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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Malla de ingeniería
Importada
Caucho único
Eje mide aproximadamente parte baja del arco
EVERUN Zona de aterrizaje, EVERUN Plantilla, ingeniería malla, construcción ISOFIT. Offset: Tacón de 8 mm Altura de la pila: 30 mm Pila de antepié Altura: 22mm
EVERUN Plantilla para continuo amortiguación a lo largo tu carrera
Diseñada malla elástica para un ajuste dinámico
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

Saucony 19986 Triumph marino ISO 3 Running Saucony Sneaker para mujer Azul marino 9d5559b

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,

Sandalia de Sarasota KEEN para mujer Bisonte , Sandalia Drew Element para mujer Nubuck negro , Propet Women s TravelActiv Ss Sandal Púrpura / negro OluKai Ohana Blanco negro , BONJOMARISA 2018 Verano Nuevo Elegante Sandalias de malla de la marca Decoración de metal Zapatillas Peep Toe Tacones bajos de mujer Tamaño grande 34-43 Calzado Oro Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro Sandalias Bali KEEN para mujer Red Violet / Boysenberry Sandalia deportiva Tirra para mujer Teva Negro / Blanco Multi , Sandalias Newport H2 de KEEN para mujer Cáñamo / Gorra blanca , Sandalia deportiva ECCO Terra 3S para mujer Negro / Teaberry , Sandalias Bali KEEN para mujer Higo de verano / Crabapple , OluKai Kulapa Kai Buganvillas / negro , Sandalia deportiva Tirra para mujer Teva Atlántico Norte Sandalia deportiva Tirra para mujer Teva Chispas de chocolate Sandalia de susurro de mujer Keen Cascada / Stone Blue , Sandalia Terrell Convertible II Merrell para mujer Negro Propet Women s TravelActiv Ss Sandal Azul negro , Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Monumento Sandalia KEEN Ana Cortez W para mujer Brisk / Espresso Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro , Sirena de mujer Merrell Flip Q2 Athletic Sandal Aluminio , Sandalia Teva Terra-Float Nova para mujeres Gris Sandalia Drew Angela para mujer Beige-multi Sandalia Paula Fisherman de Haflinger para mujer Piedra BZees Sandalia con cuña de beso para mujer Blue Multi Print , Sandalia Teva Women s W Hurricane Xlt2 Sport Lago Coral Correa de sirena de mujer Merrell Q2 Sandalia de atletismo Aluminio Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink ,
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 para correr Saucony Freedom Iso Unisex Aceituna , Brooks Womens Revel Diva Pink / Plum Caspia / Black Correa de plataforma PUMA para mujer Satin En Pointe WN Sneaker Puma Black-puma Black Zapatillas de running ASICS Gel-Contend 4 para mujer Podar / Plata / Negro Zapatillas de running NIKE Zoom All Out Low para hombre Crimson total / Crimson total Zapatillas de running NIKE Mens Lunarglide 8 Multicolor / multicolor Zapato para caminar WW813 para mujer New Balance Gris Easy Spirit Solana Oxford para mujeres Gris oscuro Nubuck múltiple , Zapatos de paso Valeris Backstage para mujer de Skechers Negro , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Mango brillante negro / blanco Zapatillas de running ASICS GT-2000 6 para mujer Brillante rosa / negro / blanco , Zapatilla deportiva Kailee de mujer Ryka Insignia Blue / Yucca Mint / Summer Grey , Sandalias de gamuza Birkenstock Unisex de Arizona, color café oscuro - 36 N EU (Mujeres de EE. UU. 5-5,5 de la UE) - Sandalia de mujer Corkys Float Negro Zapatillas de deporte Xelero Matrix Comfort Therapeutic Extra Deep Sneaker Shoe Leather con cordones Negro New Balance WW813 Hook and Loop Walking para mujer Gris Clásicos de base Slip-on (Tm) para hombres de Vans (Tablero de ajedrez) Negro / Negro Sam Edelman Sandalia Bernice Slide para mujer Candy Red Sandalias Jerusalem Sandalias Deborah Slide Miel , 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2829- Blanco y plateado, tacón de aguja GUESS Zapatillas bajas con adornos en las mujeres Black Multi , Sandalia de vestir Taimi de Naturalizer para mujer Negro , Sandalias de vestir Patti de Sam Edelman para mujer Blue / Multi Festival Floral Print Zapato de pista Md800v5 New Balance para mujer Oro rosa / negro D-Story Fashion Lace up Tall Punk Dancing Canvas Botas largas Sneakers Zapatos para mujeres Multicoloured16 Zapatillas UGG Womens Florencia Violeta gris Baffin Unisex Cush Insulated Slipper Oscuridad Bota de nieve SOREL Women s Out N About Leather Rain Barro Negro , Eastland mujeres Eden Oxford Nuez moscada Arcopedico Womens Lolita Slip-On ciruela

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 Saucony 19986 Triumph marino ISO 3 Running Saucony Sneaker para mujer Azul marino 9d5559b 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.

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

Geometry Construction Reference

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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.