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
Caucho único
Eje mide aproximadamente parte baja del arco
Fresco Entresuela de espuma
Bootie Construcción
TPU Contador externo del talón
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

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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.


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

NIKE Mujeres Air Zoom Pegasus 33 Platino / Negro / Gris / Rosa explosión Zapatillas de running NIKE Unisex Flyknit Racer Negro / gris oscuro / blanco Calzado de running rápido para mujer Reebok, tejido flexible Cool Shadow / Porcelain / Blue Lagoon / White / Blue Zapatillas de running Adizero Prime Sp de adidas Originals Núcleo Negro, Ftwr Blanco, Naranja Calzado de running rápido para mujer Reebok, tejido flexible Blanco / gris ceniza / negro / polvo del desierto , Skechers Women s Go Ride 7 Running Shoe Blanco negro Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco negro Zapato de running ASICS GEL-Kayano 22 para mujer Onyx / Plata / Carbón de leña , Sandalia deportiva NIKE Kawa Slide para hombre Lobo gris / antracita / gris lobo , Zapatillas Nike Women s Flex Trainer 5 Antracita / Blanco / Gris Oscuro / Hipervioleta , Zapatillas de running NIKE Roshe Two para mujer Blanco / Blanco-Puro Platino Converse Chuck Taylor All Star Seasonal Color Hola Brillo de sol Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , Zapatillas NIKE Women s Revolution 3 para correr Negro / Metálico Gris Oscuro / Antracita Zapato de senderismo KEEN para mujer Versago Dahlia roja / coral profundo Zapatillas de running adidas Alphabounce Cr W, mujer Core Black / Core Black / Core Black Zapatillas de running New Balance Coast V3 para mujer Silver Mink Zapatillas de running ASICS Gel-Scram 4 para mujer Negro / gris oscuro , Zapatillas de running Spira Phoenix para mujer con resortes Salmón / blanco Zapatillas de running Adidas Performance Ultra s Boost M para hombre Negro / Negro / Gris Sólido NIKE Kids gratis Rn (grande) Hyper Violet / Black / White / Hyper Turquoise , Zapatillas de running A7000WX12 para mujer Apex gris Converse Chuck Taylor All Star Seasonal Color Hola Azul , adidas MUJERES ORIGINALES CLIMACOOL 02/17 ZAPATOS # BY9289 Tactile Rose / Footwear Blanco , Zapatillas NIKE Women s Revolution 3 para correr Blanco / Púrpura Tierra / Lobo Gris , Sandalias OLKAKAI Ohana - Hombre Profundidad / Negro Zapatillas de running Ultra Kalani 3.0 de Zoot para mujer Splash / Flame / Gulf Stream , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Pizarra negra / negra / oscura Zapatillas náuticas para mujer Saucony Originals Gris / Azul / Citron
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 Roxy Women s Set Session Zapatillas atléticas para caminar Armada , Lanzamiento de mujeres Brooks 5 - Boston Marathon, Boston Blue / Vida nocturna / Blanco Boston Blue / Vida Nocturna / Blanco Zapatillas de deporte de moda Kanye West Desert Rat 500 Zapatillas de correr 3M para hombres Mujeres DB2908 DB2966 Sonrojo Zapatillas de running NIKE Air Huarache City para mujer Vela / Luz Menta-lt Racer Blue , Propet Women s TravelActiv Ss Sandal Azul negro Colchoneta de amortiguación 620v2 New Balance para mujer Silver Mink / Gunmetal Zapatillas Slip-On Niyama de mujer Teva borgoña , EQUICK Water Shoes Aqua Sports Sneakers Slip on Quick Dry para Hombres Mujeres Niños Pesca 1y.black Calzado de espuma fresca New Balance para mujer 1165v1 Gris blanco , Zapatillas MBT Hani 6s para mujer Patente negra Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , Sandalia Verona mujer Alegria Pearl Rose Skechers Performance para mujer Go Step Lite Deluxe Carbón ASICS Womens Weldon X Negro / Carbono / Blanco CLARKS Leisa Lakia Sandalia para mujer 9 B (M) US Tan oscuro - Zapatillas de skate Heathrow DC para mujer Rosa brillante , Vans Unisex Old Skool Zapatillas clásicas de skate Blanco negro , Zapatillas NIKE Roshe One Nylon para mujer Coral Stardust / Blanco Zapatillas de deporte Adidas Adipure Sport para mujer Rosado , Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Cuero negro de acción Sandalia Maci de mujer Dansko Negro Full Grain , Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero desnudo , 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 6005- Tan satinado Mocasines con suela de piel de oveja Minnetonka para mujer - 3341 Gris Zapatilla de ante mocasín con forro de piel Alena de OZZEG para mujer Negro Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro , Mocasín Sabrina Moc II Lamo para mujer Blue Island , Naturalizer Michelle s Dress Dress Pump Serpiente negra ED Ellen DeGeneres Mujeres Lilliane2 Negro / Oro Animal Avalon Nappa / Eco Sheep , Zapatos de mujer de Babe Oxford de zapatos deseados Azul marino / Multi

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 New de running New Balance Fresh 3947 Foam Foam Lazr V1 Melocotón espolvoreado/ Melocotón espolvoreado c217c4d 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.

Corriendo
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.