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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Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
MONO-SOCK Fit System: An interior elástico manga que reemplaza una lengua tradicional proporcionando un calcetín ajuste.
sin costura Construcción: El uso de inconsútil materiales reduce el potencial para irritación y fricción causada por tradicional puntadas y costuras.
Retropié GEL acolchado Sistema: Atenúa choque durante el impacto fase y permite para una transición sin problemas a midstance.
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

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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.


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

Zapato Ryka INFLUENCE2.5 Cross-Trainer para mujer Negro / púrpura Skechers Sport Skech Air 2.0 City Love Fashion Sneaker Azul marino / rosa , Entrenador cruzado Grafik Flow Ryka para mujeres Frost Grey / Bright Chartreuse / Soft Blue , Nike Tech Xtreme Cadet Glove Negro / Negro / Gris Fresco / Volt , Zapato Ryka Women s Vestige RZX Cross-Trainer Gris / rosa , Empuje cargado de Under Armour para mujer Bayou Blue (953) / Blue Infinity , Zapatillas de entrenamiento cruzado PUMA Women s Pulse XT V2 FT WNS Cantera / Asfalto Adidas Mujeres Aleki X Cross Tainer Black / Bliss Coral / Intense Blue Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Cantera / Puma Blanco / Poliuretano Zapato Cross-Trainer X Conviction Women de ASICS Aqua Splash / Diva Blue / White Entrenador cruzado 20v7 de New Balance para mujeres Vivid Coral / Fiji , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Black-paradise Pink PUMA Mujer Ignite Flash Evoknit WN Sneaker Island Paradise-whisper White-puma White , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Pure Platinum / Night Purple / Wolf Gray , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Lava Glow / Blanco / Universidad Rojo / Negro Inov-8 Mujer F-Lite 195 V2 (W) Cross Trainer Gris oscuro , Entrenador cruzado Minimus 20v7 de New Balance para mujeres Negro , New Balance Women s Wx715cw3 Carbón , New Balance 611v1 Cush + Cross Trainer para mujeres Negro NIKE Entrenador flexible para mujeres 6 Blanco / Brillante Crimson / Pure Platinum Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Blanco / Plata Metálica / Lobo Gris / Sigilo , Under Armour UA Team Ultimate Speed Rojo blanco , PUMA Women s Fierce Solstice Wn Sneaker Aquifer-whisper Blanco , Plataforma de satén mono de ante PUMA para mujer Suede Light Blue 1666 Skechers Sport Women s Burst City Scene Zapatillas de moda Natural , Adidas Adipower Weightlift Cross Trainer Trace Cargo / Trace Cargo Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Gris oscuro Zapatillas de entrenamiento NIKE Women s Air Bella Azul marino / Blanco-Wolf gris New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Fiji / 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 Drew Womens Cascade Nubuck azul marino , Drew Womens Cascade Nubuck negro Zapatillas para correr Nike Women s Flex Experience Rn 5 Pink Blast / Negro / Blanco Zapatillas de running adidas Alphabounce HPC W para mujer Negro / Negro / Onix , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Láser Naranja / Gamma Azul / Cool Grey / Black Dr. Comfort Victory Profundidad extra terapéutica para mujer Athletic Shoe leather / mesh cord-up Rosado , Zapatillas Isla de las mujeres Softinos Amarillo , BareTraps Kaylyn Sandalia para mujer Rosa SOREL Womens Ella Lace-Up Negro Zapatos de running Flyknit de movimiento libre para mujer de Nike, negro / blanco - 7 B (M) US - Romika Tahiti 03 Sandalia de vestir para mujer Oceano , Sandalias Harper Huarache de SoftWalk para mujer Beige / Tan Tecnología Vionic Men s Orthaheel White / Navy Walker Blanco negro Sandalia Tiffany de BareTraps para mujer Castaño , Sandalia Bella Vita IMO-Italia para mujer Cuero negro , Calzado para caminar de mujer Mephisto Cross Patente blanca suave / verde Me Too Women s Lightning 2 Pump Cuero Multi Floral Blanco , Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Ciruela / Plateado / Rojo Rouge Zapatilla de deporte KAANAS Santa Fe Fashion Skate Slip-on Casual Sneaker Arena , Zapatillas de golf Skechers Women s Go Golf Birdie Gris / Azul de punto , Sam Edelman Sandalia Bernice Slide para mujer Cuero plateado metalizado / pulido Mini Croco de plata suave , Sandalia Jamaika de Finn Comfort para mujer Mármol Sandalias Skechers Bumblers-Bright Start Slide Carbón , Sam Edelman Women s Bay Slide Sandal Tangelo Vaquero Saddle Leather , Zapatos de bicicleta Louis Garneau Carbon LS-100 2 para mujer Blanco / amarillo brillante Toms Womens Classics Black / Black Canvas 10002472 Negro , CLARKS Cheney Madi Slip-on Loafer para mujer Cuero Tan oscuro Easy Spirit Women s Traveltime Mule Blanco / Verde oscuro / Cuero Multi TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Navy Slub Chambray , Zapato Lites Bates para mujer Blanco ,

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 ASICS/ Gel-Fit B0140WYVF8 Sana 3 Cross Trainer para mujer Trainer Invierno Bloom/ Plata/ Violeta 06cc229 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.

Fitness y entrenamiento cruzado
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 360759
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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.