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.

Sandalia Chaco Sandalia Juniper Negro Mujer Chaco Negro 246700b




Cuero
Importado
Goma único
grano completo parte superior de cuero
Antique acabado de hebilla de metal y forro de cuero
Mujer LUVSEAT específico Entresuela de EVA
cuero plantilla envuelta con bordado gráfico
No marcado ChacoGrip Eco 15% caucho reciclado compuesto
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


Sandalia Chaco Sandalia Juniper Negro Mujer Chaco Negro 246700b - 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

Sandalia Chaco Sandalia Juniper Negro Mujer Chaco Negro 246700b

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.


Sandalia Chaco Sandalia Juniper Negro Mujer Chaco Negro 246700b

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

Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave OLUKAI Hiona Sandal - Mujer Negro Huracán XLT Mujeres Teva - Avellana / Negro Hazel / Black mobils by Mephisto - Sandalias de tacón Getha para mujer Camel Savana / Platinum Magic Leather BZees por Naturalizer Dream Negro Wolky Comfort Jewel Brandy Smooth Leather Earth Origins Sophie Women s Sandal Negro , Sandalia KEEN Women s Exo-W Uneek Etherea / Plaza Taupe , Sandalias Columbia Sorrento Flip Athletic para mujer Salmón salvaje / ballena Sandalias Ahnu Women s W Tilden V Athletic Negro , Sandalia deportiva Speedy de Charles David para mujer Negro , BZees Sandalia con cuña de beso para mujer Blue Multi Print , Sandalia Teva W Terra-Float Nova de Teva para mujer Tacion Gray / Multi Chaco Women s Z1 Classic Athletic Sandal Negro BZees Sandalia Deportiva Jive para Mujer Gris Amalfi by Rangoni Sandalia deportiva Cestello para mujer Perla / Taupe Parmasoft Teva Terra-Float Active Lace - Mujeres Azul Sandalias Newport H2 de KEEN para mujer Rama de Malaquita / Plata , Adidas Women s Adilette CF + Logo W Sandalia deslizante Blanco / Negro / Choque Rosa KEEN - Sandalia de agua EVOFIT ONE para mujer para aventuras al aire libre, 6.5 M US, azul claro - Sandalia Savannah de Ryka para mujer Azul marino / verde , Sandalias Therafit Summer Women s Slingback Sport - para fascitis plantar / dolor en el pie Rosado , Sandalia para mujer Terrell Ari Wrap de Merrell Negro , NAOT Women s Prescott Slide Sandal Blanco Lthr / Silver Mirror Lthr / Cork Lthr Sanda Verra Teva para mujer Mar de Bering Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón , KEEN Hombres Newport H2 Sandal Negro , Chaco Women s Z1 Classic Athletic Sandal Trellis Teal ,
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 Nfinity Vengeance Cheer Shoe (par) Blanco , Sandalia deportiva Tirra para mujer Teva Mar de Bering , Wolky Comfort Jewel Bronce , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Blanco , Zapatillas de running adidas Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco Zapatillas de running adidas Women s Cf Racer Tr W, mujer Trace Pink / Pearl Gray / Mystery Ruby Ballet de niña de flores Aerosoles femenino Negro exótico , Sandalia de vestir Bray de Lucky Brand para mujer Vidriado , Chaco Mujer Zcloud X2 Sandalia Deportiva Angora Webb Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Tan británico Zapato de malla Nohea de OLUKAI - Hombre Negro Negro , Drew Womens Ariana Dusty Gold , Zapatillas para caminar RYKA Women s Devotion Plus Blanco / plata cromo / almendra helada , Sam Edelman Women s Gigi Marfil moderno Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Baya , Easy Spirit Deiny Women s Walking Azul oscuro Zapatillas deportivas sintéticas Drew Shoe para mujer Black Sport Mesh Easy Spirit Solana Oxford para mujeres Negro , Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Blanco azul Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Blanco invierno Sandalias de mujer Ros Hommerson Mellow Negro Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Masilla pálida , Zapatos de golf Sandbaggers Madison para mujer Ónix Sandalia de diapositivas de ocio para mujer de Seychelles Broncearse Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero desnudo Sandalias de mujer Mia de Sandalias de Jerusalén Blanco , Zapato de bailarina Swing Tic-Tac-Toes para mujer rojo Mizuno Women 9 Spike Swift 4 Fast Pitch Metal Softball Cleat Azul marino Georgia Boot para mujer Pendleton Romeo Marrón / gris oscuro Zapatillas Ida Vionic para mujer Plaid rosa ,

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.

Sandalia Chaco Sandalia Juniper Negro Mujer Chaco Negro 246700b - 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 Sandalia Chaco Sandalia Juniper Negro Mujer Chaco Negro 246700b 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 4128204
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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.