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
Hecho en EE. UU. o importado
la marca sugiere ordenando un lleno medir como esto el estilo corre pequeño en comparación con el tradicional Talla europea.
Estilo y la comodidad viene en forma rara y es posible con la joya Slingback Mary jane de Wolky®.
Ofrecido en una variedad de cueros, la joya ofrece un gancho y lazo sistema de cierre para asegurar fácil encendido y apagado.
Suave forro de cuero mantiene los pies sintiendo cómodo y seco todo el día.
Extraíble corcho anatómico y plantilla de látex masajes el pie con todos y cada uno paso. Poliuretano la suela asegura durabilidad liviana en una variedad de superficies.La joya es aceptado por el podiatra americano Asociación Médica.Importado.Medidas: Altura del tacón: 1 3 & frasl; 4 in Peso: 12 Plataforma oz Altura: 1 1 y frasl; 4 en mediciones de productos fueron tomadas usando tamaño 40 (mujeres de EE. UU. 8.5-9), ancho B - Medio. tenga en cuenta que las mediciones puede variar según el tamaño. Peso de foo
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

Wolky Comfort Jewel Choque B00W5K3D5M de Jewel Wolky Multi Picasso b492285

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.


Wolky Comfort Jewel Choque B00W5K3D5M de Jewel Wolky Multi Picasso b492285

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

Sandalia deportiva Chaco Zvolv X2 para mujer , CLARKS Sandalia Saylie Jade para mujer Cuero caído negro Wolky Comfort Jewel Cuero de brandy , Zapato de senderismo KEEN Mujer Arroyo II Chocolate Chip / Sap Green , Sandalia de mujer Drew Alana Naranja amarillo , Sandalia de vestir Tillman de SoftWalk para mujer Broncearse OluKai Ohana Paraíso / negro , Sandalia Savannah de Ryka para mujer Gris oscuro Sandalias Bali KEEN para mujer Nórdico azul / azul ópalo , Sandalia Yucatan ECCO Mujer Irritante Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose Calzado casual Slip-On de Bernie Mev para mujer Bronce Multi , Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Sodalita Sandalias Newport H2 de KEEN para mujer Rosa negra / brillante , Wolky Comfort Jewel 70000 canales negros , Chaco Women s ZX2 Classic Athletic Sandal Negro Sandalia de mujer Drew Alana Naranja amarillo , Sandalias Drew de mujer de Drew Piel azul pizarra , Chaco Women s ZX2 Classic Athletic Sandal Trullo angular Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose Wolky Comfort Jewel White Stone Print , Sandalia NIKE Benassi JDI para mujer Light Bone / Sai , Sandalia de mujer Drew Alana Blanco negro , CLARKS Sandalia de pescador de melón Tuvia para mujer Tan sintético , Sandalia Terrell Ari Convert Sport de Merrell para mujer Aluminio , Sandalias de Gladiador Mujeres Fretwork Sandalias de Tacón Extrañas Mujeres Zapatos de Cuña de Verano para Mujer Negro Sandalia Shannon del Dr. Comfort Women Camello Skechers Performance On The Go 400 Tropical Flip Flop Natural / Coral Tropical Amalfi de Rangoni Women s Candito Sport Sandal Rame / Grigionabuk Liz
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 Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7643 Negro , Zapatillas de running NIKE Lunar Skyelux para mujer Golpe caliente / Marrón nocturno Zapatillas para correr Saucony Freedom Iso Unisex Oro blanco , Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa , Adicción a las mujeres de Brooks 13 Ébano / Plata / Pagoda Azul , Zapatillas de running Mizuno Women s Wave Kazan Trail Gunmetal / Yellow Cream , Zapato acuático Merrell Capra Rapid Sieve para mujer Negro Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Tan Heritage / Diamond Tribal Sandalia Donald J Pliner Mujer Maui Sillín Zapato KEEN Presidio MJ para mujer Negro Vivien Flat de Bella Vita para mujeres Champagne cuero Propet Ghillie Walker Sandal de las mujeres Azul , David Tate para mujer Nelly Turquesa Comfortiva Mujer, Faye Sandalia de tacón medio Rubí Zapato de entrenamiento WX1267 para mujer New Balance Negro / rosa Zapatillas de senderismo Pro Pro 3D Pro de mujer Ryka Gris / rosa FitFlop Womens F-Sporty II Python Print Blanco Urbano Zapatillas de deporte Sneak Name de FLEXX para mujer Corda Elba Intreccio Zapatillas de tenis 696v3 New Balance para mujer Blanco / Alpha Pink , Sandalia Sharpie de mujer Steve Madden Color de malva Sandalia para mujer FRYE Ally Criss Cross Slide Negro , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2829- Blanco y Plata Bomba de vestido Pointe para mujer Easy Street Patente desnuda 50 sombras de blanco 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) 5517- Satén blanco Zapatilla de piel sintética Cally Minnetonka para mujer Canela , Dr. Martens Womens Eastern Art Pascal Boots Grano de paja revés oriental Me Too Arena Sandalia para mujer New Luggage Goat Spore Tiempo libre para mujeres de SAS Moca Bota de agua caliente Belleville 390 Broncearse , Mocasín de conducción Minty de Minnetonka para mujer Brown Ruff

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 Wolky Comfort Jewel Choque B00W5K3D5M de Jewel Wolky Multi Picasso b492285 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2211772
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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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