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 NIKE/ Air Max Torch 4 Rosa/ para mujer Negro/ Plata Metálico/ Rosa Flash 3927ccb




malla
Hecho en EE. UU. O importado
TPU superposiciones proporcionan soporte dirigido para un ajuste seguro en forma.
Transpirable malla en la parte superior para la ventilación y enfriamiento.
Phylon entresuela en el antepié para peso ligero amortiguación.
Visible Unidad Max Air en el talón maximiza amortiguación.
Waffle relleno de suela para durable, multi-superficie tracción.
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 NIKE/ Air Max Torch 4 Rosa/ para mujer Negro/ Plata Metálico/ Rosa Flash 3927ccb - 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 NIKE/ Air Max Torch 4 Rosa/ para mujer Negro/ Plata Metálico/ Rosa Flash 3927ccb

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 NIKE/ Air Max Torch 4 Rosa/ para mujer Negro/ Plata Metálico/ Rosa Flash 3927ccb

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 deporte con cordones Sneakers de hombre Carson Runner de PUMA Blue Atoll-white NIKE Mujeres Downshifter 7 Negro / Mtlc Hematita Antracita Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Nube de lluvia / Artic Fox , Zapato de running ASICS GT-1000 6 para mujer Porcelana Azul / Humo Azul / Blanco Zapatillas de running adidas Springblade Razor para mujer Power Teal / Frost Mint / Sol Gold , Zapatillas de running NIKE Lunartempo 2 para mujer Rosado Zapatillas de running NIKE Airstream Sequent 3 para mujer Sequoia / Light Bone / Metallic Silver , Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Negro / Fantasma Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Deep Porcelain Blue / Overcast , Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / Campanula / Carbono , Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla del océano / Cirrus Blue / College Navy , Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / Hola Lite , Zapatillas de running Wstro para mujer New Balance Negro / gris / Lime Glow Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Baya del Saúco / Coral Vivo , Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Cubierto Gris / Blanco / Cristal , Zapatillas de running ASICS GT-2000 6 para mujer Gris medio / plata / carbono Calzado de running NIKE RN 2017 2017 para mujer Black / Hot Punch / Aurora Green Zapatillas de running NIKE Air Zoom Vomero 12 para mujer Lava Glow / Black-racer Pink Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosado Zapatillas de running Fresh Foam 1080v6 de New Balance para mujer Gris azulado , Zapatillas de running Questr Cc W de mujer adidas Aero Blue S, Aero Blue S, Colegiata Marina , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Deep Borgoña / Pure Platinum Calzado deportivo New Balance Vazee Urge v1 para mujer Blanco negro Zapatillas de running Saucony para mujer Hurricane ISO 2 Azul / Plata / Citron Zapatillas para correr Nike Zoom All Out Flyknit para mujer rojo Zapatillas de running NIKE Air Max 90 para mujer Negro / Negro / Gran Gris Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Blanco Aurora Verde 104 ,
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 Uneek O2-w de mujer KEEN Anemone / Bright Rose Correa de plataforma PUMA para mujer Satin En Pointe WN Sneaker Puma Black-puma Black , Sandalia Drew Angela para mujer Bone Smooth Leather , Corredor Merrell para mujeres, agilidad Peak Flex Trail Armada Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / gris Salomon Women s Speedcross 4 CS W Trail Runner Sombra Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Estaño , Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow , Sandalia Tamara Trotters para mujeres Negro , CLARKS Sandalia Saylie Moon para mujer Cuero azul marino , NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco Bota de chukka Bootie Bootie de cuero de mujer Comfort de Cara Comfort para mujer Negro , Calzado de mujer de fila Cress Walking Negro / Negro / Negro , Naturalizer Samantha Pointed Toe Flat para mujer Oro negro , Diadora Unisex Intrepid NYL Vermillion / Naranja , Sandalia de la diapositiva Adalea de las mujeres de Dolce Vita Cuero Ivory Sandalia de viaje Via Spiga Women s Porter Pavo real gamuza Sandalia de cuero Nirveli de Nine West para mujer Cuero blanco 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) 6030-gold Stardust & Gold Leather , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 6006- Tan Pu , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Negro / Blanco Gunsmoke Oil Grey Mocasines casuales Alpargata Deconstructed de TOMS para mujer Pale Pink Suede , Zapato de bailarina Swing Tic-Tac-Toes para mujer champán Zapatillas Zeppa en suela francesa para mujer Impresión roja del remolino SOREL Women s Out N About Slippers Negro , Bomba de mujer David Tate Dainty Armada , Botas de tacón alto de mujer Kendall + Kylie Brett Tela multi negra Bota de campo Timberland para mujer Camdale Nubuck de trigo , Zapatos holgazanes Joleen de mujer Alegria Cenizas a las cenizas

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 NIKE/ Air Max Torch 4 Rosa/ para mujer Negro/ Plata Metálico/ Rosa Flash 3927ccb - 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 NIKE/ Air Max Torch 4 Rosa/ para mujer Negro/ Plata Metálico/ Rosa Flash 3927ccb 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 91000
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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.