , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul Zapatillas de running Reebok Print Smooth Ultk para mujer Negro / polvo de asteroide / blanco Zapatillas de running Kinvara 5 Saucony para mujer Blanco / Crepúsculo / Puesta de sol , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Platinum / Wolf Gray / Metallic Platinum / White Zapatillas de running adidas Aerobounce W para mujer Indigo crudo / plata metálica / naranja de alta resolución Zapatillas de running NIKE Fs Lite 2 para mujer Chalk Blue Racer Blue White Zapatillas para correr Nike Flex Experience RN 5 Sigilo / Mtlc Plata / Antracita , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Negro / Fantasma Zapatillas running para mujer Saucony Liteform para mujer Negro Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Negro / sal marina , Calzado de running para mujer New Balance Cypher Castlerock / White Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco negro , Zapato deportivo ASICS Gel-Kayano 24 para mujer Blanco / azul violeta / amarillo de seguridad , Zapatillas de running NIKE Air Max Torch 4 para mujer Cool Grey / Hyper Violet / Hyper Punch , Zapatillas para correr NIKE Womens Sock Dart Negro / Vachetta Tan-negro-blanco Brooks Womens Adrenaline GTS 18 Grey / Rose Gold ,
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-Kayano 25 para mujer, blancas / blancas, 8.5 B (M) US - Sandalias KEEN Mujer Whisper-w Ágata gris / azul ópalo , Sandalia Terrell Post II Femenina de Merrell Negro , Zapatillas de running adidas Aerobounce Pr W para mujer Núcleo Negro, Plata Met., Ftwr Blanco , Zapatillas de running NIKE Air Max Flair para hombre Obsidiana oscura / azul marino / blanco , Sandalia Yucatan ECCO Mujer Shadow White Zapato de running Mission XR para mujer Salomon Celedon / Papaya / Pop Green Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Laguna Negra / Blanca-Volt-Blue , Zapatillas de running ST-2 para mujer Topo Athletic Verde gris , Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco / Plata / Melón brillante WMNS Zoom Fly Hielo Azul / Zorro Azul Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Negro , Hush Puppies Walker clásico de las mujeres Blanco , Tecnología Vionic Men s Orthaheel White / Navy Walker Azul / verde azulado , Zapatillas Cross-Trainer de PUMA Carson Hatch Wn para mujer Puma negro brillante Sandalia de mujer Aravon Beaumont Peep Slide Paloma , Zapatillas de deporte Skech Air Infinity Fashion Skechers Sport para mujer Plata blanca Ahnu Women s Yoga Flex Snow Melt Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Carbón Converse Chuck Taylor All Star Core Ox Salsa roja / blanca , Vans Auténtico Azul Imperial Onlymaker Mujeres High Peep Peep Puntera plataforma Stiletto Tobillo cruzado Correa Hebilla Snap Vestido fiesta tacón Sandalias Brillo de plata Bota de mujer Vince Camuto Estiva Negro Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Black-patent CLARKS Womens Un Cruise Lace (Wave) Sneaker Cuero negro , Piso Frankie Oxford de Shellys London para mujer Bright Multi , Hush Puppies Ekko Gwen Oxford para mujer Coral Suede , Skechers for Work Cottonwood Goddard Twin Gore Slip Resistant Slip On Negro , TOMS Clásicos de mujer Beige

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 New Balance para mujer 940v3 Azul mujer running para negro 23cfc45 - 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 New Balance para mujer 940v3 Azul mujer running para negro 23cfc45 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 1929976
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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.