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.

Kensie Sandalia Sandalia para Negro mujer Kensie Digby Slide Negro a7a1a1c




Sintético
Importado
Suela sintética
plantilla acolchada
slip on
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


Kensie Sandalia Sandalia para Negro mujer Kensie Digby Slide Negro a7a1a1c - 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

Kensie Sandalia Sandalia para Negro mujer Kensie Digby Slide Negro a7a1a1c

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.


Kensie Sandalia Sandalia para Negro mujer Kensie Digby Slide Negro a7a1a1c

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

Birkenstock Granada Sandalia Moca , Sandalia Birkenstock de cuero Florida para mujer Negro Carlos by Carlos Santana Sandalia mujer Slide Liliana Negro Sandalia Cleo Birkenstock para mujer Cuero rosa metalizado Sandalia Vince Westcoast para mujer Vince Caucho azul Sandalia Aerosoles Midday Women Slide Tela de mezclilla Birkenstock Mujer, Madrid Sandalia deslizante Plata Sandalia Mason para mujer Naturalizer Slide Blanco Sandalia Heather Slide de mujer Vía Spiga Ladrillo de gamuza Sandalia Merrell Terrano Ari Slide para mujer Armada , Sandalia de la diapositiva de junio de las mujeres de Propet Mezclilla Sandalias Sandalias Bashan moldeadas Sandalias Sandalias Jerusalem rojo , Stuart Weitzman Landslid femenino Jeans ante Sandalias con tachuelas Gale para mujer Donald J Pliner Armada , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Cuero negro , Vince Camuto Sandalias Ellanna Slide para mujer Hot Berry Pink , Sandalia Victoria Slide de mujer SCHUTZ Perla , Kensie Sandalia para mujer Digby Slide Negro , Birkenstock Arizona Sandalia de ante con suela suave Cuero Habana Aceitado , Sandalia de diapositivas Henlee de Dolce Vita para mujer Tela roja , Sandalia de diapositivas de ocio para mujer de Seychelles rojo Sandalias Fifi19 Slide de Donald J Pliner para mujer Oro ALMA NATURAL Sandalia Adalia Slide para mujer Puesta de sol Sandalia del vestido de la inversión de LifeStride de las mujeres Lux Navy , Sandalia Dolce Vita Celaya Slide para mujer Jade Fringe , Sam Edelman Women s Slide Sandalia Cuero de gamuza amarillo , Sandalia Dolce Vita Halle Slide para mujer Tela natural / negra , Sandalias FRYE Carla Criss Cross Open Toe Casual Slide Negro , Sandalia de diapositivas de ocio para mujer de Seychelles 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 Zapatillas para correr Nike Flex Experience RN 5 Gris / antracita / verde fantasma / verde resplandor CLARKS Sandalia plana Brizo Ravena para mujer Coral Sandalia Yucatan ECCO Mujer Irritante Zapatillas de running New Balance Coast V3 para mujer Blanco , Chaco Mujer Zcloud Sport Sandal League Gold , PUMA Men s Roma Basic Sneaker Blanco / Nueva Marina Zapatillas de trail running WT910V3 para mujer New Balance Azul negro Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / Begonia rosa , Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Rose Gold Specchio / Hemp , Sandalias de tacón de soñador Easy Street para mujer Negro Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Mora , Naturalizer Samantha Pointed Toe Flat para mujer Gris pardo Entrenador cruzado 20v7 de New Balance para mujeres Vivid Coral / Fiji Drew Shoe Women s Tulip Comfort Brown Croc Zapatillas de deporte Xelero Genesis Comfort Comfort Therapeutic Extra Deep Sneaker Leather / Mesh Lace-up Blanco azul Vans Auténtico Negro / Blanco verdadero Sam Edelman Women s Bay Slide Sandal Cuero de silla Sam Edelman Women s Bay Slide Sandal Denim Blue / New Blue BareTraps Womens Jimina Open Toe sandalias de diapositivas casuales Cepillo marrón , Zapato de voleibol Gel-Rocket 8 femenino de ASICS Blanco / plata / blanco , Zapatillas de deporte Trinity Sneaker Dance de Bloch Negro , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer S92307 2 pulgadas Talón + Plegable Paquete de cepillo Escala plateada de cuero plateado Bomba de vestido exquisita Aerosoles mujeres Bone Combo Pantuflas para mujer Cincasa Menorca Mahon Ladies / Slippers rojo UGG Womens Riley cerveza negra Trotters Lucy Ballet para mujer Armada Zapatilla deslizante Loki II para mujer BEARPAW Tela de leopardo dorado , Zapatillas de running NIKE Men s Train Speed ​​4 Negro / Blanco - Negro Drew Shoe Women s Drew Slip Resistant Leather, caucho, espuma, oxfords Negro Trotters Liz Loafer para mujer Hueso ,

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.

Kensie Sandalia Sandalia para Negro mujer Kensie Digby Slide Negro a7a1a1c - 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 Kensie Sandalia Sandalia para Negro mujer Kensie Digby Slide Negro a7a1a1c 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

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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.