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.

Zapatillas Gris de 6 running Saucony Saucony para mujer Echelon 6 Gris rosado 8feb188




Textil y sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
EVERUN Plantilla para continuo amortiguación y energía mejorada regresar
SAUC-FIT para un altamente adaptable, ajuste dinámico en el pie medio
8mm PU Sockliner entrega Mayor durabilidad añadida y resiliencia a pie huelga
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


Zapatillas Gris de 6 running Saucony Saucony para mujer Echelon 6 Gris rosado 8feb188 - 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

Zapatillas Gris de 6 running Saucony Saucony para mujer Echelon 6 Gris rosado 8feb188

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.


Zapatillas Gris de 6 running Saucony Saucony para mujer Echelon 6 Gris rosado 8feb188

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 deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro Zapatillas de entrenamiento NIKE Women s Metcon 4 Blanco / Negro-total Crimson Zapatillas de running Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño Calzado casual NIKE Flyknit AF1 para mujer Black / Black-team Red-clr Jade NIKE Women s Revolution 4 Running Shoe Wide Gunsmoke / Ocean Bliss / Dark Grey / White , Zapatillas de running New Balance 990V3 para mujer Aguamarina Zapatillas de running Nike \ Volt / Black / White , Zapatillas de running Reebok Women s Run Supreme 2.0 Aleación / Gris Estaño / Orquídea Celestial / Blanco Zapatillas de running ASICS Gel-Nimbus 19 para mujer Insignia Blue / Glacier Sea / Crystal Blue , Zapatillas para correr Altra Women s Intuition 3.5 Hemlock / Peltre Zapatillas de running Reebok para mujer Faux Indigo / Sheer Blue / White / Colegiate Navy , NIKE W Air Max 97 Se Mujeres Aq4137-101 - , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Max naranja / negro / blanco , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Gris / sal marina , Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres gris , Zapatillas para correr mujer Gaviota HOKA ONE ONE , Zapatillas de running New Balance 990V3 para mujer Blanco azul , Zapato de running ASICS Gel-Quantum 360 cm para mujer Melocotón negro / Onyx , Short NIKE Tempo para mujer Negro 010 Saucony Women s Ride 10 Running Shoe Azul marino / mezclilla , Zapatillas de running adidas Mana Bounce de adidas Performance para mujer Bold Pink / Prism Blue / Shock Pink , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Negro Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Utiblk, Ftwwht, Mgsogr , Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura Zapatillas de running NIKE Lunarglide 9 para mujer Negro / blanco / gris oscuro / gris lobo , Rieker womens Slipper oldgold / fango-silver Oro viejo / Fango-plata ,
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 Chaco Mujer Zcloud Sport Sandal Agrupado azul , Vibram FiveFingers Mujeres Vi-S Zapatos descalzos y paquete de Toesock Negro , Zapatillas de running con amortiguación W560v7 para mujer New Balance Silver Mink Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral , Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Blanco negro Inov-8 Mujeres X-Claw 275 Chill Trail Runner Plata / Verde azulado / Gris CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Forest Sun XYD Prom Dancing Shoes Elegant Open Toe Strappy Sandalias de tacón de tobillo Wrap vestido de bombas para las mujeres Plata , Chancleta Yoga Sling 2 Sanuk para mujer Plateado / Tan claro YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Gris Arche Women s Drick Flat Nacre / Brume Fast Metal Under Armour Women s Street Precision Low Lux Downtown Green (330) / Marfil Zapatos OLUKAI Pehuea - Mujer Dusty Olive / Palm Zapatos de mujer Spira Classic Walker 2 con resortes Negro Zapatillas de skate casual unisex Heathrow Prestige para hombre de DC Piedra Zapato de skate Etnies Jameson Vulc para mujer Denim negro , Sandalia de viaje Via Spiga Women s Porter Patente naranja caliente Sandalia de mujer Blakely Suede Slide de Jack Rogers Negro Sandalias clásicas para mujer del Dr. Scholl s Shoes Black Global Stripe , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro Kaepa Youth Stellarlyte Cheer Shoe (par) Blanco Sandalias planas con cordones Pinch Montauk de Cole Haan para mujer Cuero acolchado negro Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 6829BEB Cómodo-Muy Fino 2.5 \ de 5] Cuero Marrón Beige , Zapatilla de mujer Old Friend Scuff Rubí , Minnetonka piel de venado mujer Gore Front mocasín Champagne Deerskin Baabuk Wool Sneaker - Mujer Gris claro / negro , Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse Bota de media caña Georgia GB00096 marrón Zapatillas de deporte Arizona Trotters para mujer Negro , Bomba de cuña Etenia Total Motion Rockport para mujer Serpiente negra / múltiple ,

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.

Zapatillas Gris de 6 running Saucony Saucony para mujer Echelon 6 Gris rosado 8feb188 - 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 Zapatillas Gris de 6 running Saucony Saucony para mujer Echelon 6 Gris rosado 8feb188 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

Thirteen elementary straightedge 1620648
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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.