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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Textil / Sintético
Importado
Sintético único
Eje mide aproximadamente parte baja del arco
Runner tipo: neutral
BOUNCE proporciona energizado comodidad para todos deportes todo el día
sin costura FORGEDMESH superior diseñado con áreas de apoyo y estirar para ayudar asegurar una costumbre ajuste que se adapta a cada movimiento
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 Amarillo de running para blanco adidas Alphabounce Em W para mujer Amarillo hielo/ Semi Solar amarillo/ blanco 0d00c79 - 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 Amarillo de running para blanco adidas Alphabounce Em W para mujer Amarillo hielo/ Semi Solar amarillo/ blanco 0d00c79

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 Amarillo de running para blanco adidas Alphabounce Em W para mujer Amarillo hielo/ Semi Solar amarillo/ blanco 0d00c79

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

Zapato deportivo ASICS Gel-Kayano 24 para mujer Blanco / azul violeta / amarillo de seguridad Zapatillas de running adidas Cloudfoam Ultimate para mujer Blanco / Blanco / Gris Tres Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Verde Zapatillas de running adidas Ultraboost W para mujer Gray Five / Carbon / Ash Pearl Zapato de running ASICS Gt-2000 5 para mujer Negro / Bay / Diva Pink Zapatillas deportivas NIKE Lunar Apparent para hombre Negro rojo antracita blanco Zapatillas de entrenamiento NIKE Metcon Dsx Flyknit para mujer Negro / Blanco-total Crimson-medium Blue New Balance Women s w990v4 Running Shoe Negro / Plata , Zapatillas de running NIKE Air Max Thea para mujer Gym Red / University Red , New Balance W880gb6 para mujeres Gris azul Zapatillas de running ST-2 para mujer Topo Athletic Fucsia / Negro , Zapatillas de deporte PUMA Womens Speed ​​600 Ignite Low Top con cordones rojo , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Indigo Blue / Paradise Green / Pink Glow Zapatillas de running NIKE Air Huarache para hombre Mango brillante / blanco Zapatillas de running ASICS Gel-Kayano 23 para mujer Blanco / Plata / Acuario Pantalón largo clásico NIKE Golf para mujer Tech Essentials Ember Glow Team Red Black 802 , Zapatillas de running Wstro para mujer New Balance Alpha Pink / Black / Lime Glow Zapato de correr ideal para rejilla de mujer Saucony Negro / Gris / Pr Zapatillas para correr Saucony Women s Cohesion 10 Azul marino Zapatillas de running ASICS Gel-Contend 4 para mujer Negro NIKE Men s Benassi Just Do It Athletic Sandal Blanco / puro platino / negro / blanco , Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco / Bahía / Plata Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Negro / Gris Sólido , Zapatillas de running New Balance Boracay V3 para mujer Supercell / Ozone Blue / Bleached Sunrise Brooks Womens Neuro 2 Noche Azul / China Azul / Coral Fiery Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Acero / Trueno , Zapatillas de running Saucony Women s Freedom ISO Azul Zapatillas de running Gel-Moya para mujer de ASICS Azul blanco
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 Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Lobo Gris / Blanco-Puro Platino-Negro Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Ture White Zapatillas de trail running Hierro V3 para mujer New Balance Coral vivo / Pigmento Sandalia Savannah de Ryka para mujer Gris oscuro Calzado deportivo New Balance Vazee Urge v1 para mujer Gris / rosa , Zapatillas de trail running Inov-8 Trailtalon 235 (W) para mujer Teal Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Gris negro , Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Armada , Zapatillas para mujer Cloud 2568 Caliber Slide Mystere , Sandalias de mujer Ros Hommerson Mellow Armada MBT Women s Sport 2 (Le) Zapatillas atléticas para caminar Blanco CLARKS Sandalia Arla Elin para mujer Textil sólido negro , Chaco Mujer Fallon Sandal Arena , UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Púrpura / blanco , Zapatillas clásicas Vionic para mujer Walker Taupe / Taupe Drew Shoe Women s Tulip Comfort Brown Croc Zapatillas tenis NIKE Air Zoom Ultra para mujer Hidrógeno Azul / Metálico Gris Oscuro / Blanco Zapato Dantera, para hombre, L Low Waxed Court Oro blanco / ámbar Converse Chuck Taylor All Star Core Ox Blanco Colección casual para mujer FootJoy Azul profundo Dakota femenina de UGG Ante negro Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Malt Taupe Action Leather , 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) 1692- Cuero dorado y malla de carne Zapato ecuestre Federica Suede de Marvin K. de Aquatalia by Marvin K. Café exprés , Ringor Bandit 2.0 Spike- Pitching Rojo 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 Stardust de oro y cuero Bota holgazana novata Fergalicious para mujer Negro Wolverine 1883 Botines con botines Jean Chelsea, para mujer Negro , Bota de lluvia Pamp Sport Cuff Wpn para hombre de Palladium Brindle Brown Hush Puppies Womens Chowchow WT Oxford Gamuza gris hielo ,

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 Amarillo de running para blanco adidas Alphabounce Em W para mujer Amarillo hielo/ Semi Solar amarillo/ blanco 0d00c79 - 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 Amarillo de running para blanco adidas Alphabounce Em W para mujer Amarillo hielo/ Semi Solar amarillo/ blanco 0d00c79 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 4575566
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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.