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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Viene en caja original
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


Zapatos 15447 ASICS para para hombre Gel-Lyte-V Gel-Lyte-V Negro Negro de4076e - 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

Zapatos 15447 ASICS para para hombre Gel-Lyte-V Gel-Lyte-V Negro Negro de4076e

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.


Zapatos 15447 ASICS para para hombre Gel-Lyte-V Gel-Lyte-V Negro Negro de4076e

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

Calzado de running NIKE Kids Roshe One (GS) Negro / Max naranja Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / Blanco Zapatillas de running ASICS Gel-Cumulus 19 para mujer Gris medio / Carbono / Amarillo de seguridad , Zapatillas de running New Balance para mujer 490v4 Negro / Guayaba , PUMA Men s Roma Basic Sneaker Blanco / Llama escarlata / Glaciar gris , En la ejecución de calzado de mujer en la nube Todo blanco Zapatillas para correr ASICS Gel-DS Trainer 22 para mujer Acuario / Aqua Splash / Flash Coral Under Armour Women s Charged 24/7 Low NM Acero (035) / Gris Nublado , Zapatillas para correr Mizuno Women s Wave Creation 16 Mulberry Purple / Silver , Zapatillas de running Kinvara 5 Saucony para mujer Menta / cereza Zapatillas para correr Saucony Women s Cohesion 10 Gris / Coral Zapatillas de running New Balance 1340v3 para mujer Blanco / morado , Zapatillas de running Kinvara 8 Saucony para mujer Azul nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Hematita Metálica / Negro-gris oscuro Under Armour Women s Dash 2 Mediterráneo / Blanco / Blanco Zapatillas de running Mizuno Wave Rider 20 para mujer Gunmetal / Fuchsia Purple Zapatillas de running mujer HOKA ONE ONE Clifton 4 Blanco negro NIKE Men s Benassi Just Do It Athletic Sandal Blanco / puro platino / negro / blanco Zapatillas de running Nike Kaishi para hombre Gunsmoke / Negro-blanco , Zapatillas de running ASICS Gel Flux 2 para mujer Ónix / rosa fuerte / azul , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco azul Zapatillas de running NIKE Mens Air Zoom Structure 19 Gris blanco Zapatillas de running NIKE Air Max 90 para mujer Negro / Negro / Gran Gris , Brooks Women s Ravenna 8 Negro / Diva Pink / Plum Caspia Brooks Women s PureFlow 6 Bluebird / Peacoat / Lime Punch , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Phantom / Black / White New Balance Women s w990v4 Running Shoe Gris / rosa , Zapatillas de running Nike Kaishi para hombre Equipo oscuro Rojo / Gris oscuro-negro , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal
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 NIKE Benassi JDI para mujer Light Bone / Sai , Zapatillas NIKE Femenil Flex 2016 Rn Running Azul Brillo / Blanco / Azul Racer / Azul marino medianoche Zapato de running ASICS GEL-Kayano 22 para mujer Púrpura / Plata / Rosa Brillo , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Binario Azul / Cerúleo / Negro / Violento , Sandalia Atlética de Cuero Caprizee Columbia para Mujer Stone / Oxford Tan , Sandalia Pavi de estilo suave para mujer Niebla blanca , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Plata / rosa L Artiste de Spring Step Womens TUTTIFRUTTI Armada Zapatillas de running Mizuno Wave para mujer Inspire 14 Folkstone Gray / Pink Glo , Sandalia Harman Bandolino para mujer Café latte Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory Alpargata de tela \ Lino de piedra Skechers Women s Go Step Lite-Indulge Loafer Flat Chocolate , Taipán de mujer de David Tate Verona Cordero negro , ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Goma negra / nubuck gris medio , Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer gris Slingback de Seleste de las mujeres naturales del alma Cuero negro , Sandalia de diapositivas para mujer LifeStride Remix Negro , Sandalia Beau Slide Kate Spade New York para mujer Negro Koolaburra de UGG Women s W raychel Slide Sandal Sonrojo Calzas de la dinastía Real / Blanco , Nuevas zapatillas de punta redonda de mujer Mordenmiss Estilo 4 Café , JOE N JOYCE Paris Sandalias de suela suave SynSoft Rosa roja BEARPAW Boo Carbón , Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Teal | Jacques Levine Bel Esprit Pom Pom Negro Mocasín de gamuza Kilty de Minnetonka para mujer marrón Zapatilla Newbie de SOREL para mujer Alce Bootie con cordones Lunata de Toms para mujer Trigo / Gamuza , Zapato femenino Keli de Alegria Jet Lustre ,

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.

Zapatos 15447 ASICS para para hombre Gel-Lyte-V Gel-Lyte-V Negro Negro de4076e - 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 Zapatos 15447 ASICS para para hombre Gel-Lyte-V Gel-Lyte-V Negro Negro de4076e 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.

Carretera en funcionamiento
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.


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