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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100% poliéster Malla
Importada
Caucho suela
Ajuste de forma y sandalia abierta aduladores superiores el pie
estiramiento cierre ajustable sistema con fácil lanzamiento lateral
Serrado diseño del terminal de la cuchilla suela
Zero Gota
Máquina lavar frio seco
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


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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 Vibram VI-s Cross-Trainer Vibram para mujer Nightshade Violet/ B00JPQL9RS Violet 4dcc101

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 Vibram VI-s Cross-Trainer Vibram para mujer Nightshade Violet/ B00JPQL9RS Violet 4dcc101

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

Entrenador cruzado New Foam Crush V1 de New Balance para mujer Silver Mink / Pigment Zapatillas de entrenamiento cruzado WOM s Tazon 6 Accent WN de PUMA para mujer Quarry-puma White-aruba Blue , New Balance - Braguitas Nergize V1 FuelCore Cross Trainer para mujeres Negro , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Poción de amor negro Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Black / Puma Black , Zapatillas de deporte Skech Air Infinity Fashion Skechers Sport para mujer Negro / rosa / blanco Zapatos de entrenamiento Nike Free TR 6 para mujer Wolf Gray / Smokey Blue / Mica Blue / White Zapatillas de deporte Skech Air Infinity Fashion Skechers Sport para mujer Azul marino / azul claro , Sandalias Casual Slide de Michael Kors MK rojo , Under Armour Charged All Around Speedknit True Ink (918) / Blanco , adidas Athletics 24/7 Tr M Cross Trainer Core Black / Core Black / Hi-res Blue , Zapatillas Superga 2750 Cotu para mujer Rosa / Multi , Zapatillas de entrenamiento Puma Mujeres Feroz Nubuck Naturals Negro Zapatillas de deporte Fit Fitness GEL de ASICS para mujer Taupe / Cotton Candy / Coral Rose Zapato de entrenamiento cruzado TR-4 Trainer Gel-Craze TRY de ASICS para mujer Gris medio / gris medio / blanco Zapato de Cross-Trainer de PUMA Women s Burst Multi WN Puma Black-puma White , adidas Women Cloudfoam Lite Racer Slip on Shoes Azul marino / blanco , Zapato de entrenamiento WX824 para mujer New Balance Negro , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Wolf Gray / Black / Cool Gray / Total Crimson Under Armour Charged All-Around Neutral Nordic Green (903) / Nori Green , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Ultra Magenta-puma Blanco , Zapatos de entrenamiento Nike Free TR 6 para mujer Negro / Carmesí total - Blanco , adidas Athletics 24/7 Tr M Cross Trainer Acero sin Procesar / Acero Crudo / Núcleo Negro Zapato de entrenamiento WX824 para mujer New Balance Flint Gray Under Armour Charged All-Around Neutral Nordic Green (903) / Nori Green , K-Swiss Tubos de mujer Infinity CMF Cross Trainer Titanio / Oro Artesanal , Zapatillas PUMA Tazon 6 Wide Wn para mujer Puma White-island Paradise-puma Silver Zapatos de entrenamiento Nike Free TR 6 para mujer blanco Zapatillas adidas Performance Crazytrain Pro-Mid Cross-Trainer-Shoes, para mujer Negro / Negro / 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 Saucony Women s Ride 10 Running Shoe Gunmetal / rosa , Zapatillas de running adidas Ultraboost W para mujer Misterio Azul / Misterio Azul / Vapor Gris , Zapatillas de running Reebok Zpump Fusion 2.0 para mujer Silver Metallic / White / Opal / Electric Peach / Celestial Orchid Chaco Women s ZX2 Classic Athletic Sandal Negro , Sandalias Swiftwater Mesh de Crocs para mujer Negro , Sandalia original Teva Women s Original Miramar Fade Coral Sand Multi New Balance Wrl247fc para mujeres Silver Min , Zapatillas de running NIKE Air Max 2017 para mujer Té Berry Puerto Vino Lobo Gris 601 , Zapatos OLUKAI Pehuea - Mujer Bronce / Dk Java , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Gray 789 Sandalias EVA Gizeh de mujer Birkenstock Antracita Metálica Drew Smiles Slip para mujer Azul / Microdot , Sandalias de cuero de Viv Viv de TOMS Chambray negro texturizado Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Blanco , Sandalia de tira de Ghillie Bre-Italy Bella Vita para mujer Cuero de ante de tabaco , Jambu Women s Spain Flat Harina de avena Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Astilla-c (talón cónico) LifeStride Women s Camille Sandal Rosa caliente , Sandalias de cuero de Viv Viv de TOMS Rose Gold Specchio , Zapato para caminar WW813 para mujer New Balance Gris Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Rosado Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker borgoña Sandalia de mujer de Worishofer 251 rojo Zapatillas de ciclismo X-Road Fuel IV Pearl iZUMi para mujer Vino Púrpura / Shadow Gray PADDERS Women s Repose Super Soft microfibra Plain Toe Slipper (azul marino) EE Wide , Bella Vita Womens Mel-Italy Camel cuero italiano , Spring Step - Chino para mujer Negro , ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Pisos tachonado Zapatillas Zapatos Azul claro , Mocasín Huntley de Naturalizer para mujer Cuero de pimienta roja , Slip clásico de mujer de SAS en Armada ,

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 Vibram VI-s Cross-Trainer Vibram para mujer Nightshade Violet/ B00JPQL9RS Violet 4dcc101 - 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 Vibram VI-s Cross-Trainer Vibram para mujer Nightshade Violet/ B00JPQL9RS Violet 4dcc101 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 595437
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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.