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 Classic s Sandal Z1 Chaco Classic Athletic Sandal Negro 51c6634




Poliéster
Importado
Goma suela
poliéster tejido jacquard envoltura superior alrededor el pie y a través la entresuela para un ajuste personalizado
Ajustable y durable de alta resistencia correas del talón de la correa
Mujeres LUVSEAT específico Entresuela de PU
Antimicrobiano solicitud de control de olores
No marcado Caucho ChacoGrip suela con 3 mm zapato profundo
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 Classic s Sandal Z1 Chaco Classic Athletic Sandal Negro 51c6634 - 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

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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 Classic s Sandal Z1 Chaco Classic Athletic Sandal Negro 51c6634

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

Mordenmiss - Sandalias Huarache de cuero, para mujer, de verano, zapatos planos de tacón Estilo 2-verde , Northside Womens Savannah Sport Closed Toe Sandal Negro / gris , Sandalia Omnium de mujer Teva Pizarra , CLARKS Sandalia Saylie Jade para mujer Armada Sandalia Drew Angela para mujer Dusty Multi Metallic Leather , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 rojo OluKai Kulapa Kai Contton Candy / Black , OluKai Ohana Burbujeante / negro , Sandalia Drew Angela para mujer Dusty Multi Metallic Leather Chaco Mujer ZX3 Classic Sport Sandalia Duna de arena azul Zapato de senderismo KEEN Mujer Arroyo II Chocolate Chip / Sap Green Sandalia deportiva Tirra para mujer Teva Simplemente Taupe , Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva Chaco Mujer Z / 1 Yampa Sandalia Negro , Zapatilla de deporte Sina 11 de Josef Seibel para mujer Platino , CLARKS Womens Wave Grip Smokey Brown , Sandalias de cuña Hailey 33 de Josef Seibel para mujer Camello Chaco Women s ZX2 Classic Athletic Sandal Camper Purple BZees Sandalia Deportiva Jive para Mujer Gris , Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Chaco Women s ZX2 Classic Athletic Sandal Motif Eclipse Wolky Comfort Jewel Gray Picasso Crash , Chaco Mujer Diana Sport Sandalia Avo florentino , Calzado casual Slip-On de Bernie Mev para mujer Gris mobils by Mephisto - Sandalias de tacón Getha para mujer Black Queen / Magic Leather Sandalia Belmar Athletic de Ryka para mujer Blanco azul Chaco Mujer Diana Sport Sandalia Biselado negro KEEN Sandalias Newport Eco-W para mujer Victoria / Dress Blues Sandalia para mujer Teva Terra-Float Travel Knit 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 Zapatillas de running PUMA para mujer Ignite Omphalodes / Coral caliente / Ultramar , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta Skechers Performance Women s The Go 400 Radiance Flip Flop Negro , Zapatillas Drew Shoe para mujer Paige Ternero blanco Sandalia Archie Fisherman del Dr. Scholl Monumento / Sugar Rose , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Acero , Zapatillas de senderismo Air Mesh de Ahnu para mujer con malla de aire Aura astral , Earth Origins Womens Gaven Gris , Sandalia P-Oudry de Yellow Box para mujer Gris pardo , NIKE W Roshe One Womens 844994-601 Partícula rosa / vela negra Calzado SCARPA Women s Crux Walking Camel / Poppy Red , Atletismo femenino de Apex Blanco / Vinca Zapato de malla Nohea de OLUKAI - Hombre marrón Zapatillas de entrenamiento CUSH + WX577V4 New Balance para mujer Negro / Heather , Entrenador cruzado NIKE Flex 7 femenino Negro / Plata Metálica - Antracita - Blanco Under Armour Women s Charged 24/7 Low Gris Nublado (941) / Blanco , K-Swiss St329 CMF Sneaker para mujer Angel Wing / Blanco , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Seafoam / Blanco , Supra Vaider Skate Shoe Negro / negro blanco Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Gris , Zapatillas de deporte Xelero Genesis Comfort Comfort Therapeutic Extra Deep Sneaker Leather / Mesh Lace-up Blanco azul , Sandalia de vestir Lauren de Jack Rogers para mujer Oro rosa Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Oro marino Zapatos muy finos EKSA1154 de la danza de salón de señoras de las señoras con el talón de 3 \ Satén negro , Liana de mujeres de Arcopedico Ante negro Haflinger mujeres Ar Beethoven Slip on Slipper Gris plateado Wolky Comfort Sneakers Ewood Blanco negro , Bota de mujer FLY London Mong Sludge / Olive Oil Suede / Rug Womens Fly London Yama plataforma cuña talón botín aceite gamuza medio talón Expresso , Rockport Cobb Hill Women s Paulette Flat Gris ,

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.

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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 Classic s Sandal Z1 Chaco Classic Athletic Sandal Negro 51c6634 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

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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.