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.

Bomba Sadie Sadie Cole Wedge 40MM de Cole Haan para mujer 19993 Fig Gamuza 4118fcd




Hecho por el hombre
Importado
Caucho único
Disponible en una variedad de parte superior.
Slip-on diseño.
Hecho por el hombre forro.
Ligeramente plantilla acolchada.
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


Bomba Sadie Sadie Cole Wedge 40MM de Cole Haan para mujer 19993 Fig Gamuza 4118fcd - 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

Bomba Sadie Sadie Cole Wedge 40MM de Cole Haan para mujer 19993 Fig Gamuza 4118fcd

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.


Bomba Sadie Sadie Cole Wedge 40MM de Cole Haan para mujer 19993 Fig Gamuza 4118fcd

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

Giesswein Unisex Acadia Antracita 206 Cojín de deslizamiento Roosevelt Shearling de mujer Collective Ante negro Bomba Parigi de LifeStride para mujer rojo Zapatilla Dakota de las mujeres de Australia UGG Rosa magenta , TOMS Clásicos de mujer Armada Bomba de cuña Farrow para mujeres LifeStride Negro 2 , Bomba de vestido de pasión Easy Street para mujer Patente rojo brillante Zapatillas UGG Womens Florencia Negro Zapatilla Abela de UGG para mujer Niebla de Lavanda Trotters para mujer Dea Slingback Pump Blanco / negro Soft Kid / patente , Zapatillas EMU Platinum Albany para mujer Chocolate Zapatillas FitFlop para mujer Audrey Python Print Smoking M97 Blanco Urbano Zapatillas OLUKAI Upena - Mujer Café Konda Trotters para mujer Dea Slingback Pump Armada , Zapatilla Klogger Klogger de Mujeres Tempur-Pedic Hashbrown Bomba de vestido de pasión Easy Street para mujer Patente negra , Foamtreads Barry Masculina Negro Ara Zapatillas Aster para mujer Black Lamfell , Naturalizer Michelle s Dress Dress Pump Kettle Red Leather Haflinger Kris Chocolate , Botas Karel para mujer de UGG Violeta gris Bomba de cuña Farrow para mujeres LifeStride Broncearse Mocasín de tela de hombre viejo amigo castaña , MUK LUKS Suede Jane Moccasin Slipper Negro , Bomba de Angel II para mujeres estilo suave Elegancia de hueso , Bota Deia para mujer de TOMS Black Suede / Glimmer , Bomba Sadie Wedge 40MM de Cole Haan para mujer Ante de piedra de hierro Bomba de vestido de pasión Easy Street para mujer Patente rojo brillante , Furfurmouton Genuine Piel de oveja australiana Pantuflas botines Super Grueso para Mujeres Piel de oveja premium Zapatilla Chocolate ,
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 Mujer adidas Cf Qt Racer W Gris Uno / Blanco / Gris Dos Converse Chuck Taylor All Star Seasonal Color Hola Icy Pink , Zapatillas de running Brooks Women s Glycerin 16 Road Negro / Rosa / Gris , Zapatillas de viaje Proather para mujer II Malla de plata , Sam Edelman Women s Gigi Rose Gold Metallic Boa Snake Print Zapatillas de running ASICS GT-2000 6 para mujer Brillante rosa / negro / blanco Zapatillas de running adidas Supernova Sequence 9 W para mujer Súper Púrpura / Metálico / Plateado / Gris Medio , Drew Shoe Athena para mujeres Blanco azul , Sandalia Terra FI Lite de Teva para mujer Luces de la ciudad negro pastel , Calzado de entrenamiento Under Armour UA Strive 6 para mujer Acero / Blanco / Rosa Chroma Zapatillas de tenis NIKE Women s Air Zoom Prestige Hardcourt - Blanco / Hot Lava / Pure Platinum / Blue Nebula , New Balance 611v1 Cush + Cross Trainer para mujeres Negro , DC Womens Rebound Hightop zapatos de skateboard perforados Oro / negro Vans Unisex Old Skool Zapatillas clásicas de skate Negro / peltre Supra Vaider Skate Shoe Blanco Blanco , Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro , Zapatillas de skate auténticas Vans Kids Vestido (sobrante) Blues / Grafito azul Vans VN0003B9IC6 Authentic Unisex Low Sneakers (Chckrbrd) clscwt / Ciudadela , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Burdeos profundo , Converse Chuck Taylor All Star Core Hola Mono negro , Zapatillas de golf Skechers Women s Go Golf Birdie Punto negro / blanco , Sandalia de la diapositiva Indi Kate Spade New York para mujer Denim azul claro Sandalia para mujer Ivanka Trump Earin Slide Ante rojo UGG Mujer W Joni Bronceado 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) S9283 Trenza negra y dorada , Zapatilla de deslizamiento Effie de mujer BEARPAW Lino Converse Chuck Taylor All Star Core Ox Mono blanco , Botas de moda veganas Castel Indigo para mujer de Dr. Martens Índigo , Botín de tobillo Cyra Catelyn de Hush Puppies para mujer Leopardo Natural Gamuza , Slip clásico de mujer de SAS en Negro Negro

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.

Bomba Sadie Sadie Cole Wedge 40MM de Cole Haan para mujer 19993 Fig Gamuza 4118fcd - 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 Bomba Sadie Sadie Cole Wedge 40MM de Cole Haan para mujer 19993 Fig Gamuza 4118fcd 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 1782053
1837455
6566921
8255721
8924541
10039363
12908669
14387197
18452422
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.