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.

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Gamuza, cuero, o superior hecho por el hombre
Importado
Sintético suela
Medidas de talón aproximadamente 4
Talón bloqueado
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


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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 negro vestir Carrson de mujer Steve Madden 19045 Ante Madden negro b954566

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 negro vestir Carrson de mujer Steve Madden 19045 Ante Madden negro b954566

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

Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Cielo azul , Bomba Francesca de mujer Adrianna Papell Mosaico de oro Lame , J.Renee Womens Bilboa Beige / Oro Oscurecimiento de la habitación de montaña Wide Curtains European Macizo rojizo Piedra en el francés Alps Range en Hot Día de verano con Sun Art personalizado Cortinas Naranja , Mocasines con suela de piel de oveja Minnetonka para mujer - 3341 Golden Tan Zapatillas deslizantes para mujer Applaud de Merrell Negro Rieker Mirjam 11 bomba para mujer Blanc (Weiss / Bian) , Cobb Hill Rockport - Aubrey Dress Pump para mujer Amarillo Bomba de Angel II para mujeres estilo suave Oro / Plata Cósmica Mocasín Step-In para mujer de Old Friend Castaño con lana gruesa , Dualyz Fit Unisex 100% lana Zapato con suela extraíble Gris oscuro / gris claro Bomba de vestido Cindie Easy Spirit para mujer Nubuck negro Zapatillas de señora Shepherd Genuine Sheepskin Rolled Cuff Bootie - Flory castaña , JOE N JOYCE Soft Slippers Zuecos Zapatos de cuero estrecho - Para hombres y mujeres Negro , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy de arena Zapatilla gráfica clásica unisex Crocs Armada , Dualyz Fit Unisex 100% lana Zapato con suela extraíble Gris claro / salmón , Vionic Women s Tide Ii Negro Negro CLARKS Womens Quartz Sandalia Nubuck negro Curan la suela negra a cuadros tradicional de las mujeres Cory 4Cs de las mujeres Negro / chispa Bomba de vestido LifeStride Sevyn para mujer Piedra , Mocasines de piel de caribú de mujer Laurentian Chief , Giesswein Unisex Brixlegg Gris pardo , Zapatilla de vals femenina Foamtreads Terciopelo lila , TOMS Clásicos de mujer (Armada) Zapatillas de casa de lana de invierno U-lite cálido cálido Shearling de las mujeres, zapatillas de lana caliente para las mujeres Gris (Gamuza de piel de oveja) , UGG Tamara Slipper para mujer Negro , Zapatillas de mujer Vionic Relax Castaño 9 M - Trotters para mujer Dea Slingback Pump Lavar florales
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 Sandalia deportiva ECCO Terra 2S para mujer Titanio Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Brgd Bl / Brgd Bl / Sqdrn Bl / Pr Pl , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de running NIKE Juvenate para mujer Lobo gris / blanco / gris lobo / blanco , Merrell Siren Edge Hiker para mujer Aqua blanqueado Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Acero , Zapato deportivo ASICS Gel-Excite 4 para mujer Aluminio / Plata / Aqua Splash , Sandalias con plataforma SAS de cuero con punta abierta para mujer Negro , Sandalia Tonya de mujer Trotters Negro Trotters Mujer Simmy Mary Jane Flat Gris claro , Zapatillas de viaje Proather para mujer II Malla de plata Easy Spirit Women s Motion Lace up Oxford Natural , Zapatillas de paseo Skechers Go Walk Lite-Luna para mujeres rojo , Zapatillas para caminar Go Walk 4 Achiever de Skechers Performance para mujer Difuso: negro / negro , Zapato Ryka INFLUENCE2.5 Cross-Trainer para mujer Negro / púrpura Zapatilla para mujer Dr. Comfort Katy Rosa y azul , Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Azul oscuro-blanco , Supra Vaider LC Sneaker Borgoña-blanco , Zapatos de golf de ocio FootJoy para mujer Imprimir-m Golf de mujeres y rsquo; s Peso ligero genuino Cuero hecho en Huerto de Brasil Street Golf Performance Marc Joseph NY Zapatos de moda Blanco granoso , Sandalia ecológica ECCO Women s Offroad Lite Slide Caliza , Sam Edelman Sandalia Bernice Slide para mujer Tangelo Sandalia de mujer Lucienne Suede Slide de Nine West Gamuza rosa , Bota de montar Bailey-xwc para mujer Brinley Co Ternero extra ancho negro , Bota de montar de felpa forrada de mujer Staheekum Annika Black Patricia Green Womens Mayfair Azul , Sandalia Birkenstock de cuero Florida para mujer Cuero aceitado negro Zapatos con cordones Daphne de Bernie Mev para mujer Jeans ligeros , Bota Chelsea Duckfeet Roskilde Cuero marrón Drew Shoe Women s London II Slip On Textil Casual Flats Armada

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.

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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 negro vestir Carrson de mujer Steve Madden 19045 Ante Madden negro b954566 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 1359154
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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.


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