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.

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Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
I.G.S. (Orientación de impacto) Sistema) Tecnología: Filosofía de diseño ASICS que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.
FluidRide Entresuela: FluidRide proporciona lo último combinación de recuperarse y propiedades de amortiguación con peso reducido y excepcional durabilidad.
Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.
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


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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 running de running ASICS Gel-Cumulus 19 para hombre Carbon 13633 para/ Black/ Hot Orange ed7b123

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.


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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 NIKE Kids Revolution 3 (GS) para correr Tarta / Blanco / Volt / Negro , Zapatillas de running Mizuno Wave para mujer Wave Horizon Plata / rosa , Zapatillas de running A7100w Apex para mujer Armada Zapatillas de running NIKE Free Run para mujer - SU16-10 - Blanco Ocean Fog / Turq / Mango / Navy , Lanzamiento de Brooks Womens 4 Flor de Cactus Púrpura / Diva Rosa / Negro , Calzado de running New Zealand Vazee Pacev2 para mujer Negro / Lime Glow , Calzado de running NIKE Kids Roshe One (GS) Dark Stucco / Light Bone ASICS GelGlyde Shoe para mujer Negro / Blanco / Coral Zapatillas de running NIKE RN 2017 para mujer gratis Blanco EN la zapatilla de deporte Running Cloud para mujer Blanco negro , MBT Shoes Mujeres GT 17 Athletic Shoe Cuero / Malla Con cordones Blanco / morado Zapatillas de running adidas Cloudfoam Super Flex W para mujer Misterio Azul / Blanco / Naranja Brillante Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow Zapatillas de running Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño , Zapato deportivo ASICS Gel-Kayano 24 para mujer Flash Coral / Negro / Blanco , Zapatillas de running Saucony para mujer Echelon 5 Gris / rosa Zapato de running ASICS para mujer Torrance Carbon / Pink Peacock / Black , Zapatillas de running NIKE Air Max Flair para hombre Universidad Gold Black 700 Zapato deportivo ASICS Gel-Kayano 24 para mujer Concha Rosa / Blanco / Begonia Rosa Zapatillas de running adidas Aerobounce W para mujer Núcleo Negro / Aero Verde / Hi-res Verde , Calzado casual NIKE Air Max 90 Ultra 2.0 Flyknit para mujer Blanco / Lt Armory Blue Under Armour Women s Charged 24/7 Low NM True Ink (400) / Blanco , Zapatillas de running W775V2 para mujer New Balance Naranja / Plata Zapato para correr Saucony Women s Cohesion 9 Plata / azul marino / verde azulado , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ecru Tint / Ash Pearl / Ash Pearl , Zapatillas de running Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul En la ejecución de calzado de mujer en la nube Denim / Blanco , Zapatillas de running NIKE Lunar Skyelux para mujer Golpe caliente / Marrón nocturno
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 NIKE Women s Revolution 3 para correr Blanco / Metálico Gris oscuro / Gris lobo , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Multi , Wolky Comfort Jewel Black Croco Salomon Women s Speedcross 4 W Trail Runner Poseidón , Zapato de running ASICS Gel-Quantum 360 cm para mujer India Ink / Flash Coral / Mid Grey , Zapatillas de running NIKE Mens Air Zoom Structure 19 Cool Gray White Black 018 , Zapatillas deportivas NIKE Lunar Apparent para hombre Negro / Racer azul-antracita , Zapatillas de running NIKE Flex Run 2018 para mujer Cool Gray / White Zapatillas de trail running WT10v1 Minimus para mujer New Balance Azul negro El corredor de trail impermeable Xa Pro 3D Cs para mujer de Salomon Sombra Comfortiva Mujer, Faye Sandalia de tacón medio Rubí , Easy Spirit Women s Motion Lace up Oxford Armada , SAS Womens Sunny Teal , Zapatillas para caminar RYKA Women s Devotion Plus Azul marino Sandalia de mujer Aravon Beaumont Peep Slide Azul Zapatilla deportiva Skechers Go Walk 4-Attuned Armada Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Navy Skechers Women s Go Run Mojo-Mania Sneaker Blanco negro Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Pacífico / Plata Baile Bella Me de Bernie Mev para mujer Bronce Tierra Everwood Negro Gamuza / resistente al agua APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Plata Metálico / Blanco , Zapatillas de skate Vans Unisex Old Skool (Double Light Gum) Hielo de albaricoque Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Monocromo negro Sandalia de viaje Via Spiga Women s Porter Ante negro 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2829- Oro Sparkle & Snake 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2707- Satén blanco y adornos de plata # 119 , Zapatillas altas Zumba Fitness Flex Classic - Zapatos de mujer de Babe Oxford de zapatos deseados Aceituna , Hi-Tec Women s Celcius Waterproof Oxford Acero gris / gris / liquen ,

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.

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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 running de running ASICS Gel-Cumulus 19 para hombre Carbon 13633 para/ Black/ Hot Orange ed7b123 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 1735596
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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.