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.

La sandalia la de la diapositiva de la del las plataforma de la correa cruzada de Unda Fix de las mujeres del arreglo Rubor de pétalos 623cba8




Gamuza
Importada
Goma suela
Medidas de talón aproximadamente 1
Medidas de la plataforma aproximadamente 0.5 pulgadas
Un contraste sierra de diente de sierra presta ventaja a esto diapositiva de gamuza rosa sandalia con entrecruzamiento correas.
Contorneado plantilla
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


La sandalia la de la diapositiva de la del las plataforma de la correa cruzada de Unda Fix de las mujeres del arreglo Rubor de pétalos 623cba8 - 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

La sandalia la de la diapositiva de la del las plataforma de la correa cruzada de Unda Fix de las mujeres del arreglo Rubor de pétalos 623cba8

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.


La sandalia la de la diapositiva de la del las plataforma de la correa cruzada de Unda Fix de las mujeres del arreglo Rubor de pétalos 623cba8

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

Vince Camuto Sandalias Ellanna Slide para mujer Hot Berry Pink Plataforma de campo Leisa para mujeres de CLARKS Cuero de arena , Sandalias Ava Sandalias Sandalias de Jerusalén Miel Sandalia de cuña Serra de mujer Athena Alexander rojo , Sandalia Donald J Pliner Gess Slide para mujer Almendra , Sandalia de la diapositiva de la Pascua de las mujeres de Sigerson Morrison Blush desnuda , Sandalia Pour La Victoire Mujer Mallory Slide Tropic Trip Sandalias clásicas para mujer del Dr. Scholl s Shoes Tela de mezclilla Sandalia Raelyn Wedge Slide de Easy Street para mujer Tan Pebble Sandalias planas Cammie de Jessica Simpson para mujer Desnudo Blush Sandalia Taryn Rose de mujer Iris Knit Slide Sonrojo Birkenstock Arizona Sandalia de cuero con plantilla suave Plata Birko-flor , Koolaburra de UGG Women s W raychel Slide Sandal Sonrojo Sandalia para mujer J.Renee Maribeth Slide Gris oscuro Sandalias planas de plataforma tobillera con plataforma Lavern de Gentle Souls Negro , Sandalia Mason para mujer Naturalizer Slide Cuero metalizado lila , Vince Camuto Women s Haydan Slide Sandal Karat Gold La Plume Womens Jessica Broncearse , Sandalia Shanny Slide de Jessica Simpson para mujer Rojo / Multi Sandalias Jerusalem Sandalia Nuri Slide Broncearse , Sandalia de cuña Balen de Charles by Charles David para mujer Desnudo Sandalia de la diapositiva de junio de las mujeres de Propet Plata , Zaxi Sandalia con cierre a presión para mujer Zaxy Plata Sandalia Dolce Vita con diadema para mujer Cuero Ivory Sandalias Skechers Bumblers-Bright Start Slide Negro , Sandalia Helina Slide de mujer kensie Raya azul , Sandalia tobillo de piscina con correa individual Iona de Gentle Souls Oro rosa Sandalia Dolce Vita de mujer Kaira Slide Cuero dorado Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Tela natural / negra ,
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 Saucony Freedom Iso Unisex Oro blanco Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Glicina / Rosa Violeta NIKE Men s Flex 2017 RN Sequoia / Black-medium Olive-dust Short NIKE Tempo para mujer Noble Rojo / Brillante Carmesí / Blanco , Zapatillas de running adidas Performance Adizero Prime LTD Negro / Negro / Blanco , Zapatillas de running NIKE Juvenate para mujer Negro / Blanco / Negro / Blanco , Zapatos descalzos CVT-Hemp de mujer Vibram FiveFingers y paquete Premium Toesock Negro , Sandalia Drew Angela para mujer Beige-multi , Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia gris , Zapatillas de deporte Onitsuka Tiger Mexico 66 Azul oscuro / gris vaporoso Zapatillas de running Altra Lone Peak 3 Mid Neo - Mujer Gris / morado Zapatillas de running Pure Cloudfoam de adidas para mujer Negro / Negro / Blanco , Sandalia Avenoir de mujer amarilla Negro OLUKAI Honoka a Sandals - Mujer Tapa / Sahara , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Armería Marina / Menta Espuma glaciar Gris-negro , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Asfalto , Sandalia de cuña Karmen de mujer Alegria Diamantes para siempre Easy Spirit Solana Oxford para mujeres Gris oscuro Nubuck múltiple , Avec Les Filles Mujeres Bella Mule Negro / Multi Furgonetas CLASSIC SLIP EN ZAPATOS DE PATINAJE Negro (lienzo) Sandalia de diapositivas para mujer LifeStride Remix Lux Navy Sandalia Bobbi Slide de Dolce Vita para mujer Gamuza naranja Sansha Zapato de grifo en T para mujer Negro , Ovation Rhona Ladies Country Boot marrón TOMS Clásicos de mujer rojo Calcetines de tacón alto para mujer, botines de media pierna con cordones, botines de tacón medio, botines de tacón de ante elástico para mujeres Red Suede-6.5cm , Bota corta con botones Jackie de mujer FRYE Borgoña-76593 , Mocasines de vestir de punta estrecha para mujer de Me Too Women Negro TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Blue Slub Chambray Arcopedico Womens Lolita Slip-On Verde claro

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.

La sandalia la de la diapositiva de la del las plataforma de la correa cruzada de Unda Fix de las mujeres del arreglo Rubor de pétalos 623cba8 - 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 La sandalia la de la diapositiva de la del las plataforma de la correa cruzada de Unda Fix de las mujeres del arreglo Rubor de pétalos 623cba8 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 2645769
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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.