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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tela y sintético
Importado
Nylon-Canvas
Sintético suela
Tirante trasero
Cordones bota de combate floral presentando acanalado entresuela y falsa forro de piel
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


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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,

Kaitlyn Pan Block Heel Microsuede Botas sobre la rodilla ajustadas Caramelo Botín de tobillo de Dallas para mujeres de Naturalizer Negro , Rocky FQ0001570 Botas altas hasta la rodilla Brown / Mossy Oak Break Up , Botas occidentales bordadas Cleff de Corral para mujeres Turquesa Cordani Mujeres Sarita Borgoña apenado Jeffrey Campbell Womens Leona Boot Taupe Suede , Wolky Comfort Lace Up Shoes Holandés 10870 Blue Nubuck Sandalias con cordones Joanie de SOREL para mujer Sal marina La Canadienne Womens Jennifer Cuero negro , Piel expuesta de W Kilmer de las mujeres de UGG Negro , Blundstone 1357 Combat Boot Secoya , Bota de moda de cuero de cosecha negra Dr. Martens 101 Negro Botas de piel de oveja real EMU Australia Womens Bells Beach Lo Winter Negro , Botines de tacón de goma Medina III de Sorel para mujer Negro / verde limpio FRYE arnés de mujer Carson Coñac lavada antigua Pull Up , Ellie Shoes E-609-Nicky 6 Booty con detalle de purpurina Oro , Bota de mujer Pikolinos Le Mans 838-8550C1 Dirigir Bota occidental Everglades SQ de Dan Post para hombre Cuero Camel , Bota de chukka Pampa Hi Knit Lp paladio para mujer Hierro negro / forjado , Western Chief Women Impermeable de cuatro ojos con cordones Duck up Buffalo Red , Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 9 Muck Boot Womens Wetland Boot Ladrar Bota de tiro de cuña Leigh Suede para mujer Crocs Negro , Botín de mujer Kike Fixed Kitten de The Fix para mujer Mushroom Python Print Wolky Jacquerie Black Dessin Suede con cuello negro , Calzado de chukka de ante de Clarks Originals Desert Boot para mujer , Botín tobillo para mujer de Seychelles Estaño , FRYE Botines Julia Bootie para mujer Forest Soft Nappa Lamb , Wolky Comfort Rio Azul ,
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 Adidas Performance Ultra s Boost M para hombre Core Black / Footwear White Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Sandalia Drew Element para mujer Nubuck negro Zapatillas deportivas 574S Sport New Balance para hombre Azul negro Zapatillas de running Pureboost X de adidas Performance para mujer Vista Gray S15 / Plata metalizada / Gris medio S14 Skechers Performance Women s The Go 400 Radiance Flip Flop Carbón Zapatillas de running Adidas Performance Ultra s Boost M para hombre Azul / Misterio Azul / Negro , Zapatillas de running definitivas Ryka para mujer Gris negro , Chaco Mujer Zcloud Sport Sandal Agrupado azul Zapatillas de trail running Hierro V3 para mujer New Balance Coral vivo / Pigmento , NIKE Men s Benassi Just Do It Athletic Sandal Equipo oscuro Rojo / Solar Rojo / Negro , Sandalia Terra FI Lite de Teva para mujer Maat Multi Drew Shoe Women s Hope cuero terapéutico moda Oxfords Denim / Nubuck Entrenador cruzado Hydrosphere Ryka para mujeres Puerto / Yucca Mint , Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro , LI-NING Zapatillas de baloncesto Samurai III Wade Culture para hombre Zapatillas de deporte Mujeres grises Therafit Eva Women s cuero ajustable correa Slip On Sandal para Fascitis plantar / Dolor en el pie Negro , Sandalia de cuña Balen de Charles by Charles David para mujer Desnudo Sandalias Jerusalem Sandalia David Slide Broncearse Sandalia para mujer SCHUTZ Margarit Slide Negro Sandalia Mephisto Lise Slide Mujer Patente negra , Sandalia Dolce Vita de mujer Kaira Slide Cuero dorado Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ marrón , 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 \ 7032- Silver Pu , TOMS - Slip-On de lienzo clásico para hombre Ceniza , Bomba Parigi de LifeStride para mujer Taupe Glory , TOMS - Slip-On de lienzo clásico para hombre Plata Zapatilla Glamour de mujer verde Daniel Bone Kidskin Bota de ante FRYE Sacha Moto Shootie para mujer marrón , Timberland Womens Lakeville Oxford Marrón medio grano completo ,

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 Botas para mujer InterestPrint Botas InterestPrint 5175 exclusivo con cordones con diseño exclusivo Multi 6 1966533 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 902577
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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.