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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Nylon ^ Sintético
Importado
Goma único
Eje mide aproximadamente Parte baja del arco
Hacer una declaración de estilo audaz cuando tu deporte el N9000 Nyl.
Durable nylon y sintético parte superior.
Acolchado lengua y cuello
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,

Zapatillas de tenis New Court 896v2 Hard Court para mujer Armada , Zapatillas de tenis adidas Adizero Defiant Bounce W, de mujer, adidas Tiza Azul / Blanco / Tiza Coral , adidas Originals Mujer Superstar Blanco / Fairway / Metálico / Dorado , Zapatillas de tenis adidas Women s Barricade 2018 W Blanco / escarlata / escarlata Zapatillas para mujer Del Rey TOMS Lona Poly gris acero , Converse Chuck Taylor All Star Core Ox Salsa roja / blanca , Zapatos de aproximación Arc teryx Acrux SL - Mujer Negro Negro Converse Chuck Taylor All Star Core Ox Mono blanco Converse Chuck Taylor All Star Core Ox Negro Negro , Zapatos de aproximación Arc teryx Acrux SL - Mujer Abyssal Blue / Gentiane , ASICS Tiger Womens Gel-Lyte V Aluminio / Aluminio Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Blanco , Onitsuka Tiger Asics Mexico 66 Blanco Blanco , Zapatos de salón para interiores Salming Kobra, blanco Blanco Zapatillas de deporte Xelero Matrix Comfort Therapeutic Extra Deep Sneaker Shoe Leather con cordones Negro , Zapatilla para mujer Dr. Comfort Katy Verde y Turquesa , Zapatillas de tenis adidas Adizero Club 2 de mujer Hi-res Aqua / Blanco / Shock Pink , Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Blanco-turquesa , Zapato de tenis 7 de ASICS Gel-Resolution femenino Red Alert / Silver , Zapatillas para tenis Junior All Propulse Fury de Babolat Kid Azul Zapatillas adidas Adizero Ubersonic 3 w Clay Tennis, de adidas, para mujer Ecru Tint / Noble Indigo / Chalk Coral Zapatillas de running Golden Spike de Altra para hombre Negro / Rojo Zapatillas de tenis Yonex Power Cushion Eclipsion All Court para mujer, rosa oscuro Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco , Diadora Unisex N9000 III Blanco / rosa melocotón APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Negro , Onitsuka Tiger Asics Unisex Tiger Corsair Aciano Azul / Azul Oscuro , Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Pigmento
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 deportiva Zvolv X de Chaco para mujer Lavender Liberty , mobils by Mephisto - Sandalias de tacón Getha para mujer Black Queen / Magic Leather Under Armour Charged Reactor Run Run Bayou Blue / Overcast Gray / Black KEEN Hombres Newport H2 Sandal Azul marino / gris medio Sandalias sintéticas Teva para mujer Terra Fi Lite Atlántico Norte Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Acuífero , Zapatillas deportivas 574S Sport New Balance para hombre Cáñamo / incienso , Carro de mujer Brooks Crema / Rojo verdadero / Gris tormentoso , Calzado de running NIKE Kids Flex 2017 RN (GS) Pure Platinum / Black-wolf Gray NIKE Women s Revolution 4 Running Shoe Ocean Bliss Blue Glacier Blue , Brooks Mujeres Transcend 5 Negro / Púrpura / Multi , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Lyl Blue / Lyl Bl / University Red / Wht Zapatillas de deporte trail running Hiero V2 para mujer New Balance Typhoon / Supercell / Alpha Pink Columbia Montrail Women s Trans Alps F.K.T. Jade Lime, Splash , Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Gris , Reebok Zapatillas para caminar de estilo libre para mujer Plata blanca , Flip-flop Oliver de Sam Edelman para mujer Oro Remato Liquid Metallic , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro Sandalia Aerosoles Rose Gold para mujer Mid Blue Suede Easy Spirit Women s Traveltime Mule Luz Natural / Blanco , Sandalia Cato Dolce Vita para mujer Lt Blue Denim Sandalia Sloan Slide de Naturalizer para mujer Gunmetal , Sandalia de mujer Lucienne Suede Slide de Nine West Gamuza rosa Sansha Zapato de grifo en T para mujer Negro Bomba de vestido animado de LifeStride para mujer Lux Navy 401 , Mulz Unisex House Shoes - Zapatillas interiores y exteriores de lana - Made in Europe Fuchsia Multi , Jessica Simpson Reiki de mujer negro Stretch gamuza montada rodilla Micro Wedge Boots Western Chief Women Impermeable de cuatro ojos con cordones Duck up Marina de guerra profunda Dr. Martens Women s 1461 GLTR Oxford Blanco / morado , TOMS - Slip-On de lienzo clásico para hombre Faded Rose

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 Diadora B01CQVF4JM N9000 19907 Men Toe Snthetic Blue Blue Sneakers Blue Angel Falls/ Dresden Blue 1c79f7a 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.

Tenis y deportes de raqueta
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2698794
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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.