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 tenis adidas adidas Universo Asmc Asmc Barricade 2017 para mujer Blanco/ Universo/ Electricidad b662410




Sintético
Importado
Caucho suela
resistente a la abrasión ADITUFF se envuelve el dedo del pie y medial antepié para ayudar proteger contra pie arrastre que ocurre durante los servicios, voleas y extrema movimientos laterales
360 grados Soporte de TPU para durabilidad en la llave áreas de abrasión y lateral superior estabilidad; proporciona adaptativo soporte del mediopié; adidas por Stella Logotipo de McCartney en la lengua y el talón
Longitud completa ADIPRENE + para optimizado amortiguación y rebotes; debajo del talón para una amortiguación superior en el impacto
ADIWEAR 6 ofertas de suela lo último en durabilidad de alto desgaste
de dos capas malla de ingeniería superior con mono-malla arriba para máximo transpirabilidad; Botín sin costuras construcción para último ajuste y comodidad
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 tenis adidas adidas Universo Asmc Asmc Barricade 2017 para mujer Blanco/ Universo/ Electricidad b662410 - 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 tenis adidas adidas Universo Asmc Asmc Barricade 2017 para mujer Blanco/ Universo/ Electricidad b662410

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 tenis adidas adidas Universo Asmc Asmc Barricade 2017 para mujer Blanco/ Universo/ Electricidad b662410

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

Teva Hydro-Life Ballet de mujeres Teva Gris , Zapato Dantera, para hombre, L Low Waxed Court Negro / Blanco / Negro Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Blanco / Tricolor HEAD Sprint Pro 2.0 Junior s Tennis Shoe Marino / Coral K-Swiss Women s Bigshot Light 3 Blanco / Acuario / Plata , adidas Adizero Club 2 Shoe Unisex Tennis Leyenda Ink-white-tech Ink Zapatillas de tenis NIKE Women s Court Lite Blanco / Lava Glow-vast Grey , Zapatillas de tenis ASICS Gel-Solution Speed ​​3 para mujer Flash Coral / Plum / Flash Coral Zapatillas de deporte Fenty X High Top Trainer de mujer PUMA Whisper White / Whisper White Zapato de entrenamiento Ryka Women s Dream Frost Grey / Summer Grey / Calypso Coral Converse Chuck Taylor All Star Core Ox Negro Negro Zapatillas de tenis Adidas Adizero Club w, para mujer Chalk Blue / Semi Frozen Yellow / Noble Indigo Zapatillas de tenis adidas Barricade Classic Wide 4E, Negro / Plateado Metálico / Blanco, 6 M US Blanco / Core Black / Mid Gray , Zapatillas de tenis 696v3 New Balance para mujer Pigmento / coral vivo , Phenma femenino Phenom Satin EP Puma Black / Puma Black , Converse Chuck Taylor All Star Core Ox Shale Grey / Pure Silver / White , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco , Diadora Unisex Intrepid NYL Blanco / negro-frost gris Lakai Limited Calzado para hombre Daly Lienzo blanco , Converse Chuck Taylor All Star Core Ox Mono blanco Zapatillas de tenis NIKE Women s Court Lite Gris / rosa Adidas Adipower Stabil 10.1 Women s Indoor Court Shoe Night Shade / White / Vivid Berry Ryka Womens Divya Gris / Menta / Plata , Zapatillas de tenis adidas Originals Barricade Court para mujer Blanco / Tech Ink / Hi-res Aqua Reebok Zapatillas de deporte Royal Aspire 2 Fashion para mujer Negro / Plata Metálico , Entrenador cruzado original de 2 ojos PUMA Womens Authentic Rosa de cobre HEAD Zapatillas para mujer Sprint Pro Court Negro / rosa , Tanga para mujer Sunrell de Merrell Sodalita , Converse Chuck Taylor All Star Core Ox Jester Red
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 Sandalias sintéticas Teva para mujer Terra Fi Lite Atlántico Norte Zapatillas de volante Nike Women s Zoom Hyperface 2 Blanco negro , Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Calzado de running adidas Pureboost X Element para mujer Ash Pearl / Ash Pearl / Hi-res Blue Zapatillas de running Saucony Women s Cohesion TR11 Gris / menta Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre Zapatillas de deporte Onitsuka Tiger Mexico 66 Amarillo / negro , Zapatillas de running ASICS GEL-Foundation 12 para mujer Plata / Pistacho / Pink Glow NIKE Men s Flex 2017 RN Negro / Metálico Cool Gray / Light Carbon Skechers Sport Skech Air Run High Fashion Sneaker Carbón de leña / verde , Sandalia de cuero Maya New Balance para mujer Negro , UB-ONEMIX Cool Sports Shoes Running Malla transpirable Zapatillas de deporte con cordones Tamaño EU 35-45 Rosa Rosa Zapato para caminar WW411v2 New Balance para mujer Negro L Artiste de Spring Step Caiman Women Slide Camello , BareTraps Womens Jozie Low Top Lace Up Running Sneaker Seta NIKE Women s WMNS Air Max 270, Blanco / Blanco-Blanco Blanco Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura , Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer Difuso: negro / negro Chancleta Yoga Sling 2 Sanuk para mujer Manta Bayridge Natural , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Mushroom Nubuck Skechers Performance Women s On The Go City 3 Optimize Negro , Zapatillas para caminar Ryka Women s Dash 2 Gris / plata / lima Sandalia para mujer Alegria Mixie Slip On Sand Safari Vans Auténtico (Vintage Aloha) Negro / Blanco Verdadero CLARKS Mujer Caddell Ivy Slide Sandal Azul marino sintético Sandalia FLY London Women s Whin176fly Slide Reef Cupido Mocasines casuales Alpargata Deconstructed de TOMS para mujer Bloom Nubuck Zapatilla de deslizamiento Loki para mujer BEARPAW Nuez dura Dr. Martens 1460 Shamrock LZ Fashion Boot Verde oscuro , Ovation Celebrity Slim Secret Asiento completo Blanco ,

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 tenis adidas adidas Universo Asmc Asmc Barricade 2017 para mujer Blanco/ Universo/ Electricidad b662410 - 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 tenis adidas adidas Universo Asmc Asmc Barricade 2017 para mujer Blanco/ Universo/ Electricidad b662410 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.

Tenis y deportes de raqueta
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1764175
2462127
5200925
5762350
6174451
7427896
10174880
16940581
18164018
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.