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.

Botines Negro Landsman de Botines mujer Cole Haan 5066 Botines ocasionales cortos de cuero causantes Negro c60c498




Cuero
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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


Botines Negro Landsman de Botines mujer Cole Haan 5066 Botines ocasionales cortos de cuero causantes Negro c60c498 - 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

Botines Negro Landsman de Botines mujer Cole Haan 5066 Botines ocasionales cortos de cuero causantes Negro c60c498

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.


Botines Negro Landsman de Botines mujer Cole Haan 5066 Botines ocasionales cortos de cuero causantes Negro c60c498

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

Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 27 , Bota de trabajo Emerson Roper para mujer marrón Botas cortas de cuero Mordenmiss para mujer Zapatos nuevos Estilo 5 Marrón , Bota de mujer L19 de Arcopedico Reptil negro Joules Women s Rockingham Botines Marina de guerra francesa Dandelion Geo , FRYE Arnés con cinturón de mujer 12r Negro , Bota de moda BEARPAW para mujer Abigail Negro , Bota de media caña Georgia GB00102 Marron oscuro , Bota Caterina Easy Spirit para mujer Dark Taupe Suede Bota de combate FRYE Julie Hook para mujer Grano completo suave guijarroso negro , Moneysworth y el mejor botín Welly de goma para mujer Negro , Botín Neponset de mujer White Mountain Marrón / cuadros Bota de hebilla Emery para mujer Merrell Moho , QZUnique Mujeres Botines planos con borlas de cuero genuino con cremallera Martin Boots Negro CORRAL E1279 Bone Glitter Embroidered Boots Bota de mujer FRYE Nova Flower Tall Coñac , Bota de media pantorrilla Georgia GB00126 Marrón claro , MICHAEL Michael Kors Mujeres Kendra Peep Toe Botas de cuero de moda Bota de piel de becerro Emma 608T BEARPAW (Niño pequeño) Negro / Plata , No incluido en la lista por Kenneth Cole Drop Bot Ankle Bootie para hombre Gris , Belleville 650 Waterproof Steel Toe Sage Green Bota corta con botones Jackie de mujer FRYE Slate-76593 , Botas Columbia Bugaboot Plus Titanium Omni-Heat Outdry - para mujeres Negro / Violeta , Bota de imitación para mujer SoftWalk Marron oscuro Belleville 700V WP Black Combat Flight Boots para hombre Negro Calou Stockholm Women s Hanna Negro , Regimiento de mujeres de Lugz Hi Fleece Water Resistant Golden Wheat / Cream / Gum Thermabuck La Sportiva Boulder X Shoe - Mujer Azul hielo , Bota corta con botones Jackie de mujer FRYE Negro - Cuero antiguo pull up lavado-76593
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 Vans ISO 2 ligero calzado deportivo Blanco negro , Zapatillas Mizuno Running Women s Wave Creation 19 Imán / Plata Zapatillas de running Mizuno Wave para mujer con sombra Fiery Coral / Blanco , Sandalia Uneek O2-w de mujer KEEN Etherea / Whitecap , Zapatillas de deporte casual Gazelle de adidas para hombre Blanco OluKai Kulapa Kai Contton Candy / Black , Zapatillas deportivas 574S Sport New Balance para hombre Nimbus Cloud / Navy , Sandalia Drew Angela para mujer Dusty Multi Metallic , Calzado de running ft. Perpetua ft de memoria Fila Women s Castlerock, blanco, negro , Zapatillas de running Saucony para mujer Omni 16 Teal / Citron Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Lt Gray , Sandalias de diapositivas casuales Easy Street Womens Conca Fabric Open Toe Negro Sandalias de diapositivas casuales Easy Street Womens Conca Fabric Open Toe Negro , Sandalia Sintética Mydebut de Nueve Mujeres del Oeste Rosado , Sandalias de mujer Wolky Caviar beige , Zapatillas DC Mujeres Chelsea TX SE Sneaker Negro / antracita , Golf de mujeres y rsquo; s Peso ligero genuino Cuero hecho en Brasil Spring Street Rendimiento de golf Marc Joseph NY Zapatos de moda Azul marino granulado Sandalia de girasol Aerosoles para mujer Negro Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Denim azul claro Sandalia de la diapositiva de China Girl My Girl Silver Metallic Adtec 9 \ castaña , TOMS Clásicos de mujer Birch / Camo , Bota de mujer Serra de TOMS Oxford Tan Suede , Bota de tobillo Juniper Bree de Sperry Top-Sider para mujer Negro , Chanclas Deportivas Cooper Slide de Marc Jacobs para mujer Purple Multi Tamaris Women s Fee 22425 Pump Armada Bota de mujer FRYE Natalie Short Lug Engineer Black Soft Oiled Suede Cicadillo Lax Finisher Under Armour para mujeres Blanco (101) / Blanco , Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer Black Mix , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Infinity Blue Heritage Canvas ,

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.

Botines Negro Landsman de Botines mujer Cole Haan 5066 Botines ocasionales cortos de cuero causantes Negro c60c498 - 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 Botines Negro Landsman de Botines mujer Cole Haan 5066 Botines ocasionales cortos de cuero causantes Negro c60c498 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 7325800
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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.