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.

Chaco Women s Zx2 Chaco Classic Athletic Sandal Classic 10539 Fusion Rose 25a1923




Poliéster
Importado
Goma suela
con doble correa poliéster jacquard correas superiores alrededor del pie y a través del entresuela para un personalizado encajar con el dedo del pie para un ajuste seguro
Ajustable y durable de alta resistencia correas del talón de las correas
Antimicrobiano solicitud de control de olores
Mujeres LUVSEAT específico Entresuela de PU
Non-marking Caucho ChacoGrip compuesto con 3mm profundidad del asa
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


Chaco Women s Zx2 Chaco Classic Athletic Sandal Classic 10539 Fusion Rose 25a1923 - 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

Chaco Women s Zx2 Chaco Classic Athletic Sandal Classic 10539 Fusion Rose 25a1923

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.


Chaco Women s Zx2 Chaco Classic Athletic Sandal Classic 10539 Fusion Rose 25a1923

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

Sandalia deportiva Kana para mujer ECCO Negro Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco , Sandalia Drew Element para mujer Nubuck negro Teva Terra Sandal para hombre Madang Blue , Sandalia deportiva Nolita para mujer Anne Klein Tela negra Sandalias KEEN Venice II H2-W para mujer Cornstalk / Safari Sandalia de mujer Drew Alana Blanco negro , Wolky Comfort Jewel Beige , Chaco Women s Wrapsody X Sandalia Key Teal Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador , Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón , Sandalias KEEN Venice II H2-W para mujer Paloma / Turquesa Pastel OluKai Kulapa Kai Coral / Negro OluKai Kulapa Kai Pale Jade , Sandalia Terra-Float Livia de Teva para mujer Negro , Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador Mordenmiss - Sandalias Huarache de cuero, para mujer, de verano, zapatos planos de tacón Estilo 2-amarillo , Chaco Women s ZX2 Classic Athletic Sandal Motif Eclipse Sanda Verra Teva para mujer Atlántico Norte , Skechers Performance On The Go 400 Vivacity Flip Flop Blanco / azul claro , Nike Womens s Benassi JDI Print Slide Negro / Fireberry- Blanco Chaco Women s ZX2 Classic Athletic Sandal Credo Golden Sandalias KEEN Mujer Whisper-w Ágata gris / azul ópalo Sandalia de mujer KEEN Rose Aluminio / gris neutro , Chaco Women s Z2 Classic Athletic Sandal Florista Chaco Mujer ZX3 Classic Sport Sandalia Shiver Pine Wolky Comfort Jewel Fantasía de serpiente blanca / negra posters , Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador Sandalia de mujer KEEN Rose Boysenberry / Red Violet
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 Zapato deportivo para acolchado 635v2 New Balance para mujer Gris negro Zapatillas de running Nike Air Huarache Run Ultra para hombre Blanco Zapatillas deportivas 574S Sport New Balance para hombre Gris con pigmento Skechers D Lites-Retro Glam-Rhinestone River Style Deportivo Sport Sport Sandalia Negro Negro Zapatillas de running adidas Women s Element V (9.5, Grey Three) - , Guantes Trail Trail Merrell Women s Glove 4 Negro Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker GTX W para mujer Negro / carbono / verde ceniza Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Azul marino / gris Sandalia CLARKS Tri Clover para mujer Black Combi , CLARKS Sandalia plana Brizo Sammie para mujer Microfibra perforada negra , Zapatillas para caminar Brooks Women s Addiction Walker Hueso , Taipán de David Tate Doll para mujer Ternero desnudo natural Bota Deia para mujer de TOMS Broncearse , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Platino Blanco / Blanco-puro , Dr. Comfort Marla Mujer diabético terapéutico Extra Profundo Calzado Lycra Velcro Negro , Hush Puppies Mujeres Power Walker Ii Loafer Cuero negro Diadora N9000 Men Toe Snthetic Blue Sneakers Blue Angel Falls / Dresden Blue , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Alto riesgo rojo / Puma blanco Golf de mujeres y rsquo; s Peso ligero genuino Cuero hecho en Libertad de Brasil Rendimiento de golf Marc Joseph NY Zapatos de moda Celeste floral Sandalia para mujer Layne Slide de Pour La Victoire Natural Sandalia de la diapositiva de Stella de las mujeres de Rachel Zoe Negro Zapatillas de voleibol adidas Crazyflight X 2 de mujer Blanco / Plateado Metálico / Gris , 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 1682- Tan satinado Trotters Lucy Ballet para mujer Desnudo Zapatilla de plataforma Limón de mujer Soludos Chambray , Blundstone Unisex Original 500 Series Stout Brown / Navy Tartan Zapatillas NIKE Roshe Two Hi Flyknit para mujer 861708 Sneakers Boots Deep Borgoña / Deep Borgoña-team Red , Bota alta de mujer Teva W DE LA Vina Dos Bisonte Zapatillas de fútbol adidas X.2.2 Fg W, para mujer Negro / Platino Core Red S , Calzado con cordones láser mujer Mephisto Indigo Bucksoft / Bleu ,

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.

Chaco Women s Zx2 Chaco Classic Athletic Sandal Classic 10539 Fusion Rose 25a1923 - 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 Chaco Women s Zx2 Chaco Classic Athletic Sandal Classic 10539 Fusion Rose 25a1923 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 884439
952635
2016424
2315532
10184461
14170499
14304254
14356345
16088047
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.