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 gris Comfort Ride running 575v2 New Azul Balance para mujer Azul gris/ Coral c402f4f




Cuero y malla
Importado
Caucho suela
Con cordones Zapato de running con malla superior con collar agregado y lengua
tela forro
Cush + plantilla
Goma suela
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 gris Comfort Ride running 575v2 New Azul Balance para mujer Azul gris/ Coral c402f4f - 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 gris Comfort Ride running 575v2 New Azul Balance para mujer Azul gris/ Coral c402f4f

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 gris Comfort Ride running 575v2 New Azul Balance para mujer Azul gris/ Coral c402f4f

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 Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño , Zapatillas Reebok Women s Runner 2.0 MT Track Negro / carbón / ácido rosa / blanco Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco , Zapato deportivo ASICS Gel-Excite 4 para mujer Uva / Plata / Begonia Rosa , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Negro / Fantasma , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Negro / Onyx / Plata Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White Zapatillas de running adidas Alphabounce Beyond para mujer Blanco / Gris / Gris , Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino Zapatillas de running NIKE Air Zoom Vomero 13 para mujer Ocean Bliss / Noise Aqua , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Shadow Brown / Metallic Pewter / Rush / Maroon , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Rojo de alto riesgo / rojo de alto riesgo (hombres) Zapatillas para correr W630V5 para mujer New Balance Gris oscuro Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Gris oscuro / Barely Volt / Wolf Gray , Zapatillas para correr Go Run Forza Boston 2016 de Skechers Performance para mujer Azul marino / verde azulado , Zapatillas para correr NIKE Womens Sock Dart Midnight Navy / Golden Beige , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul , Calzado de running NIKE RN 2017 2017 para mujer Solar Red / Black / University Red , Zapatillas para correr Mizuno Women s Wave Rider 21 High Rise / Gray Stone Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Coral vivo / negro / blanco Munsell Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Negro / Plata , Camiseta adidas Swift Run W para mujer Traza púrpura / Traza púrpura / Traza púrpura , Zapatillas para correr Zante V3 para mujer New Balance Poisonberry / Dark Mulberry , WMNS Zoom Fly Apenas gris / Sequoia , Zapatillas para correr Pearl NZUMi Women s W EM Tri N 1 Blanco / Paradise Pink Zapatillas para correr Mizuno Women s Wave Creation 16 Mulberry Purple / Silver , Zapatillas de running Under Armour para mujer Toccoa Soldadura (300) / Gris opaco , Zapatillas de running ASICS Gel-Contend 4 para mujer Gris Medio / Mar Glaciar / 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 Zapato deportivo ASICS Gel-Cumulus 18 para mujer Silver / Pink Glow / Lapis Zapatillas adidas Ultraboost X All Terrain LTD Shoe para mujer Negro Carro de mujer Brooks Caramelo , Sandalia para mujer ECCO Yucatan II para mujer Sombra oscura , Zapatillas de running Brooks Women s Glycerin 16 Road Negro / Rosa / Gris , Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Universidad Rojo / Negro-blanco Zapatilla deportiva SICS Gel-Classic retro Negro Negro Orthofeet Probado Alivio del dolor Comodidad Serene Womens Orthopaedic Artritis extra ancha Diabetic T-Strap Zapatos Negro NIKE Men s Benassi Just Do It Athletic Sandal Oro negro / metálico , Skechers Women s Go 4-14918 Calzado para caminar Blanco negro , Zapatillas de deporte adidas Originals SL Loop Racer W Lifestyle para mujer Tomate / Running Blanco Ftw / Multi Solid Gray Zapatillas de running NIKE Lunarglide 9 para mujer Rosado , Bomba de plataforma FLY London Yadi718fly para mujer Pearl Cool Zapatilla deportiva Sperry para mujer Seacoast Core Cuero blanco Zapato para caminar Skechers Women s You Blanco negro , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Estaño , Sperry Womens Fathom Negro Colección casual para mujer FootJoy Negro Sandalia Dolce Vita Jene Slide para mujer Denim azul claro CLARKS Sandalia Leisa Lacole Slide para mujer Cuero blanco Very Fine Dance Shoes 2707 (Grado de competencia) 3 \ Plata blanca , Bomba de Angel II para mujeres estilo suave Putty Elegance , Pantuflas Mahabis Classic para hombre gris Gris oscuro / Larvik gris claro Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada , El deslizador caliente Smokin de mujer FLEXX Gunmetal / Black Bruga / Cashmere Bomba de vestido animado de LifeStride para mujer Tierno Taupe Bota de mujer Vince Camuto Estiva Valleywood Sorel - Bota de mujer con cordones Emelie Premium sin conchas Cordobés / Mayor , Dr. Martens 1561 4 Eye Gibson azteca , Calzado para caminar de mujer Mephisto Cross Cuero gris Bucksoft

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 gris Comfort Ride running 575v2 New Azul Balance para mujer Azul gris/ Coral c402f4f - 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 gris Comfort Ride running 575v2 New Azul Balance para mujer Azul gris/ Coral c402f4f 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 493061
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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.