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
Goma único
anatómico moldeado por compresión Entresuela de EVA con un suave drop-in de ICEVA plantilla para ambos comodidad y soporte
No marcado goma o caucho de goma suela con maquinilla de afeitar bordes podados para agarre en superficies mojadas
No marcado Caucho: termoestable el caucho es un compuesto de natural y caucho sintético eso no se irá marcas o un \\ u20ac scuffsa \\ u20ac en superficies de cualquier tipo. suelas de caucho, como un material natural, siempre será naturalmente ser no marcado
Vegano calzado amistoso ofrece la durabilidad, tracción y soporte esperas de OluKai mientras usas 100% de alta calidad materiales sintéticos, 0% productos animales
Agua Resistente: Ligero, secado rápido, transpirable y materiales duraderos construido para adentro y alrededor del agua
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

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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.


OluKai Ohana Ohana Black Ohiared// Black b659c8d

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

Sandalia KEEN Clearwater CNX para mujer Negro / Resplandor , Sandalias Bali KEEN para mujer Nórdico azul / azul ópalo , Columbia Women s Barraca & shy; Sunlight Athletic Sandal Barro / Guijarro , OluKai Ohana Ohia Rojo / Negro Sandalia deportiva Chaco Zvolv X2 para mujer Verdure Peach , OluKai Ohana Blanco negro , Sandalia deportiva Chaco Mujer Zong X Ecotread Helix Eclipse , Wolky Comfort Jewel 70500 canales rojos , Sandalia deportiva Tirra para mujer Teva Mar de Bering , OluKai Kulapa Kai Contton Candy / Black Wolky Rio Sandal Black Crash Suede Sandalias con tira en el tobillo INFINITY 935 para mujer Pleaser Negro Negro , Sandalia Yucatan ECCO Mujer Biscaya Sandalia Drew Element para mujer Nubuck marrón , Sandalia de Hollyleaf de las mujeres de Merrell Negro Skechers Performance On The Go 400 Vivacity Flip Flop Taupe / Blanco , KEEN Sandalias Newport Eco-W para mujer Negro / Imán , Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Helecho OluKai Kulapa Kai Contton Candy / Black , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Negro , Sandalia Drew Angela para mujer Dusty Multi Metallic , Earth Origins Sophie Negro , Earth Origins Sophie Women s Sandal Frost Grey , Columbia Women s Barraca & shy; Sunlight Athletic Sandal Barro / Guijarro Chaco Women s Z2 Classic Athletic Sandal Paz azul , OluKai Kulapa Kai Fucsia / Negro , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso , Revs New Classic Masaje de reflexología Sandalias para hombres & Mujer. Mira. Diseño. Terapia. Cojín Comfort Con plantilla de masaje & Absorción de impactos Único. Negro , Sandalias de tiras Santori Hi-Tec para mujer Jadeita / gris cálido
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 Sandalia Unisex Slide de Telic Negro , Converse Chuck Taylor All Star Seasonal Color Hola Papiro Zapatillas de running Adidas Performance Ultra s Boost M para hombre Collegiate Royal / Collegiate Royal / Collegiate Navy , Sandalias KEEN Mujer Whisper-w Sterling Blue / Vestido Azul Orthofeet 872 Women s Comfort Diabetic Extra Depth Sandal , Zapatillas Altra Women s Superior 3.5 Sneaker Gris / morado , Salomon Women s Speedcross 4 W Trail Runner Rojo tibetano / sangría / negro , WHITE MOUNTAIN Bomba de Eryn para mujer Desierto , Zapatillas Reebok Women s Runner 2.0 MT Track Guava Punch / Negro / Plata Metálico Salvatore Ferragamo Women s Groove Nero / Oro Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Lt Gray Sandalia universal Teva para mujer Negro Drew Shoe Women s Tulip Comfort Cobre Metálico Zapatos de caminata SAS Low Me de cuero para mujer Blanco Hush Puppies Mujeres Power Walker Ii Loafer Cuero negro , Clásicos de base Slip-on (Tm) para hombres de Vans Tumble Black , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Negro Sandalias planas Nori de Easy Street para mujer Pewter Metallic , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1620- Cuero plateado Mocasines casuales Alpargata Deconstructed de TOMS para mujer Hojas de flor JOE N JOYCE Soft Slippers Zuecos Zapatos de cuero estrecho - Para hombres y mujeres Noche azul The North Face Womens Thermoball Microbaffle Bootie II Shiny Coffee Bean Brown / Dune Beige (Temporada pasada) Zapatillas de running NIKE Men s Train Speed ​​4 Cumbre Blanco / Gris Fresco Finn Comfort Women s Ikebukuro Oxford Plata , Rockport Cobb Hill Women s Paulette Flat rojo , Kenneth Cole REACTION Women s Loop There Is Flat Notch Suede Tobillo Botín Hormigón , Timberland Womens Lakeville Oxford Marrón medio gamuza , Betunia Loafer de mujer Aerosoles Leopardo Tan Zapatillas de barco Bluefish Sperry Top-Sider para mujer Broncearse , Ballet Tali de mujer Cole Haan Bow Bow Serpiente Roccia Natural Imprimir

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 OluKai Ohana Ohana Black Ohiared// Black b659c8d 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 4940325
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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.