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.

Wolverine 1883 de Women s Rosie pulgadas Bota borgoña occidental occidental con cordones Kiltie de 6 pulgadas borgoña 0964ef5




100% Cuero
Importado
Goma único
Eje mide aproximadamente 7.5 desde el arco
Plataforma mide aproximadamente 0.5 pulgadas
Finely elaborado por expertos artesanos ^ Pebbled suave parte superior de cuero ^ Extraíble Kiltie para añadido opciones de diseño ^ Caucho suela
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


Wolverine 1883 de Women s Rosie pulgadas Bota borgoña occidental occidental con cordones Kiltie de 6 pulgadas borgoña 0964ef5 - 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

Wolverine 1883 de Women s Rosie pulgadas Bota borgoña occidental occidental con cordones Kiltie de 6 pulgadas borgoña 0964ef5

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.


Wolverine 1883 de Women s Rosie pulgadas Bota borgoña occidental occidental con cordones Kiltie de 6 pulgadas borgoña 0964ef5

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

Botas New Rock para hombre Oxido M 391 Punk Goth Negro , Converse Chuck Taylor All Star Core Ox Blanco óptico Calzado para caminar CLARKS Women s Wave Andes Nubuck negro , Sneaker para mujer Earth Serval Gris pardo , Marc Fisher LTD Womens Maevel Cuero Multi de Oro Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Azul , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue marrón , Zapatillas de deporte Easy Spirit AP1 Sport marrón , Zapatilla de deporte Fara Tie de mujer ECCO Luna Zapatillas de deporte Converse Chuck Taylor All Star & lsquo; 70s Pergamino Sperry Top-Sider Coil Ivy Scale Emboss barco zapato Armada , Zapato para caminar con láser Mephisto para mujer Gris Claro Bucksoft / Sand Boa / Níquel Perl Becerro , Cole Haan Women s 2. Zerogrand Stitchlite Ox Oxford Blanco de punto óptico / Cole Haan Argento metálico / blanco óptico , Colección de artistas Franco Sarto Autumn Women s Oxford Cuero de la caja negra , Socofy Oxford Leather Zapatos, hechos a mano de las mujeres Empalme de impresión Gancho del patrón de planta Loop Flat Vintage Zapatos Burdeos , Ziera Alley Women s Lace Up Negro / Stingray , Zapatillas de moda Native Unisex Apollo Moc. Leche rosa / gris Dublín / arcilla rosa / caucho de Dublín , Zapatillas para mujer Del Rey TOMS (Explosión de luz azul marino) AdeeSu Womens Flecos Hebillas Metálicas Tacones Cuadrados Botas de Ante SXC02534 marrón Aravon Betty-AR Oxford para mujeres Piedra , Easy Spirit Women s Motion Lace up Oxford Armada Nine West Mujeres Gradskool Nubuck Oxford Flat Off Nubuck blanco 2 Lips Demasiado mujeres Riddle Oxford Negro , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón GrandEvOlution Grandiewing Oxford de Cole Haan para mujeres Vapor Grey , FRYE Mujer Terri Perf Oxford Flat Negro ED Ellen DeGeneres Womens Stella Open Toe sandalias de cuff del tobillo Tela de Bisque , Finn Comfort Women s Ikebukuro Oxford Aceituna , Sam Edelman Mujer Levine 2 Sneaker Pearl Pink Satin
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 adv W de adidas Originals para mujer Blanco / Blanco / Gris Uno , Propet Women s TravelActiv Ss Sandal Negro rojo , Zapatos OLUKAI Pehuea - Mujer Dark Plum / Dark Plum , Flip Flop de Hari Mari Fields - Mujeres Azul marino / melocotón / gris , Brooks Women s Adrenaline GTS 17 Antracita / Festival Fuchsia / Bittersweet , Bota de senderismo Gore-Tex de media caña alta para mujer Merrell Moab Beluga , Comfortiva Mujer, Faye Sandalia de tacón medio Fumar , LifeStride Womens Corinne Negro Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Azul marino / ondulado / estiramiento Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Negro , Sandalia Gora Yoga Sanuk para mujer Olivo oscuro CLARKS Womens Un Plaza Sling Nubuck negro , Zapatillas de senderismo Pro Pro 3D Pro de mujer Ryka Carbón de leña / negro , Vans Old Skool Unisex Adults Low-Top Trainers (Gamuza) Negro / Blanco Zapatillas de golf Skechers Women s Go Golf Birdie Natural / Coral Vans Unisex Old Skool Zapatillas clásicas de skate Flujo mono / hielo , Sandalia Dolce Vita Delana Slide para mujer Cuero negro Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse 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 6030- Satén negro brillo , CLARKS Mujer McKella Morris Sneaker Nubuck gris sintético Botines Emelia Slouch de mujer MIA Moca Trotters para mujer Dea Slingback Pump Armada , Bella Vita Womens Tex-Italia Tan Suede Bota Pleaser Women s Delight-5000 / B / PU Negro Sandalia Florence Slide de Vionic Women s Rest menta , Bota de chukker M-Velcra Madden para mujer Coñac , Mephisto Sabatina Tuxedo Oxford para mujer Pewter Velcalf Premium ILSE JACOBSEN Women s Tulip 139 Flat Plata TOMS Clásicos de mujer (Armada) Zapatillas Slip-On Roamer SAS para Mujer Santolina ,

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.

Wolverine 1883 de Women s Rosie pulgadas Bota borgoña occidental occidental con cordones Kiltie de 6 pulgadas borgoña 0964ef5 - 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 Wolverine 1883 de Women s Rosie pulgadas Bota borgoña occidental occidental con cordones Kiltie de 6 pulgadas borgoña 0964ef5 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 407897
3683781
3819059
6704032
7419676
8802000
13336727
13601854
18651020
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.