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.

Under G Armour Blanco Women Blanco s Micro G Limitless 2 Blanco (101)/ Blanco 226e738




Cuero, sintético, Textil, EVA, goma
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Ajuste de forma media manga de compresión envuelve tu pie en una capa apretada de apoyo
Overlay utiliza sintético material en el lado lateral para protección liviana y cuero en el lado medial para durabilidad
Cuero puntera agrega protección y durabilidad
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


Under G Armour Blanco Women Blanco s Micro G Limitless 2 Blanco (101)/ Blanco 226e738 - 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

Under G Armour Blanco Women Blanco s Micro G Limitless 2 Blanco (101)/ Blanco 226e738

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.


Under G Armour Blanco Women Blanco s Micro G Limitless 2 Blanco (101)/ Blanco 226e738

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 running New Balance Coas V3 para mujer Blanco plateado Converse Chuck Taylor All Star Seasonal Color Hola Azul polar Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Blanco negro Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Negro Under Armour Women s HOVR Phantom NC Azulejo azul (302) / grafito NIKE Men s Lunar Fingertrap Cross Trainer Negro / Blanco - Negro , Converse Chuck Taylor All Star Seasonal Color Hola rojo Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Total Naranja / Foto Azul / Gamma Azul / Blanco , NIKE Women s Tanjun Partícula rosa / blanco Zapatillas de running NIKE Roshe Two para mujer Blanco / Blanco-Puro Platino Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco , Zapatillas de running NIKE Air Huarache City para mujer Vela / Luz Menta-lt Racer Blue , Converse Chuck Taylor All Star Seasonal Color Hola Beach Glass , Zapatillas de running NIKE Juvenate para mujer Dark Loden / Olive Flapnk / Sail , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Arcilla Verde / Barely Gray , Mujer adidas Originals Ultraboost St Negro / Plata Metálico / Carbono Calzado de running rápido para mujer Reebok, tejido flexible Negro-pedernal gris-blanco Converse Chuck Taylor All Star Seasonal Color Hola Yute , Zapatillas Nike Women s Flex Trainer 5 Wolf Gray / Racer Pink / Stealth Zapato Casual de Cuero Cortez Clásico NIKE para Mujer Blanco negro , Zapatillas NIKE Air Huarache Run SE (GS) Girls Fashion 904538 Negro, antracita , Kangoo Jumps XR3 Edición especial Negro Negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco negro , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Medium Heather gris / Medium Gray Heather / Dark Heather gris , Converse Chuck Taylor All Star Seasonal Color Hola borgoña NIKE Mujeres Air Zoom Pegasus 33 Ocean Fog / Black-gamma Blue-volt Mujer adidas Originals Ultraboost St Indigo crudo / tinta noble / naranja de alta resolución Zapatillas de running New Balance Coast V3 para mujer Amanecer Glo / Plata NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Negro 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 Zapatillas de running NIKE Air Huarache para hombre Azul marino / difuso-obsidiana Sandalia Yucatan ECCO Mujer Shadow White Inov-8 Mujer F-Lite 260 (W) Cross Trainer Púrpura / negro Zapatillas de running NIKE para mujer Lunarstelos Bright Crimson Metallic Silver 602 , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco / azul oscuro Saucony Women s Ride 10 zapatillas para correr Gris Zapatillas para correr Brooks Cascadia 12 para mujer Azul marino / azul / menta , Nike Hombres Air Zoom Pegasus 33 Wolf Gray / Blanco-cool gris Sandalias EVA Gizeh de mujer Birkenstock Rosa Eva , Salvatore Ferragamo Mujer Varina Nuevo Blush Sandalia de cuña de Easy Street para mujer Serpiente Metálica Indigo Rd. Gris pardo , Easy Spirit Women s Traveltime Mule Blanco negro Zapatos de mujer RYKA Sky Walking Frost Grey / Aqua Sky / Iron Grey Zapatillas Superga 2750 Cotu para mujer Rosa / Goma Zapatillas para caminar Brooks Women s Addiction Walker Negro , Entrenador cruzado de trabajo 627v2 New Balance para mujeres Gris / rosa Zapatillas para caminar al aire libre Go Skechers Performance para mujer Caqui Sandalias de mujer Mia de Sandalias de Jerusalén Blanco , Sandalia Tilden Slide de Franco Sarto para mujer Tordillo La Plume Diapositiva de estiramiento de la mujer marrón Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Terciopelo gris (pedido personalizado, 4-6 semanas de tiempo de proceso) Conjunto adidas Performance PowerAlley 2 TPU W Softball Cleat Blanco / Carbono / Negro , Zapatillas de verano mahabis Gya Gray / Rjukan Red Zapatilla de cuña para mujer Jacques Levine # 4640 Negro Metálico , Haflinger para mujer Berenjena Mocasines de conducción Twisted X para mujer Bomber / Neon Pink - Stiched Vamp Design Outdoor Footwear , Bernie Mev Womens Chesca Pump Terciopelo de plomo / negro , Zapatillas de deporte Arizona Trotters para mujer Negro , Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker 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.

Under G Armour Blanco Women Blanco s Micro G Limitless 2 Blanco (101)/ Blanco 226e738 - 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 Under G Armour Blanco Women Blanco s Micro G Limitless 2 Blanco (101)/ Blanco 226e738 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 682937
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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.