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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75% piel de cerdo de gamuza / 25% Malla
Hecho en Estados Unidos
Caucho único
ENCAP tecnología de suela intermedia proporciona soporte y durabilidad máxima
Encap anillo de poliuretano con etileno vinilo núcleo de acetato en talón
Doble densidad espuma de cuello
12 mm de caída: debido a varianzas creadas durante el desarrollo y fabricación procesos, todo referencias a 12 mm de caída son aproximados
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


New 19847 Balance Women B075V9DXDH s w990v4 New Running Shoe Gray/ Castlerock ce0a50c - 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

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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,

Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul , Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / Blanco Zapatillas de running W1260V5 para mujer New Balance Azul plateado Zapatillas de running Mizuno Women s Wave Hitogami 4 Naranja / gris Pantalón largo clásico NIKE Golf para mujer Tech Essentials Black Crimson total , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul Zapatillas de running NIKE Air Huarache para hombre Olive Flax White 302 , Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink , Short NIKE Tempo para mujer Negro, Racer Pink Under Armour Women s HOVR Sonic NC Blanco (102) / Elemental , Zapatillas de running adidas Adizero Boston 6 W para mujer Utilidad Negro / Platino / Negro Zapatillas de running Nike \ Noble Red White Black 600) , Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Polvo negro / asteroide / peltre / blanco , Zapatillas de running Under Armour para mujer Toccoa Chambray Blue (400) / Overcast Grey , NIKE Men s Benassi Just Do It Athletic Sandal Negro Zapato de correr ideal para rejilla de mujer Saucony Gris blanco , Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul , Zapato deportivo ASICS Gel-Kayano 24 para mujer Porcelana Azul / Humo Azul / Blanco , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Rojo , Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink , Zapatillas de running NIKE Zooms Winflo 4 para mujer Blanco / Negro / Aurora Verde EN Zapatilla Cloudster para mujer Magenta / Berry Zapato de running GEL-Excite 3 para mujer ASICS Rosa fuerte / Blanco / Amarillo intermitente , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Lush / Silver Mink Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono Zapatillas de running NIKE Lunarglide 9 para mujer Gunsmoke / Solar Red / Hot Punch / Vast Grey Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Negro / Gris Sólido Skechers Performance Women s Go Run 400 Obstruct Negro Zapato deportivo para acolchado 680v5 New Balance para mujer Coral ,
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 Zapatillas de running NIKE para mujer Air Huarache Run Ultra Elemental Rose / Wolf Gray / White , NIKE Women s Tanjun Blanco negro Zapato de running COMUTORA para mujer ASICS Porcelana Azul / Porcelana Azul / Blanco , Chaco Women s Wrapsody X Athletic Sandal Key Teal , Sandalia de deslizamiento ultra Comfort NIKE para mujer Negro / Vivo Rosa-negro Sandalia NIKE Kawa Slide para mujer Hyper Turquesa / Blanco , Corredor SCARPA Neutron Wmn Running Shoe Trail Runner para mujer Laguna / Limón Calzado de running para mujer New Balance Cypher Mulberry / Angora , Zapatos OLUKAI Pehuea - Mujer Pokeberry / Pokeberry Katy Perry Women s The Alanna Sandal Arena rosa , Sandalia Terra FI Lite de Teva para mujer Agua NIKE Women s Roshe Two SI Drew Womens Element Denim Blue Nubuck Charles by Charles David Mujeres inquietantes Mary Jane Coral Microsuede Zapato de running 840v4 New Balance para mujer Gris oscuro Skechers Work Zapatillas Marbleton resistentes a los resbalones D Lites para mujer Negro , Zapatillas para caminar Brooks Women s Addiction Walker Blanco , New Balance 711v3 Heather Cross Trainer para mujeres Trueno / Poisonbery , Zapatillas de deporte Fenty x Trainer Mid de PUMA para mujer Puma Black / Puma White Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Gris New Balance WW577 Hook and Loop Walking para mujer Blanco , Sandalia Slip con nudo de mujer Soludos Flat Sonrojo , Chanclas Deportivas Cooper Slide de Marc Jacobs para mujer rojo Mephisto Women s Slidena Sandalia Negro Zapatillas de voleibol adidas Performance Volley Team 4W para mujer Blanco / Mate Plata / Negro , Zapatillas de ciclismo Smartodoors con suela de carbono o suela de tpu de nylon para carretera y MTB Azul para MTB Botín de tobillo Oceani para mujer de ALDO Verde medio Zapatillas de fútbol NIKE Hypervenom Phade III FG Soccer Dinastía Cool Gray / Purple CLARKS Enfield Senya Tobillera para mujeres Gamuza marrón oscuro , GUESS Women s Rodeo Faux-Leather botines occidentales Oro

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 New 19847 Balance Women B075V9DXDH s w990v4 New Running Shoe Gray/ Castlerock ce0a50c 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.