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 de mujer Rose KEEN mujer Rose Dark Purple/ Sage Purple Sage 4cdc5e3




Cuero / Sintético
Hecho en EE. UU. O importado
Suela de goma
Zapato para exteriores.
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 de mujer Rose KEEN mujer Rose Dark Purple/ Sage Purple Sage 4cdc5e3 - 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 de mujer Rose KEEN mujer Rose Dark Purple/ Sage Purple Sage 4cdc5e3

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 de mujer Rose KEEN mujer Rose Dark Purple/ Sage Purple Sage 4cdc5e3

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 para mujer Terrell Ari Wrap de Merrell Muy uva , Chaco Women s ZX2 Classic Athletic Sandal Credo Golden OLUKAI Luana Sandal - Mujer Sombra negra / oscura Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino , Drew Womens Cascade Sand Nubuck Sandalia Jacardia Merrell para mujer Armada Sandalia Yucatan ECCO Mujer Coral Blush , Chaco Mujer Zcloud X Sandalia Deportiva Echo Paloma CLARKS Womens Wave Grip Sabio Sandalia deportiva Tetra Cloud de Chaco para mujer Bluebell Eclipse , Sandalia NIKE Benassi JDI para mujer Rosa / Plata , Sandalia de espalda de mujer Merrell Women s Town marrón , Sandalia deportiva ECCO Terra 2S para mujer Ladrillo , Drew Womens Ariana Cuero de peltre Chaco Mujer Zcloud X Sandalia Deportiva Echo Paloma Skechers Performance On The Go 400 Tropical Flip Flop Grey Tropical , Encaje de quimera de Columbia para mujer Zuraqua Sandalia NIKE Kawa Slide para mujer Cool Grey / Hyper Turquoise / Cool Grey BZees Sandalia Deportiva Jive para Mujer marrón , Sandalia deportiva Hammond de Spring Step para mujer gris Sandalia deportiva Kankitta de Vince Camuto para mujer Combo puro , Skechers Women s on The Go 400 Jazzy Sandal Azul marino / rosa fuerte Sandalia original Teva Women s Original Campo Negro / Blanco FABULICIOUS LIP-145 Women s 5 \ Taupe Pu / Clear , Sanda Verra Teva para mujer Deep Teal , KEEN sandalia de Venecia para mujer Azul Sandalias KEEN Mujer Whisper-w Vapor / acero gris , Earth Origins Sophie Azul marroquí Sandalia Yucatan ECCO Mujer Negro / topo / negro
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 cuña Jubilee Propet para mujer Denim Multi , EN la zapatilla de deporte Running Cloud para mujer Denim / Blanco , Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose , Zapatillas de running Nike \ Volt / Black / White Skechers Sport Skech Air Run High Fashion Sneaker Malla azul marino / azul claro Zapatillas de agua Merrell para mujer All Out Blaze Sieve Brown Sugar / Blue Heaven Zapatillas de running Saucony PowerGrid para mujeres Gris / Coral , CLARKS Womens Wynnmere Avah Nubuck negro , Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower Sandalias EVA Gizeh de mujer Birkenstock Eva negro , ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Goma negra / Nueva gasolina G Nubuck , Zapatillas para caminar Omni Walker Grid de mujer Saucony Negro Chaco Zcloud 2 Sport Sandal para hombre Dapple Caribou Trotters Mujer Simmy Mary Jane Flat Ante negro Zapato de entrenamiento Ryka Women s Dream Negro / Gris hierro / blanco Zapato deportivo de skate para mujer Dickies Negro , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma True White , APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Black / 24k Melange Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Pergamino / blanco Sandalia Rockdale Lanea Woven Slide para mujer Tejido negro , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA7036 Tacón de 3 pulgadas + plegable Paquete de cepillo Dark Tan Satin Birkenstock Mules para mujeres Marron oscuro Zapatillas de mujer Vionic Relax tamaño rosa 5 M - , Bota de mujer Toms Alpa Chocolate Cafe , Jeffrey Campbell Botines de ante Cromwell para mujer Negro , Sandalia de vestir Taimi de Naturalizer para mujer Blanco Bota impermeable para excursionistas Zerogrand de Cole Haan para mujer Búfalo de gamuza de cuero negro y tango a cuadros , Pleaser - Botas para mujer, con tacón y punta abierta, 8 \ Blk Pu / Blk Matte Mercurial Superfly Quinto Triunfo FG Gold Odema Cuero Oxfords Mujer Perforado con cordones Wingtip Talón bajo Talla Brogue Zapatos de vestir Oxfords Amarillo verde

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 de mujer Rose KEEN mujer Rose Dark Purple/ Sage Purple Sage 4cdc5e3 - 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 de mujer Rose KEEN mujer Rose Dark Purple/ Sage Purple Sage 4cdc5e3 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 y toboganes deportivos
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.


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