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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Cuero y Sintético
Importado
Sintético suela
Medidas de talón aproximadamente 1
Cuero nubuck, parte superior de tela
UniFly entresuela
Vibram suela
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 de agua 19996 agua Merrell Heaven para mujer All Out Blaze Sieve Brown Sugar/ Blue Heaven 59bd175

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 agua 19996 agua Merrell Heaven para mujer All Out Blaze Sieve Brown Sugar/ Blue Heaven 59bd175

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

NRS Desperado Wetshoe Azul negro , Sandalias EVA Gizeh de mujer Birkenstock Eva rojo Zapato de agua NRS Crush - para mujeres Gunmetal , Zapato para agua anfibia Speedo para mujeres Offshore Amarillo , Zapatos OLUKAI Pehuea - Mujer Bronce / Dk Java CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Knightrider Clorts Women & Men s Water Shoe Sandalia para caminar de secado rápido WT05 Rosado Salomon Women s RX Break Sandal Estanque reflectante / Deep Peacock Blue / Lime Green , Zapato de entrenamiento cruzado deportivo Hydro Sport para mujer RYKA Detox Blue / Twinkle Blue / Chrome Silver , Zapatillas de agua para mujeres Astral Porter Azul marino / blanco Zapato de agua NRS Crush - para mujeres Azul celeste Zapatos OLUKAI Pehuea - Mujer Trench Blue / Trench Blue Zapato de agua de la Isla Komomo para mujer de Tommy Bahama Nadar de noche , SoftScience The Drift Canavas Comfort Calzado casual unisex Caqui Zapatillas de senderismo acuáticas para mujer Astral TR1 Junction Granito gris , Zapatos OLUKAI Pehuea - Mujer Negro Negro Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Negro 1786 Zapatillas de agua Vibram V-Aqua gris / azul para mujer Gris azul Zapato de agua Tetrex Rapid Crest para mujer Merrell Lima , Sandalias EVA Gizeh de mujer Birkenstock Scuba Coral Eva , Zapatos OLUKAI Pehuea - Mujer Pale Gray / Carboncillo Zapatillas de agua Ccilu Women s Link W rojo Zapatillas de agua ZipWalker para mujer Speedo Rosa / negro Sandalias EVA Gizeh de mujer Birkenstock Metallic Anthracite Eva Zapatillas de senderismo acuáticas para hombre Astral TR1 Junction Deep Water Navy , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Negro Sandalias EVA Gizeh de mujer Birkenstock Plata Astral Mary Jay - Mujeres Negro , Clorts Zapato de agua cerrado para mujer Sandalia de senderismo de secado rápido 3H025 Morado oscuro
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 Zapato de running ASICS GEL-Cumulus 20 para mujer Gris piedra / negro Zapatillas de running NIKE Flex Run 2018 para mujer Cool Gray / White NIKE Women s Tanjun Black Metallic Gold Zapatillas de running Reebok Print 2.0 para mujer Lila Ash / Whisper Gray / Fire Coral / Stellar Pink , Encuentro Under Armour Women s Street Cristal / blanco Zapatillas de running NIKE para mujer Air Huarache Run Ultra Elemental Rose / Wolf Gray / White , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Azul profundo / Cumbre Blanco-Negro NIKE Men s Benassi Just Do It Athletic Sandal Niebla de medianoche / blanco , Salming Elements Zapatillas para mujer Rosado Zapatillas de trail running WT10v1 Minimus para mujer New Balance Gris azul Zapatillas de running NIKE Air Vapormax Flyknit 2 para mujer Cactus negro / caliente Punch-blanco-polvoriento , Zapatillas de trail running Salomon para mujer Techamphibian 3 W Vintage Kaki Zapato para agua anfibia Speedo para mujeres Offshore Negro / púrpura , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro Calzado para caminar CUSH + WA365v1 New Balance para mujer Blanco negro , Me Too OLYMPIA1 Zapato plano de cuero para mujer Nude Nappa , Reggae Birko-Flor con Birkenstock Betula para mujeres Blanco básico Zapatillas para caminar adidas outdoor Terrex Voyager CW CP W para mujer Misterio Ruby / Dark Burgundy / Black , Sandalias con hebilla de botón Ridgeport para mujer Rockport Bronce , Sandalia Cyprine Slide de mujer Naturalizer Rosa Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Azul Indigo / Plata / Azul Regatta BareTraps Women s Rebecca Slide Sandal Black Stretch , Sandalia Dolce Vita Halle Slide para mujer Tela blanca a rayas Sandalia Dolce Vita Hazle Slide para mujer Fuera de cuero blanco Sandalia plana Kip de Trotters para mujer Armada Sandalia plana Sofie de Badgley Mischka para mujer Baya Sandalia Evelia Slide de Bandolino para mujer Tela dorada Sandalia de girasol Aerosoles para mujer Bone Multi , Zapatillas Dr. Mary Comfort Flauta Lycra Elasticables Diamante Mary Jane Licra Bomba de vestimenta Cambredge para mujer Jessica Simpson Desnudo ,

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.

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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 agua 19996 agua Merrell Heaven para mujer All Out Blaze Sieve Brown Sugar/ Blue Heaven 59bd175 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.

Zapatos de agua
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.