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.

The B077H3XZGC North Face Womens Shellista III/ Tall Dune Shellista Beige/ Demitasse Brown (Temporada pasada) b8cad72

Cuero, Gamuza
Madre La naturaleza no tiene nada en ti cuando enfrentar el clima frío en The North Face & reg; Shellista III Tall arranque.
Impermeable Gamuza BLC-obediente cuero y cuero superior.
Impermeable la construcción mantiene los elementos húmedos.
200g PrimaLoft & reg; Aislamiento de plata Eco mantiene los pies cálido.
Cordones sistema para un seguro fit.Fuzzy falso forro de piel ofrece un gran zapato sensación. plantilla textil para soporte todo el día y comodidad. Específicas para mujeres, moldeado por inyección, TPR a prueba de agua caparazón protege pies. Winter Grip & reg; suela de goma con sensible a la temperatura IcePick & reg; por excepcional tracción en clima frío condiciones.Importado. Mediciones: Altura del tacón: 1 1 & frasl; 2 in Peso: 1 libra Eje de 9 oz: 13 en la plataforma Altura: 1 y frasl; 2 yo
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

The B077H3XZGC North Face Womens Shellista III/ Tall Dune Shellista Beige/ Demitasse Brown (Temporada pasada) b8cad72 - 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

The B077H3XZGC North Face Womens Shellista III/ Tall Dune Shellista Beige/ Demitasse Brown (Temporada pasada) b8cad72

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.

The B077H3XZGC North Face Womens Shellista III/ Tall Dune Shellista Beige/ Demitasse Brown (Temporada pasada) b8cad72

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

FRYE Bota de Combate con Cremallera Frontal Julie para Mujer Forest Soft Oiled Suede , Ariat Womens Volant S Tall Zip Tall Riding Negro Prop & eacute; t Blizzard de mujer Zip de invierno con cremallera Marrón / nylon Botines Chelsea del Dr. Martens 2976 con cremalleras para mujer Negro Bota de combate de lentejuelas Gia Mia Dancewear para mujer Negro Nine West Jatoba Suede para mujer Negro / negro gamuza FLY London Womens Ypul Nubuck Wedge Heel Open Toe Platform Sandalia de verano Plata , Rieker Womens R2278 Kelani 78 Schwarz / Kastanie , Bota posterior Aerosoles para mujer Fabrica gris Kenneth Cole New York Mujeres Justin Engineer Boot Negro , Blundstone Super 550 Series Boot Negro Aigle Parcours 2 Iso Bronce Twisted X Boys Cowkid Work Boot Square Toe - Ycw0005 Chocolate FRYE Bota corta Renee Seam para mujer Cashew-72063 , Vuelo a prueba de agua Belleville 690 Sage Green , 5.11 Zapato con cordones para hombre Negro , Sandalia de vestir OTBT Free Spirit para mujer Negro GUESS Women s Rodeo Faux-Leather botines occidentales Blanco , Bota Darcy para mujer Easy Street Brown / Gore Bota de mujer FRYE Natalie Short Lug Engineer Black Soft Oiled Suede Bota de imitación para mujer SoftWalk Marron oscuro Bota Kikka para mujer Dromedaris Aceituna Bota Darcy para mujer Easy Street Tan bruñido Botas de seguridad Reebok RB874 Stealth SR para mujer - Negro Negro , Denunciar Hanelli Tobillera para mujeres Negro , Bota de trabajo Harland Harley-Davidson para mujer Negro , Bota de caza Kenetrek Men s Mountain Extreme 400 aislada marrón Blundstone Super 550 Series Boot Gris acero , Moneysworth y el mejor botín Welly de goma para mujer Bosque verde ,
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 NIKE Men s Lunar Fingertrap Cross Trainer Negro / Blanco - Negro Zapatillas de running adidas Women s Edge Lux W Colegial azul marino / azul claro / azul claro , Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad Chaco Mujer Zcloud 2 Athletic Sandal Trace Eclipse Brooks Womens PureGrit 6 Negro / Ebony / Diva Pink Sandalia Jacardia Merrell para mujer Gris pardo , Zapatillas de running adidas Pureboost X Tr Zip para mujer Negro / Plata Metálico / Blanco , Zapatillas de running mujer Salomon Wings Flyte 2 W para trail running Slateblue / Light Onix / Teal Blue F Zapatillas de running Trail Speed ​​Speedstride de Fila para mujer Dark Shadow / Black / Aruba Blue Zapatillas de deporte Romy de Easy Spirit para mujer Cuero blanco / azul mediano Sandalia plana Nanci de Naturalizer para mujer Beige adidas Performance Ultraboost X para mujer Borgoña / negro CLARKS Womens Un Reisel Myra Slide Sandal Cuero negro Sandalias de cuero de Viv Viv de TOMS Blue Slub Chambray Sandalias con cuña de tacón medio Lynette de Onex para mujer Estaño , Zapatilla deportiva Skechers You-14958 para mujer Blanco , Drew Shoe Women s Paradise II Velco Slip-on Taupe ternero , Traje de juego unisex Diadora Melocotón rosa Adidas Originals EQT Support Adv W para mujeres Blanco / blanco / púrpura brillo , FootJoy Colección de mujeres a medida Cierre de golf Zapatos 91690 Oro / Blanco Rumpf Scooter 1575 Zapatilla de deporte Black Dance Split Sole Men Salsa Latina de Mujeres Jazz Hip Hop Fitness Zapato de ejercicio , Zapato Pedini 321 Capezio para mujer Caramelo , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Blue Orbit / Bright Citron / Blue Void Minnetonka Leopard Cally Slipper mocasín mujer Negro Merrell Women s Encore Ice Slip On Negro , Bota Deia para mujer de TOMS Ante rojo Mocasín slip-on de Metro Stuart Weitzman para mujer Ante de Londra , Zapatillas de deporte Easy Spirit AP1 Sport Natural Fb TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Núcleo rojo , Zapato femenino Keli de Alegria White Tooled ,

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.


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.

The B077H3XZGC North Face Womens Shellista III/ Tall Dune Shellista Beige/ Demitasse Brown (Temporada pasada) b8cad72 - 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 The B077H3XZGC North Face Womens Shellista III/ Tall Dune Shellista Beige/ Demitasse Brown (Temporada pasada) b8cad72 it out, and I got a chance to try out this new graphics software.


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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3941000
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.

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