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.

Botines metálicos opciones A-Game 4074 para mujer marino Boombah - 6 opciones de color - Varios tamaños Azul marino/ rojo 90ef4a2




Nuestro A-Juego para mujeres tacos de metal utilizan grano profundo sintético superposiciones de cuero combinado con nubuck subcapas para crear un clásico y duradero superior
Se adapta fiel al tamaño. 10 onzas. y lengua aumentar transpirabilidad y bajar de peso
Duro puntera con goma refuerzo revestido protege contra punta de arrastre con adicional cobertura en medial lado del zapato para fastpitch específico durabilidad
TPU suela con un metal configuración de la cala que proporciona el mejor delantero y explosión lateral
Desde Boombah - el líder en rendimiento deportes de ingeniería vestimenta y equipo.
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


Botines metálicos opciones A-Game 4074 para mujer marino Boombah - 6 opciones de color - Varios tamaños Azul marino/ rojo 90ef4a2 - 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

Botines metálicos opciones A-Game 4074 para mujer marino Boombah - 6 opciones de color - Varios tamaños Azul marino/ rojo 90ef4a2

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.


Botines metálicos opciones A-Game 4074 para mujer marino Boombah - 6 opciones de color - Varios tamaños Azul marino/ rojo 90ef4a2

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

Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Blanco negro Mizuno Women s 9 Spike ADV Finch Elite 2 Juego de sordina de softbol moldeado rápido Azul marino , Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Azul marino / Columbia Zapatillas de beisbol Synchro MX para mujer Mizuno Blanco negro Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Royal / Red Zapatillas de beisbol de fútbol de nueva generación Fusev1 Metal Fast Pitch de New Balance para mujeres Blanco negro Calzas de la dinastía Azul marino / blanco , New Balance Sm4040r1 para mujeres Rojo blanco Mizuno Women s 9 Spike ADV Finch Elite 2 Juego de sordina de softbol moldeado rápido Blanco negro Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Negro / real Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Real / Blanco , 3N2 Mofo Turf Trainer rojo , Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Blanco negro , Zapatillas de samobal Swar 5 Fastpitch Cleat de Mizuno para mujer Blanco Blanco Zapatillas de beisbol Softball Velo V1 para mujer New Balance Azul marino / blanco , Mizuno Women 9 Spike Swift 4 Fast Pitch Metal Softball Cleat Blanco negro Zapato de césped de la dinastía Ringor Blanco negro , Calzas de la dinastía Plata blanca Mizuno Women s 9-Spike Advanced Sweep 3 Metal Softball Cleats - Blanco y Blanco Calzas de la dinastía Naranja / Blanco Ringor Bandit 2.0 Spike- Pitching Rojo blanco , Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Azul marino / rojo , Boombah Women s A-Game - Juego de calas intermedias moldeadas - 8 opciones de color - Varios tamaños Azul marino / rojo , Mizuno Women s 9 Spike ADV Finch Elite 2 Juego de sordina de softbol moldeado rápido Blanco negro Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Azul marino / Columbia Ringor Dynasty Turf Shoe- Lanzamiento Blanco negro Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Negro / Rojo Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Blanco Blanco , Calzas de la dinastía Plata blanca ,
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 NIKE Femenil Flex 2016 Rn Running Pink Blast / Blanco / Negro / Eléctrico Verde Carro de mujer Brooks Berenjena , Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Fila Navy / Wedgewood / Deep Pink Zapatillas de running New Balance para mujer Vazee Coast Azul blanco , Zapatillas de running New Balance para mujer 590v3 Negro / Castlerock , Zapato de agua NRS Crush - para mujeres Azul celeste Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Cuero blanco / malla blanca Alegria Womens Kourtney Exclusive Dama , Orthofeet más cómodo Fascitis plantar diabético zapatillas de deporte de la mujer - 941 Sandalia Huarache Easy Street Allure para mujer Blanco / Metálico Imprimir , Zapatillas de running New Balance 990V3 para mujer Gris Sandalia Chantel de mujer Steve Madden Blush Multi , Drew Shoe La alegría de las mujeres Combo de la Marina , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Oro negro Teva ALP Premier - Mujeres Armada J.Renee Womens Mcwayfalls Negro , Imperio de las mujeres de Skechers - Zapatos de entrenamiento de tela con alto tobillo Blanco negro , Zapatillas Superga 2750 Cotu para mujer Rosa / Multi , Sandalia de la diapositiva de Florene de las mujeres de Lucky Brand Macarrón Charles by Charles David Women s Victorious Slide Sandal Marfil , Sandalias de mujer de Jack Rogers Delilah, negro, 8 medios US - Sandalia Maria Slide de Nanette Lepore para mujer Negro , Salón de baile de mujeres Zapatillas Dance Salsa Práctica latina Zapatos 1670cEB Cómodo-Muy Fino 1.5 \ de 5] Cuero marrón Así que Danca CH99 3 Heel con T-Strap Character Shoe Caramelo , Zapatillas de ante negras de Laurentian Chief para mujer con mocasines con cuello de piel de conejo - Bota de invierno UGG Classic Mini II para mujer Coral vibrante , Tamaris Women s Locust 28634 Oxford Flat Pepper Combo Zapatillas de deporte y construcción Caterpillar Superstat Waterproof Nano Toe / Tater para mujer Negro , Easy Street Lucky Oxford para mujeres Ante negro , TOMS - Slip-on tejido clásico para mujer Arpillera ,

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.

Botines metálicos opciones A-Game 4074 para mujer marino Boombah - 6 opciones de color - Varios tamaños Azul marino/ rojo 90ef4a2 - 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 Botines metálicos opciones A-Game 4074 para mujer marino Boombah - 6 opciones de color - Varios tamaños Azul marino/ rojo 90ef4a2 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.

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

Geometry Construction Reference

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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.


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