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.

Zapato Capezio Mary Mary 19907 Jane Tap para mujer mujer Caramelo d18b4a9




Cuero
sintético suela
Medidas de talón aproximadamente 1
Preincorporado Tele Tone toe y grifos de talón montados en tablero de fibras
Mínimo primavera del dedo del pie; suela de goma para la fuerza y flexibilidad
espuma plantilla proporciona Absorción de impacto y mayor comodidad
Algodón calcetines y forro colgado; caucho antideslizante almohadilla
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


Zapato Capezio Mary Mary 19907 Jane Tap para mujer mujer Caramelo d18b4a9 - 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

Zapato Capezio Mary Mary 19907 Jane Tap para mujer mujer Caramelo d18b4a9

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.


Zapato Capezio Mary Mary 19907 Jane Tap para mujer mujer Caramelo d18b4a9

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 Bloch para mujer, malla naranja 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 6030- Black Pu 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) 1625- Granito dorado 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 1612- Black Pu , ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ Cuero desnudo de cobre S9216 Zapatos muy finos EK6829 de la danza de salón de mujeres de las señoras con 2.5 \ Satén negro , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7001-light Tan Velvet , 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 \ 5008mirage- Party Sparkle & Ultra Silver , Zapato de danza de gema Soleil moderno para mujer de PUMA Puma Black / Puma Black , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres SERA1683 Tacón de 3 pulgadas + plegable Paquete de cepillo Plata , 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 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 \ 2721- Silver Sparklenet & Silver Trim , 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) 5017- Satén blanco y malla blanca , Adulto 3 \ Broncearse , Zapato de bailarina Swing Tic-Tac-Toes para mujer Estaño , 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) S1001 Gold Scale & Dark Gold , Gorro adulto Gore Insert Jazz, T7602 Broncearse , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ , Así que Danca CH99 3 Heel con T-Strap Character Shoe Caramelo Zapatillas de deporte unisex de bajo perfil Negro Rumpf Scooter 1575 Zapatilla de deporte Black Dance Split Sole Men Salsa Latina de Mujeres Jazz Hip Hop Fitness Zapato de ejercicio Zapato de danza de gema Soleil moderno para mujer de PUMA Puma Black / Puma Black 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) 7013- Bronceado dorado Zapatos muy finos Dance Sneakers Series SN016 Verde 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) 5004- Gold Sparkle Zapatillas para adultos con hebilla deslizante, T9200 Broncearse CAPEZIO Mujer cambre (1128W) -Petal Pink Pétalo rosa , Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ Negro Zapatos de paloma de oro Fiesta Party 1613 Noche de confort Sandalias de bomba de vestir, Salón de baile de mujeres Zapatos de baile (2.5 \ 3 \ Tacones) 1613l- Azul
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 de running ASICS Gel-Scram 4 para mujer Peacoat / Soft Sky , Adidas Women s Adilette CF + Logo W Sandalia deslizante Negro / Blanco / Negro Zapatillas de volante Nike Women s Zoom Hyperface 2 Blanco negro Chaco Women s Z1 Classic Athletic Sandal Atardecer de Yosemite Sandalia P-Oudry de Yellow Box para mujer Claro , Mocasines y zapatillas Slip-ONS Monica de TOMS para mujer Ante de pino Chanclas de mujer Havaianas Animal Verde , Zapatillas de running NIKE Runallday para mujer negro , Walking Cradles Womens Henna Nubuck negro estampado Easy Spirit Mujer Ezcool Mule Azul marino Combo de gamuza SOREL Sandalia casual Ella de mujer Natural Zapatillas de entrenamiento WX711V2 para mujer New Balance Rosado , Vans Unisex Old Skool Zapatillas clásicas de skate Gum Mono Supra Vaider LC Sneaker Khaki / Gum , Zapatillas de skate casual unisex Heathrow Prestige para hombre de DC Blanco negro , Converse Chuck Taylor All Star Core Hola Blanco óptico , Calzado de baloncesto NIKE Hyperdunk 2014 para hombre Blanco negro Sandalia tobogán X-Band con plataforma Mikenzie de Gentle Souls para mujer Seta , Sandalia para mujer KAANAS con plataforma tejida Akumal Azul , Sandalias planas Nori de Easy Street para mujer Equipaje , Mizuno Women 9 Spike Swift 4 Fast Pitch Metal Softball Cleat Gris blanco , Botas Karel para mujer de UGG Violeta gris , Zapatilla de ante mocasín con forro de piel Alena de OZZEG para mujer Negro Botín para mujer Pleaser Asp1018 / Bpu / M Black Faux Leather / Black Matte , Bota occidental Everglades SQ de Dan Post para hombre marrón , Zapatos clásicos Rockport para hombre Nubuck negro aceitoso , TOMS Kids 10009918 Alpargata-K Cinnamon Suede Perforated , Justin Boots MSL504 Mujeres punta redonda de cuero marrón botas de lona Chocolate Barocco , Zapatillas para mujer Del Rey TOMS Blue Aster Multi Space Dye Bota de lluvia Tinsley para mujer Sam Edelman Espacio azul / negro

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.

Zapato Capezio Mary Mary 19907 Jane Tap para mujer mujer Caramelo d18b4a9 - 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 Zapato Capezio Mary Mary 19907 Jane Tap para mujer mujer Caramelo d18b4a9 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2497708
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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.