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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Malla
Importada
Sintética único
Eje mide aproximadamente Parte baja del arco
Ancho anchos
Flexible suela
Transpirable malla superior
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,

Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Gris pardo Easy Spirit - Zapatillas Faisal2 para mujer Blanco , CLARKS Janey Mae Oxford para mujeres Cuero negro Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Heather gris / azul Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Plata blanca Easy Spirit Mujer Ezcool Mule Blanco SoftWalk Women s Topeka Flat Sabio , Zapatilla S Skechers para mujer E Z Flex 3.0 Swift N Sly Coral , Zapatillas para caminar Reebok Women s Work N Cushion 3.0 Blanco / Acero_1 , Calzado de espuma fresca para mujer New Balance 1100V1 Púrpura / blanco , Finn Comfort Women s Ikebukuro Oxford Espresso / Jazmín Calzado de senderismo Go Joy 15603 de Skechers para mujer Carbón de leña / negro , Zapatillas de viaje Proather para mujer II Malla Taupe Zapato de lona KEEN Sienna MJ para mujer Nuevo negro Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Navy Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Blanco negro , Orthofeet 877 Women s Comfort Diabetic Therapeutic Extra Profundidad Calzado Lycra Velcro Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Plano extra ancho Feet Breeze Womens Zapatos para caminar Zapatillas para caminar Skechers Performance Go 4 Inspire para mujer Gris pardo , Zapatillas Skechers para mujeres Go-Sandy Boating Gris pardo , PUMA Mujer Basket Corazón NS WN Sneaker Puma Black-puma Black Zapato de agua Easy Spirit para mujer First Walker Atmósfera-topo , Zapatillas de paseo Skechers Go Walk Lite-Luna para mujer rojo Zapato para caminar Volodia para mujer Mephisto Sano Blanco liso, plata Ana / hielo , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Mezclilla Skechers Women s Go 4-14918 Calzado para caminar Blanco negro Drew Shoe La alegría de las mujeres Azul cielo de gamuza / malla , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Gris pardo Zapatillas clásicas Vionic para mujer Walker Gris pardo
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 Xero Shoes Z-Trail Sandalia ligera - Descalzo-Inspirado Ruta de senderismo, Correr sandalias deportivas - De las mujeres Multi-azul Zapatillas de running Saucony Women s Triumph ISO 4 Niebla / gris , NIKE Women s WMNS Zoom Fly, negro / blanco Provence púrpura / negro Zapatillas para correr Mizuno Women s Wave Enigma 6 Quarry-diva rosa-negro , Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Imán / Rojo verdadero Zapatos OLUKAI Pehuea - Mujer Arcilla / arcilla , Zapatillas deportivas sintéticas Drew Shoe para mujer Estampado floral fucsia , Skechers Hombres Merodeador Mershon Negro Negro Sandalia Jenley de Lucky Brand para mujer Vidriado , Zapatillas de running New Balance para mujer Fresh Foam Zante V3 Performance Gunmetal / Sea Salt I-RUNNER Maria Puntera terapéutica para mujer con profundidad extra Calzado de cuero / malla con cordones Gris / Salmón Bomba de cuña Dovehouse para mujer J.Renee Taupe / Gold Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Negro , Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow Eva , Flip-flop Oliver de Sam Edelman para mujer Patente de naranja neón , Entrenador Yui Cross de Yuki Gel-Fit para mujer de ASICS Azul Índigo / Mar Glaciar / Plata Walking Cradles Womens Michaela Cuero negro Vans Unisex Old Skool Zapatillas clásicas de skate Blanco negro Zapatilla para caminar Slip-On Go Flex de Skechers Performance para mujer Gris oscuro , Chass y eacute; Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla Romika Fidschi 22 Sandalia de vestir para mujer Black Juras , Sandalia mujer Jessica Simpson, diadema Crizma Blanco negro La Plume Jen de las mujeres Gray Print Pantuflas Wilma de Katy Perry para mujer Carbón Zapatillas de running New Spike 900 v3 para pista New Balance para mujer Azul negro Bella Vita Womens Mel-Italy Blush-gamuza Botas de cuero Blundstone para hombre 1472 Rústico Negro Rústico , CLARKS Women s Teadale Rhea Oxford Ante azul marino Nurse Mates Womens Fleet Blanco ,

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 Prop & eacute; para t Zapatillas para & Prop caminar TravelFit para mujer Gris oscuro c369532 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.

Para caminar
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1197570
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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.