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.

Vans 741 Auténtico B00SNND88W Neón/ Rosa/ Negro d2eb4ba




lienzo
sintético suela
El estilo el nombre es Auténtico. El número de estilo es VN-04MLJOG.
Marca Color: Neón Rosa / Negro (Color principal: rosa)
Material: Lona
Medidas: 0.75 talón
Ancho: B (M)
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


Vans 741 Auténtico B00SNND88W Neón/ Rosa/ Negro d2eb4ba - 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

Vans 741 Auténtico B00SNND88W Neón/ Rosa/ Negro d2eb4ba

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.


Vans 741 Auténtico B00SNND88W Neón/ Rosa/ Negro d2eb4ba

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

Vans Auténtico Pewter / True White Lakai Fremont Skate Shoe Grey Suede Supra Vaider LC Sneaker Greyviolet White P5A the Day Breakers Persona 5 Anne Takamaki Zapatos de Cosplay Botas por encargo Zapatillas de skate Vans Unisex Sk8-Hi Reissue Azul verdadero , Vans Unisex Brigata Zapatillas de skate, náuticas Zapato de barco inspirado Clásico, Cómodo y duradero en Waffle original Suela (C & l) negro / blanco verdadero , Vans Auténtico Duke / Black / Gold Foil , Supra Vaider LC Sneaker Negro / Negro - Blanco , Vans Auténtico (Vintage Aloha) Negro / Blanco Verdadero , Supra Vaider LC Sneaker Lana de carbón / blanco , Supra Vaider Skate Shoe Blanco negro DC Youth Rebound Skate Shoes Desierto Clásicos de base Slip-on (Tm) para hombres de Vans Carbón Vans Auténtico Hawaiian Floral Black Vans Unisex Old Skool Zapatillas clásicas de skate Negro / peltre Vans Unisex Old Skool Zapatillas clásicas de skate Lux Cuero Porcini Chocolate Afeitado , Clásicos de base Slip-on (Tm) para hombres de Vans Black Off White Checker , Supra Method Skate Shoe Gris oscuro / gris-blanco , Final Fantasy XIII FF13 Serah Farron Zapatos de Cosplay Botas por encargo 2 Zapato de skate Diadora B.Elite Suede Beige deletreado , Supra Vaider LC Sneaker White Perf Gunny Tuf , Vans Old Skool Unisex Adults Low-Top Trainers (Cuadros primario) Negro / blanco Vans Auténtico Crema / nuez , Vans Unisex Old Skool Zapatillas clásicas de skate Clásico Negro / Blanco Zapatillas de skate Vans Unisex Authentic Platform Tablero de ajedrez / Globos oculares 7236 Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Goma negra / clásica , Zapato de skate casual Heathrow DC para hombre Blanco Blanco Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Verdant Green , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma (Mte) Demitasse / Cuero negro ,
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 correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de deporte Onitsuka Tiger Colorado Eighty Five Azul marino / gris suave , Zapatillas de running W1540V2 New Balance para mujer Blanco azul Zapatillas de running Adidas Performance Ultra s Boost M para hombre Energía / Energía / Negro , Earth Origins Sophie Women s Sandal Azul marroquí , Zapato de caminar infinito SMW de Ryka para mujer Azul / Lima Sandalia Pavi de estilo suave para mujer Vitello negro , Sandalia de cuña Daytona SoftWalk para mujer Multi Naughty Monkey Mujer Hey Pony Sandal Blanco , Pikolinos Womens Puerto Vallarta 655-0575 Orquídea Zapatillas de running Skechers Performance para mujer Go Run 400 Action Carbón de leña / azul Earth Origins Selby Murciélago Me Too Women s Lightning 2 Pump Nappa Metálica Rosa Dorada , Hush Puppies Mujeres Power Walker Ii Loafer Negro , Zapato Cross-Trainer X Conviction Women de ASICS Aqua Splash / Diva Blue / White , Sam Edelman Womens Barry Mulberry Pink Vaquero Saddle Leather Vans Herren Authentic Core Classic Sneakers True White Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker rojo Zapatillas de golf Skechers Women s Go Golf Birdie Carbón de leña / azul , Sandalia UGG Australia Royale Slide para mujer Bebe azul , BareTraps Women s Rebecca Slide Sandal Cameo Rose Sandalia de la diapositiva Adalea de las mujeres de Dolce Vita Lino negro , Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero desnudo Bomba Parigi de LifeStride para mujer Crucero Marina , Mavic Women s s Elite Echappee Trail Elite Shoe Fig / hibisco , Corral Women s Patch Western Snip Toe Boots Marrón, coñac Sandalia Birkenstock de cuero Florida para mujer Negro Easy Spirit Women s Traveltime Mule Cuero negro / negro , Trotters Liz Loafer para mujer Blanco TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Lana estampada tribal natural ,

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.

Vans 741 Auténtico B00SNND88W Neón/ Rosa/ Negro d2eb4ba - 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 Vans 741 Auténtico B00SNND88W Neón/ Rosa/ Negro d2eb4ba 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 184482
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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.