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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Cuero
Importado
Eje mide aproximadamente Low-Top del arco
Aerocore Sistema de energía con Acción de alto rebote Y excepcional Comodidad
Ligero Eva Foam para alto Amortiguación de rebote
Deportes Inspirado aerodinámico Diseño
Durable Entresuela está cubierta Con contacto en el suelo Caucho
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.


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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 de running para exteriores adidas outdoor Terrex Tracerocker W para mujer Vapor Azul / Gris Cuatro / Icey Blue Zapatillas de trail running Hiero V2 para mujer New Balance Espacio ultraterrestre / negro / amanecer blanqueado , Zapatillas de trail WT610V4 para mujer New Balance Gris / rosa , Zapatillas de trail running Inov-8 Mudclaw 300 Negro / Rojo Salomon Sense Ride Running Shoe - Mujer Parachute Purple, Purple Opulence, Beach Glass , Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Azul marino / azul claro Zapatillas de trail running Salomon Women s Speedcross 3 Burdeos / rosa fuerte / rosa de loto , Zapatillas de running adidas Women s Rockadia W Trail Negro / Blanco / Naranja Fácil New Balance WT510RS3 Trail Running Shoes para mujer Gris oscuro Zapatillas de running New Balance para mujer 590v3 Imán / vórtice Zapatillas de trail WT10V4 para mujer New Balance Libélula Zapato de trail running ASICS GT-2000 4 para mujer Azalea / Melon / Perfect Plum , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral , Salomon Women s XA Enduro W Trail Runner Ceramic / Lime Punch./ Black Zapatillas sin cordones Flex Merit para mujer Merrell Agua , Zapatillas Merrell Women s Vapor Glove 2 Barefoot Trail Running Black / Castle Rock Zapatillas de running para trail running adidas Performance para mujer Kanadia 7 TR W Colegial azul marino / Running blanco / Lucky Pink , MT-onemix Carta de las mujeres de los hombres Print Air Cushion Mesh Running Sneakers Azul profundo / rojo Zapatillas de trail running Inov-8 Inov8 para mujer X-Talon 230 Gris / verde azulado Salomon Women s X-Mission 3 W-w Rosa virtual Zapatillas de running V-Trek Black / Gum Trail Vibram para mujer Negro / goma de mascar , Zapatillas de senderismo Air Mesh de Ahnu para mujer con malla de aire Aura astral UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Blanco negro , Salomon Women s X-Mission 3 W-w Menta / Deep Dahlia / Athletic Green X , Zapatillas de exterior Aetrex para mujer V753 Voyage Trail Runner Outdoor Gris / morado Zapatillas de running adidas Performance Galaxy Trail W para mujer Gris oscuro / Océano / Vapor azul F16 La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Esmeralda / menta Zapatillas de deporte Terrex Skychaser GTX W de mujer Adidas Sport Performance, gris, 7,5 M - , Zapatillas de running New Balance para mujer 590v3 Ciclón / Ciclón
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 adidas Running Womens Supernova Sequence 9 Vapor Blue F16 / Silver mate / Craft Blue F16 Zapato de running ASICS Gel-Quantum 360 cm para mujer Kint negro / blanco / plata , Sanda Verra Teva para mujer Negro. , Zapatillas de volante Nike Women s Zoom Hyperface 2 Blanco / Juego Real Zapatillas de entrenamiento cruzado Trainflex para mujer Reebok Blanco / Calavera Gris / Plata Metálica / Gris Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout Zapatillas de running acolchadas 420v4 para mujer New Balance Gris Bota de chukka Bootie Bootie de cuero de mujer Comfort de Cara Comfort para mujer Negro Nurse Mates Womens - Dash Arco iris negro Calzado de senderismo Go Joy 15602 de Skechers para mujer Blanco negro CLARKS Mujer Gracelin Mara Flat Berenjena , Walking Cradles Womens Michaela Cuero negro , Zapatillas de running NIKE Lunar Skyelux para mujer Negro / Blanco / Antracita Splendid Sandalia de mujer Farley Sonrojo , Marc Fisher LTD Mujeres Glenna Suede azul claro , Zapatillas para correr New Balance Rushv3 para mujer Acero / Negro NIKE Men s Benassi Just Do It Athletic Sandal Blanco / Real Academia Easy Spirit Women s Traveltime Mule Blanco / Rosa Multi Zapatillas de tenis New Court 896v2 Hard Court para mujer Coral vivo / Coral vivo , Sandalia Continental Slide para mujer de Seychelles Mid Blue , 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 \ 1683s- Cuero plateado Bota de trabajo con abertura para mujer Ariat, estampado de hipopótamo de chocolate, 10 B US Antique Tan Croc Print Dublín Ladies Wye Boots , New Balance Women s Sp4040n1 Azul marino / blanco , Zapatillas de ciclismo SPI X-Alp Launch SPD Pearl iZUMi para mujer Perla ahumada / Highrise , Trotters para mujer Dea Slingback Pump Patente perlada de oro acolchado / negro Bota Cobb Hill para mujer Penfield Cuero de almendra , Aravon Betty-AR Oxford para mujeres Piedra , Zapatillas de deporte Janet Calvin Klein para mujer Platino blanco Zapatillas Slip-On Roamer SAS para Mujer Sabio ,

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 Timberland Flyroam Go Timberland Lace Chukka para mujer Pure Cashmere Go Cashmere Escape/ Natural Sandune 8a99275 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.

Trail Running
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2097212
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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.