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.

Sandalia Amara 2 de Carlos mujer Santana Amara para mujer Carlos de Carlos Maravilla 0106b00




Tela
Importada
Sintética suela
sandalia
trenzada con tiras superior
importado
talón altura: 1 pulgada
cremallera cierre
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


Sandalia Amara 2 de Carlos mujer Santana Amara para mujer Carlos de Carlos Maravilla 0106b00 - 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

Sandalia Amara 2 de Carlos mujer Santana Amara para mujer Carlos de Carlos Maravilla 0106b00

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.


Sandalia Amara 2 de Carlos mujer Santana Amara para mujer Carlos de Carlos Maravilla 0106b00

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

Cuña con cordones Tansy Noble para mujeres Vionic Ante negro Sandalia de tobillo con correa Gomera Finn Comfort para mujer Niebla tóxica , NIKE Men s Benassi Just Do It Athletic Sandal Negro / Gimnasio Rojo Sandalia Shauna de mujer de Eastland Blanco Slingback de Seleste de las mujeres naturales del alma Negro , Sam Edelman Sandalia Gretchen Gladiador de Mujer Marfil moderno , Crocs Flip Crocband unisex Grafito / Volt Green Mercado de Tierra Santa Adultos unisex / niños piel genuina Sandalias Bíblicas / Flip fracasos (Jesús - Yashua) Jesús - Estilo Yashua II Camel Trademark Vionic Grace Samar - Sandalia Ajustable Mujer Negro , CLARKS Mujer Caddell Shine Sandal Nubuck sintético de arena , Sandalia de tobillo con correa Gomera Finn Comfort para mujer Gris G.H. Berry Pink Propet Ghillie Walker Sandal de las mujeres Azul Sandalias planas Isla de Jack Rogers para mujer Plata blanca , Sandalia Tessa de mujer Miz Mooz Trigo Sandalia de mujer Aravon Beaumont Peep Slide Azul , Sandalia floral Carly de Cole Haan para mujer Silver Crackle Metallic Sandalia Intrinsic 2 de ECCO Women Warm Gray Metallic / Moon Rock Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Chambray azul , Drew Womens Ariana Dusty Gold Leather , Bomba de cuña Dreamstride Women s Dreams Piedra , CLARKS Haley Stork Loafer para mujer Negro BareTraps Women s Dasie Platform Sandal Blanco Multi Sandalias Peep Toe de Misu para mujer Latin Salsa Tango Practice Bailarinas de salón con tacón de 3.3 \ Negro , NIKE Men s Benassi Just Do It Athletic Sandal Blanco / Negro-negro , Sandalias de tacón de soñador Easy Street para mujer Pewter Metallic Tecnología Vionic Men s Orthaheel White / Navy Walker Gris oscuro , CLARKS Sandalia de cuña Arla Jacory para mujer Elástico Fucsia Profundo Estuardo Mira Mule de Stuart Weitzman Nero Cush Nappa
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 de susurro de mujer Keen Sombra Oscura / Cerámica Sandalia Drew Angela para mujer Beige-multi , Calzado casual tejido NIKE NIKE para hombre Negro / Negro Cumbre Blanco , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Azul Cloro / Antracita / Gris Fresco , Zapatillas de running NIKE Women s Air Zoom Structure 20 Midnight Turquoise / Green Glow / Ghost Green / White Short NIKE Tempo para mujer Negro Negro , Zapatillas de running NIKE Lunar Skyelux para mujer Celadon-azul , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco , Bomba de cuña Dovehouse para mujer J.Renee Taupe / Gold Zapatillas de running NIKE Women s Air Zoom Structure 20 Racer Pink / White-pure Platinum , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Mezclilla Easy Spirit Women s Traveltime Mule Patente negra / plateada Mephisto Women s Hawai Oxford Silver Monaco / Magic , Zapato de agua Easy Spirit para mujer First Walker Atmósfera-topo , Sandalia de cuero de charol Caterine para mujer Negro Revere Miami sandalia de confort para la mujer w. Negro , Zapatillas de running Golden Spike de Altra para hombre Lima / Negro Plataforma de satén mono de ante PUMA para mujer Cesta Blanco Negro 1689 DC Youth Rebound Skate Shoes Negro / Aqua , Sneaker de moda con cordones Lynx de DC para hombre Negro / goma de mascar , Sandalia de tobogán plano de piscina Iona de Gentle Souls Laguna , Sandalia plana Kip de Trotters para mujer Blanquecino DiMichi Adult Maria Leather Elastic Strap Flamenco Shoe Blanco , Bomba de cuña Farrow para mujeres LifeStride gris Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 10 Eastland Solstice MLB Women s Oxford New York Yankees-navy , G by GUESS Mujeres Olisa Hight Top Lace Up Zapatillas de moda Azul / Medio , Mephisto Women s Yliane Oxford Gris Claro Nubuck , Propet Women s Tour Walker Strap Sneaker Blanco / Berry

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.

Sandalia Amara 2 de Carlos mujer Santana Amara para mujer Carlos de Carlos Maravilla 0106b00 - 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 Sandalia Amara 2 de Carlos mujer Santana Amara para mujer Carlos de Carlos Maravilla 0106b00 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 439249
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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.