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 de trail Vibrante running Merrell para mujer Bare Bare Merrell Access Arc 4 Elegante/ Vibrante Coral ee122df




Malla
Importada
Sintética suela
Malla y Parte superior de TPU
Tradicional cierre de encaje
Reflectante detalles para aumentar visibilidad en baja luz
Transpirable forro de malla
0mm Drop / 8mm Heel Tacón Cush / 12.7mm Altura de la pila
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 de trail Vibrante running Merrell para mujer Bare Bare Merrell Access Arc 4 Elegante/ Vibrante Coral ee122df - 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 de trail Vibrante running Merrell para mujer Bare Bare Merrell Access Arc 4 Elegante/ Vibrante Coral ee122df

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 de trail Vibrante running Merrell para mujer Bare Bare Merrell Access Arc 4 Elegante/ Vibrante Coral ee122df

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 Zante V4 para mujer New Balance Blue Iris / Pigment Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Negro / Blanco / Stealth / Cool Gray Zapatillas de trail running Salomon Speedcross 4 para hombre Blue Yonder / Black / Lava Orange Salomon Women s X-Mission 3 W-w Infrarrojo / Coral Punch / Teal Blue Fabric , Zapatillas de trail running Hiero V2 para mujer New Balance Espacio ultraterrestre / negro / amanecer blanqueado , Zapatillas de running adidas Women s Rockadia W Trail Carbono / Acero sin procesar / Trace Scarlet , adidas Performance Women s Incision W Trail Runner Negro / Metálico / Plateado / Verde Fácil , La Sportiva Akasha Running Shoe - Mujer Stone Blue / Jade Green 361 Zapatillas de trail running 361-Overstep 2 para mujer Ébano negro MT-onemix retro ligero jersey superior amortiguador de choque imprimir zapatillas de deporte al aire libre Vino Rojo / Negro Zapatillas de running adidas Women s Rockadia W Trail Trace Púrpura / Blanco / Core Black , Zapatillas de trail running 620v2 New Balance para mujer Teal oscuro / Lima Reebok Women s Ridgerider 2.0 Trail Runner Lead / Fire Coral / Ash Gray / Peltre , Columbia Rogue F.K.T. Steam / Dark Raspberry La Sportiva Akasha Running Shoe - Mujer Fiordo , Reebok Women s Ridgerider 2.0 Trail Runner Lead / Fire Coral / Ash Gray / Peltre Salomon Women s Speedcross 4 W Trail Runner Negro / Negro / Negro Metálico , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Gris / azul marino Colchoneta de amortiguación 620v2 New Balance para mujer Silver Mink / Gunmetal , Zapatillas de running Gore-TEX 910v4 New Balance para mujer Pigmento / Porcelana azul Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Reflexión MT-onemix Women Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Azul oscuro I , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Armada MT-onemix retro ligero jersey superior amortiguador de choque imprimir zapatillas de deporte al aire libre Blanco negro Calzado de running NIKE Zoom All Out Low para hombre NIKE Negro / gris oscuro / antracita Salomon Women s Speedcross 4 W Trail Runner Artic Blue , Zapatillas de senderismo V753WW085 para mujer Apex Cordura de nylon marrón , Zapatillas de trail running Merrell para mujer Bare Access Arc 4 Violet Storm Zapatillas de running New Trail Hierro V3 Fresh Foam Trail para mujer Gris ,
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 Nike Men s Air Presto Essential Juego Royal / Black-black , Zapatillas de running NIKE Roshe Two para mujer Mineral de hierro ligero / blanco / voltio Sandalia Omnium de mujer Teva Pizarra Columbia Montrail Women s Fluidflex X.S.R. Camelia Roja, Melonade New Balance W880gb6 para mujeres Gris azul , Zapatillas de running NIKE LD Runner LW para mujer Sail / Sail-lt Orewood Brn-dush , Sandalia de las mujeres de la marca de Lucky Brand Pulir , Zapatillas de running NIKE para mujer Lunarstelos Volt / White / Clear Jade Zapatillas de running Reebok para mujer Negro / Rosa ardiente / Blanco , Sandalia NIKE Comfort Comfort Slide 2 para hombre Negro / Plata Metálico / Blanco Zapatillas Nike Air Max Essential para hombre Negro / escuadrón azul / gris lobo , Zapatillas OLUKAI Hokua - Hombre Ónix Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Púrpura / rosa / verde azulado , Walking Cradles Womens Henna Taupe Nubu estampado claro Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Gris oscuro Earth Origins Tova Polvo , New Balance WW577 Hook and Loop Walking para mujer Negro Trotters Mujer Simmy Mary Jane Flat Ante azul marino , Zapatillas Hailee Ryka para mujer Azul marino , Vans Unisex Old Skool Zapatillas clásicas de skate True White / Light Gum Zapatillas Hailee Ryka para mujer Frost Grey / eggshell Blue / electric Coral Supra Vaider LC Sneaker Negro Negro , Zapato de skate casual Heathrow DC para hombre Noche de oliva / blanco Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Monocromo negro Vince Camuto Sanella deslizante de mujer Brillante plata , Mephisto Michaela Bootie para mujer Hazelnut Texas / Carla / Tobacco Bucksoft , Sandalia Birkenstock de cuero Florida para mujer Negro Blundstone Womens BL1383 Brodo Vintage Cuero de mujer Lotus Jazz Oxford Negro , Easy Street Lucky Oxford para mujeres Forest Green Super Suede

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 de trail Vibrante running Merrell para mujer Bare Bare Merrell Access Arc 4 Elegante/ Vibrante Coral ee122df - 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 de trail Vibrante running Merrell para mujer Bare Bare Merrell Access Arc 4 Elegante/ Vibrante Coral ee122df 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 537733
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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.