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 running rosa Fresh running Foam 1080v6 de 16002 New Balance para mujer Plata/ rosa b308661




75% sintético, 25% Malla
Importado
Caucho único
Fresco Entresuela de espuma
Ortholite sockline
Bootie construcción
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 running rosa Fresh running Foam 1080v6 de 16002 New Balance para mujer Plata/ rosa b308661 - 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 running rosa Fresh running Foam 1080v6 de 16002 New Balance para mujer Plata/ rosa b308661

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 running rosa Fresh running Foam 1080v6 de 16002 New Balance para mujer Plata/ rosa b308661

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 adidas Duramo Lite W para mujer Easy Mint / Ocean / White adidas outdoor para mujer Terrex Swift R2 GTX Trace Scarlet / Carbon / Chalk White , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Sunset Pulse Arctic Punch Wht , Zapatillas de running Nike Kaishi para hombre Lobo gris / blanco-blanco Zapatillas de trail running Inov-8 para mujer Road Claw 275 Plata / azul / rosa , Brooks Womens Neuro 2 Rosa Glo / Negro / Blanco , Zapatillas de running Saucony para mujer Hurricane Iso 4 Gris / Negro / Vizi Rojo , Zapatillas de running Bondi LTR HOKA ONE ONE para mujer Vientos alisios , Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea , Zapatillas de running adidas Women s Edge Rc W Clear Aqua / Silver Metallic / Shock Purple , NIKE Women s Tanjun Blanco Blanco Negro , Zapatillas de running New Balance 1340v3 para mujer Gris oscuro Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro , Zapatillas de running NIKE Zooms Winflo 4 para mujer Blanco / Negro / Aurora Verde Brooks Womens Adrenaline GTS 18 Púrpura / negro / champán , Zapatillas de running NIKE Airstream Sequent 3 para mujer Blanco / puro platino Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro Calzado de running NIKE Kids Roshe One (GS) Turquesa , Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento , Zapatillas de running W1500v3 New Balance para mujer Lime Glow / Vivid Jade / Dark Denim Zapatillas de running adidas Ultraboost W para mujer Negro / Negro / Negro Amortiguación 635v2 New Balance para mujer Plata Zapatillas de running NIKE para mujer Lunarepic Flyknit Shield Plata negra / metálica Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Platino Blanco / Blanco-puro Zapatillas de running Saucony para mujer Stretch N Go Breeze Negro Zapatillas de running NIKE Airstream Sequent 3 para mujer Blue Fury / White-glacier Blue , Zapato de running ASICS GT-2000 4 para mujer Carbón de leña / Flash Coral / Flash Yellow Zapatillas para correr Nike Downshifter 6 para mujer Negro / imán gris / blanco , Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Azul / naranja / blanco
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 KEEN Venice II H2-W para mujer Negro / gris acero , Chaco Women s ZX2 Classic Athletic Sandal Bloom Peach , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Baya del Saúco / Coral Vivo , Zapatillas Reebok Women s Runner 2.0 MT Track Negro / carbón / ácido rosa / blanco Zapatos lindos del rastro de las calcomanías del Fox de la historieta para las zapatillas de gimnasia de Nmd de las mujeres - Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Blanco negro , inov-8 Roclite 290 Trail Runner para mujeres Rosa / Negro / Blanco , Una mezcla de zapatillas de deporte ONEMIX Performance Air Cushion para hombre Dorado , EQUICK Water Shoes Aqua Sports Sneakers Slip on Quick Dry para Hombres Mujeres Niños Pesca 1y.black Zapatillas de golf Skechers Performance para mujer Go Golf Elite Canvas Blanco / rosa Chaco Mujer Zcloud X Sandalia Deportiva Patch Blues Sandalia de la paz de Easy Spirit Women Natural Zapatillas de volante NIKE Air Zoom Hyperace para mujer Wolf Gray / Volt / Black Zapatillas de running NIKE Air Max 1 Ultra Flyknit para mujer Foto Azul / Azul Real Profundo CLARKS Sandalia Saylie Jade para mujer Armada Hush Puppies Mujeres Power Walker Ii Loafer Cuero blanco Rieker para mujer D1916 Malea 16 Weiss , Zapatillas ALC-Diva Cross-Trainer para mujer Avia Gris hierro / Niebla fría gris / Plumeria Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Apenas Verde / Negro , Clásicos de base Slip-on (Tm) para hombres de Vans Baby Blue / Verdadero Tablero blanco , Sandalia Kai Dolce Vita para mujer Cuero negro , Zapato de baile con esfera de gel y cohete para mujer ASICS 7 Plata blanca , Zapatillas de salón Xavier para hombres de Bloch Dance Negro , Zapatos muy finos EK1692 de la danza de salón de baile de las señoras de las señoras con 2.5 \ Cuero negro y malla negra Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos melocotón FRYE Arnés con cinturón de mujer 12r Negro , Trotters para mujer Dea Slingback Pump Armada , Soft Style Hush Puppies Mujeres Valda Oxford Serpiente de peltre , Zapato de punta de ala 3989 de Dr. Martens para mujer Rojo cereza , Dr. Martens Women s Vegan 1461 Metallic Oxford Serpiente metálica plateada

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 running rosa Fresh running Foam 1080v6 de 16002 New Balance para mujer Plata/ rosa b308661 - 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 running rosa Fresh running Foam 1080v6 de 16002 New Balance para mujer Plata/ rosa b308661 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3869533
4396291
9279966
13143437
15097223
16389090
17412125
17592400
18801837
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.