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 para correr/ Saucony Koa St para Aqua mujer B06Y5NSHPC Verde/ Negro/ Aqua 0e7414b




Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Everun Suela interior
Resistente al agua Upper
Alternar Sistema de cordones
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 para correr/ Saucony Koa St para Aqua mujer B06Y5NSHPC Verde/ Negro/ Aqua 0e7414b - 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 para correr/ Saucony Koa St para Aqua mujer B06Y5NSHPC Verde/ Negro/ Aqua 0e7414b

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 para correr/ Saucony Koa St para Aqua mujer B06Y5NSHPC Verde/ Negro/ Aqua 0e7414b

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 Gobiv2 para mujer New Balance Berenjena / Vintage Indigo , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Saúco , Columbia Montrail Women s Trans Alps F.K.T. Jade Lime, Splash Zapatillas de senderismo Columbia Outfit para senderismo ATS Trail LF92 para mujer Titanio Mhw, blanco , Zapatillas de trail WT610V4 para mujer New Balance Gris / rosa Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral Zapatillas de trail running TIMP IQ de Altra para mujer Carbón de leña / azul Zapatillas de running para correr adidas outdoor Kanadia 8 TR para mujer Negro / Core Pink / Trace Gray , Dyzing Cancerribbon y mariposas Hope Women Casual Sneakers Running Shoes , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Armada , Zapatillas de trail running Salomon Speedcross 4 para hombre Poseidón El corredor de trail impermeable Xa Pro 3D Cs para mujer de Salomon Sombra / Negro / Ártico , Reebok Women s Ridgerider 2.0 Trail Runner Negro / Rosa Solar / Plata / Estaño , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / Rojo , Zapatillas de running Salomon Women s Speedcross 4 Wide W Trail Poseidón Zapatillas para correr Salomon S / Lab Sense Ultra 2 para hombre Racing Red, Maverick, Blanco Zapatillas de trail running Salomon Speedcross 4 para hombre Poseidón , Zapatillas de running NIKE Zoom All Out Low para hombre Negro / Plata Metálico / Concord ASICS GT-2000 6 Trail Mujer corriendo Trébol de cuatro hojas / negro / negro , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Columbia / Navy , Zapatillas de trail running Xa Pro 3D W para mujer Salomon Azul Medieval , Zapatillas de running Salomon Women s Sense Mantra 3 Gentiane / Igloo Blue / Firefly Green Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / Naranja Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Saúco Zapatillas de mujer Salomon X Ultra 3 GTX para mujer Imán , Zapatillas trail running mujer Asics Fujitrabuco 5 Fl.coral / Saf.yellow / Black , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris oscuro / negro / Shock Pink S Under Armour Women s Horizon KTV Neon Coral (601) / Blue Formation Zapato de mujer Columbia de Redmond Trail Boulder, azul cielo ,
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 para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo , Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Negro / Rojo Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco , Calzado de running NIKE Kids Roshe One (GS) Negro / Plata Metálico / Blanco , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Chaco Mujer ZX1 Classic Sport Sandalia Burst Blue Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth Sandalias Swiftwater Mesh de Crocs para mujer Nuez , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 melocotón CLARKS Womens Wave Grip Gris Claro Nubuck , NIKE Men s Roshe Run Gym Blue / Anthracite-pure Platinum Sandalia Sharene de Calvin Klein para mujer Arena del desierto Zapato deportivo ASICS Gel-Cumulus 18 para mujer Negro / Plata / Negro , Zapatillas de viaje Propet para mujer Taupe Suede , Sandalia Vanessa de mujer Alegria Chrysalis Black , UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Verde blanco Teva Hydro-Life Ballet de mujeres Teva Negro , Zapatillas de skate DC Mujeres Heathrow IA Negro / Aqua , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Negro / Medio oliva-blanco , Converse Chuck Taylor All Star Core Hola Blanco / rojo / blanco Sandalia Pour La Victoire Mujer Mallory Slide Azalea Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / púrpura , BareTraps Women s Cherilyn Slide Sandal Blanco , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 7017- Serpiente dorada , Belleville 700V WP Black Combat Flight Boots para hombre Negro , Bomba antideslizante a medida Nolan para mujer Franco Sarto Negro Belleville Hot Weather Desierto Botas y vehículos de combate (cisterna) Botas, Hombres, Zapatos Broncearse Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Lobo Gris / Blanco / Hematita Metálica Braguitas de tobillo Bartonn para mujer de la marca Lucky Caramelo Zapatillas Slip-On Roamer SAS para Mujer Sabio

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 para correr/ Saucony Koa St para Aqua mujer B06Y5NSHPC Verde/ Negro/ Aqua 0e7414b - 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 para correr/ Saucony Koa St para Aqua mujer B06Y5NSHPC Verde/ Negro/ Aqua 0e7414b 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 4459417
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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.