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.

Sandalias casual B07BVPDZLB Debra para 19 2 Strap Strap de Josef Seibel para mujer Castagne a180848




Cuero
sintético Parte superior
Parte superior: Cuero
Forro: Cuero
Cierre: Rip Tape
Tacón: 2cm
Fabricante Código: 76719
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


Sandalias casual B07BVPDZLB Debra para 19 2 Strap Strap de Josef Seibel para mujer Castagne a180848 - 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

Sandalias casual B07BVPDZLB Debra para 19 2 Strap Strap de Josef Seibel para mujer Castagne a180848

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.


Sandalias casual B07BVPDZLB Debra para 19 2 Strap Strap de Josef Seibel para mujer Castagne a180848

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

Sandalias de tacón de soñador Easy Street para mujer Blanco Sandalia Sintética Mydebut de Nueve Mujeres del Oeste Negro , Sandalia de gamuza Poppy de TOMS para mujer Cuero negro J.Renee Womens Soncino champán Sam Edelman Sandalia Dariel para mujer Estampado de algodón negro / blanco , Sandalia de mujer de nivel medio Aerosoles Tela negra Sandalia adidas Duramo Slide Blanco / Poder Azul / Rojo , Toalla de vestir Dulce para mujer Touch Ups Silver Shimmer , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Estaño , Charles by Charles David Mujeres inquietantes Mary Jane Microsuede negro Bomba de vestido Nika BeautiFeel para mujer Negro , Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Red-d , Sandalia de cuña Santorini Jambu para mujer Armada , Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro Sandalias de diapositivas casuales Easy Street Womens Conca Fabric Open Toe Tela elástica blanca , Sandalias de mujer de Worishofer 251 Armada , Flip-Flops de The North Face para hombre Base Camp Lite Olive Nite Yellow y Four Leaf Clover Skechers - Sandalia deportiva de 600 brilliancy para mujer Carbón Chancleta Yoga Sling 2 Sanuk para mujer Natural Multi Radio Love , XYD Prom Dancing Shoes Elegant Open Toe Strappy Sandalias de tacón de tobillo Wrap vestido de bombas para las mujeres Desnudo Wolky Womens Nomad Cuero Cartago Rojo Naturalizer Womens Canary Cuero negro Marc Fisher LTD Womens Alida Espadrille Wedge Azul medio gamuza Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra Sandalia Tallis de mujer VANELi Black Summer Vip , Splendid Women s Brittani Sandal Plata , Salvatore Ferragamo Mujer Varina Nerón Sperry Top-Sider Women s Sunkiss Pearl Sandalia Marfil / Rosa Sandalia Pavi de estilo suave para mujer Vitello negro
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 Sandalia Uneek O2-w de mujer KEEN Opal azul / azul provincial , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Negro Zapatillas running para mujer Saucony Liteform para mujer Negro Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Nube de lluvia / Artic Fox Camiseta adidas Swift Run W para mujer Maroon / Maroon / Blanco , Calzado de running Fila Core s Memory Core Callibration 3 Negro / Monumento / Sugarplum , Zapatillas de running Questr Cc W de mujer adidas Aero Blue S, Aero Blue S, Colegiata Marina , Zapatillas de deporte Skechers Dreamstep Cool Cutie Fashion para mujer Carbón , DAWGS Womens Arch Support Z Sandals Globos , Zapato para caminar WW577 para mujer New Balance Negro , Under Armour Threadborne Push para mujer Gris Nublado (102) / Rosa Tropic , Entrenador cruzado Agility V2 de New Balance para mujeres Fiji / Blanco NAOT Mujeres Fiona Noguck terciopelo negro / Cuero Beige color caqui / Cuero espejo plateado , nativo AP Mercury Liteknit Leche Rosa / Shell Blanco / Caucho Natural , adidas Originals Hombres Superstar 80s PK Gris Phenma femenino de PUMA EP Rock Ridge / Beige metalizado Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker borgoña Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Green Palm Print Sandalia de vestir Lauren de Jack Rogers para mujer Blanco Blanco Sandalia Maria Slide de Nanette Lepore para mujer Blanquecino 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) S92307 Gold Flower & Gold Bomba de cuña Farrow para mujeres LifeStride Gris oscuro , Saucony Women s Showdown 4 Zapatillas de atletismo Negro / rosa Sandalias de vestir Patti de Sam Edelman para mujer Patente negra TOMS - Slip-On de lienzo clásico para hombre Lana negra / Faux Shearling TOMS Clásicos de mujer (Rojo) , Mocasines de vestir de punta estrecha para mujer de Me Too Women Aceituna Joules Women s Rockingham Botines Dark Leopard Rubber , Mocasín de gamuza Kilty de Minnetonka para mujer Gris , Zapatos OLUKAI Pehuea - Mujer Océanos / océanos ,

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.

Sandalias casual B07BVPDZLB Debra para 19 2 Strap Strap de Josef Seibel para mujer Castagne a180848 - 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 Sandalias casual B07BVPDZLB Debra para 19 2 Strap Strap de Josef Seibel para mujer Castagne a180848 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
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1080694
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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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