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.

Vans Herren Sneakers Vans Authentic Core Classic Sneakers Herren Armada dd4c79c




Lienzo
Unisex zapatos de diseño - B (M) para mujeres tamaño / D (M) para tamaño de los hombres (por ejemplo, 6 B (M) = mujeres de EE. UU. 6)
Parte superior de lona y suela de goma
VANS AUTÉNTICO
Importado
Por Furgonetas
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


Vans Herren Sneakers Vans Authentic Core Classic Sneakers Herren Armada dd4c79c - 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

Vans Herren Sneakers Vans Authentic Core Classic Sneakers Herren Armada dd4c79c

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.


Vans Herren Sneakers Vans Authentic Core Classic Sneakers Herren Armada dd4c79c

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

Clásicos de base Slip-on (Tm) para hombres de Vans Black Off White Checker , Vans Unisex Old Skool Zapatillas clásicas de skate Ground Breakers Ron de pasas Marshmallow , Zapato de skate Wnt High rebote DC para mujer Armada Vans Unisex Old Skool Zapatillas clásicas de skate (Lienzo) Negro / Blanco verdadero Zapatillas de deporte Unisex Skate Trainers de Vans Authentic Negro Negro , Vans Auténtico Neón Verde / Azul Zapatillas de skate Heathrow DC para mujer Rosa brillante , Calzado Lakai Limited para hombre Bristol Ante negro / blanco Vans Auténtico Pale Khaki / True White Vans Unisex Old Skool Zapatillas clásicas de skate Lux Leather Porcini negro Zapatillas de skate DC Women s Villain SE Slip-on Oro rosa , Zapatillas de skate Heathrow DC para mujer Rosa brillante , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco Vans Auténtico Port Royale Red True White , Supra Scissor Skate Shoe Gris claro - Rojo-m Zapatillas DC Evan Hi Le Skate para mujer Crema Zapatillas Slip-On Skate Print de Vans para hombre Negro Negro Supra Vaider Skate Shoe Blanco Blanco Final Fantasy 10 FF10 Rikku Cosplay Shoes Boots por encargo , Zapatillas Supra Titanium Skate Negro Negro , Clásicos de base Slip-on (Tm) para hombres de Vans (Denim C & l) Negro / Goma , Clásicos de base Slip-on (Tm) para hombres de Vans (Mono Canvas) Asfalto , Clásicos de base Slip-on (Tm) para hombres de Vans (Bum de goma) True White / True White Vans Unisex Old Skool Zapatillas clásicas de skate (Negro / Negro (Lienzo) DC Mujer Trase TX SE Chambray , Supra Vaider Skate Shoe Blanco Blanco , Supra Scissor Skate Shoe Negro - Carbón vegetal-m , Zapatillas Supra Titanium Skate Azul marino / blanco Vans Unisex Old Skool Zapatillas clásicas de skate Jengibre glaseado / 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 Brooks Womens Ravenna 9 Azul marino / azul / verde Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Pedernal Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro / Porcelana Azul-blanco NIKE Women s Tanjun Paramount Blue White Lava Glow , Teva Terra Sandal para hombre Cross Terra Black Sandalias Teva Mujer W Terra Fi Lite Atitlan Blue Brooks Womens Adrenaline GTS 18 Negro Negro Corredor de malla Primus FG L Mesh Trail de Vivobarefoot para mujer Negro / Carbón , Zapatillas de running New Balance para mujer 590v3 Ciclón / Ciclón New Balance Women s Summit - Desconocida Trail Running Shoe Rosa / negro , Bernie Mev Women s Gummies Gem Flat Pavo real , Calzado SCARPA Women s Crux Walking Camel / Poppy Red , Crocs Flip Crocband unisex Armada , Chaco Mujer Zcloud Sport Sandal Marquise Pink Sandalia de tira de Ghillie Bre-Italy Bella Vita para mujer Cuero de ante taupe Zapato de malla Nohea de OLUKAI - Hombre Carbono / rojo oscuro Zapatillas para correr New Balance Rushv3 para mujer Acero / Negro , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro , Sandalia de cuña Logan Walking Cradles para mujer Tabaco , Sandalias Sunstone para mujer Merrell Vertiver , Skechers Women s Go Flex Max Loafer Gris pardo Empuje cargado de Under Armour para mujer Grosella negra (600) / rojo , Vince Camuto Sandalias Ellanna Slide para mujer Moody Blues , Sandalia Maria Slide de Nanette Lepore para mujer Rosa empolvado , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1680- Cuero plateado , Adulto 2007 \ Zapatilla de deporte Kiera Fashion Kenneth Cole New York para mujer Almendra , Arcopedico Mujeres Ls Burdeos , Easy Spirit ON CUE Oxford para mujeres Gris pardo Zapatos holgazanes Joleen de mujer Alegria O yeah Blues ,

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.

Vans Herren Sneakers Vans Authentic Core Classic Sneakers Herren Armada dd4c79c - 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 Vans Herren Sneakers Vans Authentic Core Classic Sneakers Herren Armada dd4c79c 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 742213
754073
1909335
5941488
6059880
7282971
12330855
14374615
16721735
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.