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 Sandalia Kea Vent Fawn mujer Columbia Vent Iceberg, Fawn e2341db




Cuero y Textil
Importado
Sintético suela
Plataforma mide aproximadamente 0.5 pulgadas
PU cuero recubierto.
Hidrofóbico forro.
Omni-Grip tracción sin marcas caucho.
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 Sandalia Kea Vent Fawn mujer Columbia Vent Iceberg, Fawn e2341db - 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 Sandalia Kea Vent Fawn mujer Columbia Vent Iceberg, Fawn e2341db

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 Sandalia Kea Vent Fawn mujer Columbia Vent Iceberg, Fawn e2341db

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

Sandalias Newport H2 de KEEN para mujer Rama de Malaquita / Plata , Sandalia Yucatan ECCO Mujer Polvo / polvo , Skechers Women s on-The-Go 600-15315 Wide Sport Sandal Negro OluKai Ohana Sour Tart / Black Encaje de quimera de Columbia para mujer Zinsson , Sandalia Drew Angela para mujer Dusty Multi Metallic Softinos Tai - Sandalias de mujer de cuero liso negro Negro , Sandalia Deportiva Jordan Spires de Via Spiga Ante negro , Sandalias Bali KEEN para mujer Negro / Imán CLARKS Womens Wave Grip Gris Claro Nubuck , Sandalia Savannah de Ryka para mujer Azul marino / verde Skechers Performance On The Go 400 Tropical Flip Flop Natural / Coral Tropical , OluKai Kulapa Kai Piña / Negro Sandalia de cuero Maya New Balance para mujer marrón , Sandalia Terrell Ivy Wrap Sport Merrell para mujer Sodalita , Sandalia de mujer KEEN Rose Boysenberry / Red Violet Zapato de vestir uniforme al aire libre Vulc N Vent Slip para mujer de Columbia Blanco, gris fresco Sandalias KEEN Mujer Whisper-w Sterling Blue / Vestido Azul Earth Origins Sophie Women s Sandal Sedona Brown , Sanda Verra Teva para mujer Cuerda elástica , Sandalia Terrell Ari Convert Sport de Merrell para mujer Dusty Olive Sandalia para mujer Merrell, alrededor de la ciudad Azul , Wolky Comfort Jewel 80210 Cuero Antracita OluKai Ohana Coral / Negro , Sandalias KEEN Womenport Newport Hydro-W Imán / Navega por la Web OluKai Ohana Dk Brown Sandalia de cuero Maya New Balance para mujer Negro Chaco Women s Z2 Classic Athletic Sandal Negro , Sandalia de vestir Tillman de SoftWalk para mujer rojo
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 Ultraboost W para mujer Blanco / blanco / blanco cristal , Sandalia Merrell para hombre All Out Blaze Aero Sport Vapor , Zapato de running retro con diseño clásico de ASICS para mujer Coral / Coral , Calzado de running XA Amphib W Trail para mujer Salomon Potente púrpura , Zapatillas de trail running Xa Pro 3D W para mujer Salomon Azul Medieval Chaco Z / Canyon 2 Negro acolchado , Sandalia Knoll para mujer Marc Fisher melocotón , JBU by Jambu Women s Wildflower Sandal Café exprés Skechers Performance Women s Go Step Modish Negro , Toni Pons Womens Noa Multi Cordelia , Drew Shoe Women s Tulip Comfort Ternero blanco Sandalia de cambio de mujer Steve Madden Negro Sam Edelman Women s Slide Sandalia rojo Sandalia Pavi de estilo suave para mujer Niebla blanca Zapatillas Skechers Go Walk Lite-15360 para mujer Armada Hush Puppies Mujeres Power Walker Ii Loafer Negro Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Tamaño de Navy_men , Sandalia de plataforma de playa Coast de Seychelles Blanco , Sandalia KAANAS Women s Recife Bow Fashion Slide Desnudo , Sandalia Claire para mujer Dolce Vita Cuero natural , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA7036 Tacón de 3 pulgadas + plegable Paquete de cepillo Dark Tan Satin Zapatillas de ciclismo de Triatlón Air Carbon T4 para mujer Sidi Fucsia Blanco Estilo suave de Hush Puppies Glynis Ii Botín para mujer Vitello marrón oscuro , Bomba Sadie Wedge 40MM de Cole Haan para mujer Roccia Snake Lauren by Ralph Lauren Boot Tamia-BO-Cwd para mujer Camello , CLARKS Mujer Caddell Hop Boot Armada Camper Mujer Bowie K200016 Oxford Negro Mordenmiss Women s Leaf Elástico Panel Wedge Martin Boots School Style Plataforma Calzado Verde Sanuk Women s Pair O Dice Flat Cuarzo rosa , Mocasines de vestir de punta estrecha para mujer de Me Too Women Negro

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 Sandalia Kea Vent Fawn mujer Columbia Vent Iceberg, Fawn e2341db - 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 Sandalia Kea Vent Fawn mujer Columbia Vent Iceberg, Fawn e2341db 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

Thirteen elementary straightedge 534024
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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.