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.

Sandalia de Maria Lepore Slide de Nanette Lepore Maria para mujer Rosa empolvado 1be13d4




Hecho por el hombre
Importado
Suela sintética
Bordes cortados con láser
Flor 3D embellecida
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


Sandalia de Maria Lepore Slide de Nanette Lepore Maria para mujer Rosa empolvado 1be13d4 - 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

Sandalia de Maria Lepore Slide de Nanette Lepore Maria para mujer Rosa empolvado 1be13d4

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.


Sandalia de Maria Lepore Slide de Nanette Lepore Maria para mujer Rosa empolvado 1be13d4

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

Sam Edelman Women s Bay Slide Sandal Nuevo desnudo Sandalias Jerusalem Sandalia Joanna para mujer Gris , Sandalia de diapositivas de ocio para mujer de Seychelles Blanco Sandalia Aurora Slide de Marc Jacobs para mujer Azul Sandalia Aerosoles Midday Women Slide Negro Sandalia Brenda Slide mujer MIA Floral amarillo , Vince Camuto Sandalias Ellanna Slide para mujer Urban Lux , Sandalias Jerusalem Sandalia Joanna para mujer Azul , Sam Edelman Sandalia Bernice Slide para mujer Candy Red Vaquero Saddle Leather , Sandalia plana Dionne de Easy Street para mujer Negro Sandalias Fifi19 Slide de Donald J Pliner para mujer Plata , SCHUTZ Sandalia Yaslin Slide para mujer Negro Sandalia Aurora Slide de Marc Jacobs para mujer Oro Sandalia Rockdale Lanea Woven Slide para mujer Tejido azul marino , Sandalia de vestir Lauren de Jack Rogers para mujer Pewter / Pewter Sandalia de cuero Nirveli de Nine West para mujer Cuero blanco , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Brillo de plata , Sandalia Dian J Pliner Mujer Ginie Slide Plata , Sandalia Corey de Dolce Vita para mujer Rafia Azul Multi , Sandalia plana Kip de Trotters para mujer Blanquecino Sandalia plana Dionne de Easy Street para mujer Armada , Sandalia Dolce Vita Celita para mujer Fuera de cuero blanco Sandalia de la diapositiva de las mujeres de Franco Sarto de las mujeres Rosado , Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Malt Taupe Action Leather Sandalia Ilaria Slide de mujer SCHUTZ Vestido azul J Slides Sandalia de mujer Naomi Slide Estaño Sandalia New Balance para mujer City Slide Negro , Sandalia Pomopom Slide de ALDO para mujer Fushia Varios , Sandalia Juno Slide de Vionic Women s Rest Oscuridad
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 The North Face Bridgeton Slingback Mujeres Dark Gull Gray / Simply Taupe Chaco Women s Z1 Classic Athletic Sandal Prisma de menta , Zapato de running NIKE Kids Tanjun Print (GS) Antracita / Mtlc Platino Zapatillas Mujer Salomon Sonic RA Max Negro / Navegar por Internet Negro / Navegar por la Web Zapatillas de running Kineta Relay para mujer Saucony Maru / Blanco WMNS Zoom Fly Blanco negro Zapatillas de running NIKE Air Max Torch 4 para mujer Blanco / menta , Drew Shoe Women s Tulip Comfort Brown Croc Easy Spirit Solana Oxford para mujeres Negro Nubuck multi Drew Shoe Athena para mujeres Blanco azul , Zapatillas de golf Skechers Performance para mujer Go Golf Elite Canvas Natural / azul marino , Walking Cradles Womens Michaela Patente roja , Easy Spirit Women s Traveltime Mule Stone-medium Blue-leaf Calzado de entrenamiento casual Comfort WX409V3 para mujer New Balance Negro , Zapato para caminar con cordones Go Flex Ability de Skechers Performance para mujer Gris / rosa fuerte Zapatillas de skate DC Women s Villain SE Slip-on Negro / Leopardo Skechers Women s Burst 2.0 New Avenues Sneaker Negro Sandalia Lana Slide de Dansko para mujer Negro (grano completo) , Sam Edelman Women s Bay Slide Sandal Sugar Pink , Zapato Capezio Mary Jane Tap para mujer Negro Bota occidental con flecos y tachuelas Roper para mujer Black / Crystal Stud , Zapatilla de deslizamiento de Acorn Women s Spa II Buttercream , Sandalias de vestir Patti de Sam Edelman para mujer Cuero metálico yute , Zapatillas de ciclismo X-Alp Launch II Pearl iZUMi para mujer Negro / gris oscuro , TOMS Womens Desert Wedge Bootie Toffee Suede Perforated Leaf , Bomba Sadie Wedge 40MM de Cole Haan para mujer Patente negra , Botín de tobillo de Dallas para mujeres de Naturalizer Negro , Bomba Parigi de LifeStride para mujer Crucero Marina Zapatillas Nike Hypervenom Phantom III Dynamic Fit FG de Nike para mujer Nike , Timberland CAMPO LOTE GRIS / AF 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.

Sandalia de Maria Lepore Slide de Nanette Lepore Maria para mujer Rosa empolvado 1be13d4 - 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 Sandalia de Maria Lepore Slide de Nanette Lepore Maria para mujer Rosa empolvado 1be13d4 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 847805
1139242
3255813
4950660
8791523
14779002
16093573
16543680
18603530
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.