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.

Zapatillas de running adidas/ Mana Bounce 2 Bounce/ W Aramis para mujer Blanco/ Metálico/ Plateado/ Negro 1fdd767




Sintético
Importado
Suela de goma
Plantilla extraíble
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


Zapatillas de running adidas/ Mana Bounce 2 Bounce/ W Aramis para mujer Blanco/ Metálico/ Plateado/ Negro 1fdd767 - 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

Zapatillas de running adidas/ Mana Bounce 2 Bounce/ W Aramis para mujer Blanco/ Metálico/ Plateado/ Negro 1fdd767

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.


Zapatillas de running adidas/ Mana Bounce 2 Bounce/ W Aramis para mujer Blanco/ Metálico/ Plateado/ Negro 1fdd767

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 para correr Zante V3 para mujer New Balance Thunder / Artic Fox Zapato de running ASICS Gt-2000 5 para mujer Negro / Bay / Diva Pink , Zapato de running Saucony Zealot ISO 2 para mujer Té / Algodón , Zapatillas de running Vigor Bounce W de adidas Performance para mujer F16 / Utility Blue F16 / Ice Purple F16 , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Pure Platinum / Pure Platinum Zapatillas de running Vibram para mujer V Rosado rojo Zapatillas de running adidas Women s Edge Lux W Negrita Rosa / Haze Coral / Negro , Zapatillas de running Saucony para mujer Hurricane ISO 3 Gris / Rosa / Blanco Zapatillas de running Nike Kaishi para hombre Lobo gris / negro / blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Multi Color 999 , Under Armour Women s HOVR Sonic NC Formación Azul (500) / Elemental Brooks Womens Adrenaline GTS 18 Grey / Rose Gold , Zapatillas de running Saucony Women s Breakthru 3 Azul / menta , Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / Amp Pink , Zapatillas para correr de carretera Altra One V3 para mujer Azul Claro / Coral , Calzado de running NIKE Kids Roshe One (GS) Estuco oscuro Zapatillas de running NIKE Zooms Winflo 4 para mujer Gridiron / Provence Purple / Light Carbon , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul , Calzado de running NIKE Kids Roshe One (GS) Arctic Pink / Arctic Pink / Sail Brooks Women s Adrenaline GTS 17 Heather / Gray Pinstripe / Iris Orchid Zapatillas de running adidas Women s Edge Lux 2 Blanco / Blanco / Plata Metálica Zapatillas de running adidas Alphabounce Rc W para mujer Core Black / White / Utility Black Calzado de running NIKE RN 2017 2017 para mujer Pure Platinum / Black / Wolf Gray / Cool Gray En la ejecución de calzado de mujer en la nube Mulberry / Velvet , Zapatillas de running Pureboost X de adidas Performance para mujer Vista Gray S15 / Plata metalizada / Gris medio S14 , Zapatillas de running adidas Adistar Boost ESM para mujer Ceniza Púrpura / Blanco / Naranja Solar , Zapatillas de running ST-2 para mujer Topo Athletic Verde gris Zapatillas para correr ASICS Gel-Venture 6 para mujer Pavo real de carbono / negro / rosa ,
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 OluKai Kulapa Kai Negro Negro , Chaco Mujer Diana Sport Sandalia Biselado negro , Zapatillas de entrenamiento NIKE Train Ultrafast Flyknit para hombre Pure Platinum / Volt - Ghost Green - Volt Tint Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / Blanco Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker GTX W para mujer Gris Tres / Carbon / Chalk Coral Merrell Women s Mqm Flex Gris negro , Zapatillas de deporte Xelero Genesis Comfort Comfort Therapeutic Extra Deep Sneaker Leather / Mesh Lace-up Gris / Salmón Sandalias Harper Huarache de SoftWalk para mujer Negro / peltre Sandalia de henna de mujer Sam Edelman Brillante multi raya , New Balance WW577 Hook and Loop Walking para mujer Negro Chancleta Yoga Sling 2 Sanuk para mujer Tapiz de mar , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra , Zapatillas para correr ASICS Gel-Venture 6 para mujer Negro / Island Blue / Pink , Drew Shoe Athena para mujeres Crema / Combo Botas de tobillo Pallaphoenix K Palladium para mujer Gris Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro sin blanco Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Polvo morado , FootJoy EmPower BOA Zapatos de golf CERRAR Mujeres Plata / Lila Gc Shoes - Sandalias de cuña con roseta Sydney, modelo Sydney, para mujer Black Fl Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 16612EB Cómodo-Muy Fino 2.5 \ de 5] Satén negro y malla negra , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2707- Satén blanco y adornos de plata # 119 , Boombah Women s A-Game - Juego de calas intermedias moldeadas - 8 opciones de color - Varios tamaños Negro / púrpura , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Elemental Rose / Barely Rose Bota de mujer FRYE 8R-Old TWN Engineer Pueblo viejo de carbón , Bomba de vestido Pointe para mujer Easy Street Negro Vionic Womens Elevated Stanton Wedge Serpiente gris , SOREL Bota de cuero baja grande para mujer Negro / Tetera Bota de tobillo Missoula para mujer Cuero cepillado Taupe El Naturalista Duna N566 Ocean Cares Leather Zapato femenino Keli de Alegria Chrome Cube ,

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.

Zapatillas de running adidas/ Mana Bounce 2 Bounce/ W Aramis para mujer Blanco/ Metálico/ Plateado/ Negro 1fdd767 - 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 Zapatillas de running adidas/ Mana Bounce 2 Bounce/ W Aramis para mujer Blanco/ Metálico/ Plateado/ Negro 1fdd767 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

Thirteen elementary straightedge 851927
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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.