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 running de trail running B01MU0UL9R WT10v1 Minimus trail para mujer New Balance Gris azul 1f8c3f5




Sintético
Importado
Vibram único
Acteva entresuela. lazo
Antimicrobiano tratamiento para reducir olor
gota de 4 mm, Suela Vibram
Con una parte superior hecha de materiales de malla flexible y una suela Vibram con surcos flexibles.
Esto trail running femenino zapato ofrece el tracción y durabilidad necesitas correr en senderos rocosos. Cierre con 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 running de trail running B01MU0UL9R WT10v1 Minimus trail para mujer New Balance Gris azul 1f8c3f5 - 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 running de trail running B01MU0UL9R WT10v1 Minimus trail para mujer New Balance Gris azul 1f8c3f5

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 running de trail running B01MU0UL9R WT10v1 Minimus trail para mujer New Balance Gris azul 1f8c3f5

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 para correr adidas outdoor Kanadia 8 TR para mujer Lino / Col. Zapatillas de trail running Inov-8 Trailtalon 235 (W) para mujer Teal , Rookite 325 Trail Runner de mujeres Inov-8 Gris oscuro / rojo oscuro , El corredor de mujeres Xa Pro 3D W de Salomon Negro MT-onemix Women Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Verde claro / blanco I , Zapatillas de running Saucony Women s Excursion Tr11 GTX Gris morado Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / Naranja Inov-8 Terraclaw 250-U Trail Runner Verde azulado / azul marino / púrpura Zapatillas de running para mujer adidas outdoor Kanadia 8.1 W Acero sin pulir / Blanco apagado / Rosa real Vibram Spyridon para mujer Mr Trail Running Shoe Púrpura / negro , Zapatillas de running Mizuno Wave Wave Evo Ferus para mujer Azul , Zapatillas de running NIKE Zoom All Out Low para hombre Negro / Aura-oscuro Gris-puro Platino , Zapatillas de running NIKE Zoom All Out Low para hombre Gris oscuro / gris lobo / puro platino Zapatillas sin cordones Flex Merit para mujer Merrell Armada , Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer No teñido, blanco, blanco de tiza Zapatillas de trail running Salomon Speedcross 4 para hombre Blue Yonder / Black / Lava Orange , Zapatillas de running New Balance para mujer 590v3 Reflection / Vintage Indio , Salomon Women s Speedcross 4 W Trail Runner Poseidón , New Balance WT510RS3 Trail Running Shoes para mujer Negro Zapato de running ASICS GEL-Venture 5 para mujer Indigo Blue / Pink Glow / Living Coral Zapatillas de trail running Salomon Women s Sense Mantra 3 W Lotus rosa / blanco / verde abuelita MT-onemix Lovers Calzado deportivo ligero de malla Running Jogging Sneakers Blanco negro Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral Zapatillas de trail running Bere Access Arc 3 para mujer Merrell Aguanieve / Scuba Blue Zapatillas de trail running Salomon Speedcross 4 para hombre Negro Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Negro Zapatillas de trail running Salomon Speedcross 4 para hombre Nube oscura , Skechers Women s Go Trail 2 - Zapatillas de senderismo con agarre Negro Salomon Women s Wings Pro 2 W-W Trail Runner Lucite Green / Bubble Blue / Gecko Green
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 de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro , KEEN Hombres Newport H2 Sandal Brindle / Sunset Zapatillas de running Spira Scorpius II para mujer Fushia / Púrpura / Blanco Zapatillas de running ASICS Gel-Cumulus 19 para hombre Primer rojo / negro / plata , Calzado de running New Balance para mujer 530v2 de New Balance Ártico Fox / amanecer blanqueado Sandalia Terrell Ivy Wrap Sport Merrell para mujer Negro , Amortiguación 635v2 New Balance para mujer Baya , Sandalias Swiftwater Mesh de Crocs para mujer Violeta vibrante , Zapatillas de running NIKE Flex Run 2015 para mujer Prsn Violet / Blk / Fchs Glw / White , Chaco Mujer ZX3 Classic Sport Sandalia Duna de arena azul , Sandalias Newport H2 de KEEN para mujer zen Zapato de agua Tetrex Rapid Crest para mujer Merrell Lima Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Negro / antracita / blanco Zapatillas de running ASICS Gel-Noosa Tri 9 para mujer Blanco / Amarillo intermitente / Ciruela Skechers Women s Go Walk Joy-15615 Sneaker Multi Zapatilla de deporte ECM Biom Fjuel Tie Sneaker para caminar Dark Shadow / Dark Shadow Zapatillas de deporte de hombre Gel-Lyte MT de ASICS Negro Negro , NIKE Entrenador flexible para mujeres 6 Blanco negro , Vans Unisex Old Skool Zapatillas clásicas de skate Uva / negro , Zapatillas de golf Minimus Sport para mujer New Balance Rosa / gris Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Mezclilla Sandalia Taryn Rose de mujer Iris Knit Slide Cerámico , Sandalia mujer Jessica Simpson, diadema Crizma Lux Magenta 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 1620- Satén blanco brillo , mejor sentido Unisex Lana clásica Zapatilla para Adultos - Toda la lana natural - Ultra cómodo - Muchos tamaños y Colores Medoc-naranja rojo Botas Newton Worldander de Dr. Martens para mujer Negro, Malva Rosa Prop & eacute; t Blizzard de mujer Zip de invierno con cremallera Marrón / nylon , Botín de la linterna de las mujeres de Seychelles Cuero plateado Mocasín de gamuza Kilty de Minnetonka para mujer Dusty Brown ,

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 running de trail running B01MU0UL9R WT10v1 Minimus trail para mujer New Balance Gris azul 1f8c3f5 - 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 running de trail running B01MU0UL9R WT10v1 Minimus trail para mujer New Balance Gris azul 1f8c3f5 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 791314
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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.