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.

Northside Womens B00VHVXUQM Savannah Sport Closed/ Toe Sandal Closed Tan/ Coral f7c1f60




Cuero sintético / Nylon
Robusto parte superior flexible características soft faux gamuza con rápido secado de neopreno respaldo / transpirable inserciones de nylon tejido y resistente a la abrasión protector de dedos
Esto cordones al aire libre hasta sandalia atlética ofrece resistente al agua construcción / humedad wicking neoprene forro para todos confort diario
Ligero plantilla de espuma de EVA / acolchada respaldo garantizar amortiguación para tus pies; resistente lavable materiales ayuda resistir extendido use
Elástico cordones con cordones sistema / gancho y correa del talón del lazo mantén tus pies seguro y
duradero multi direccional suela de goma asegura confiable tracción / choque absorción sobre ambas superficies mojadas / secas
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

Northside Womens B00VHVXUQM Savannah Sport Closed/ Toe Sandal Closed Tan/ Coral f7c1f60

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.


Northside Womens B00VHVXUQM Savannah Sport Closed/ Toe Sandal Closed Tan/ Coral f7c1f60

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

Sandalias KEEN Venice II H2-W para mujer Skydiver / Blue Opal Sandalia Dusty Sport mujer muy volátil Gris Skechers Performance On The Go 400 Vivacity Flip Flop Azul marino / verde azulado , Sandalia Yucatan ECCO Mujer Irritante , Sandalia de susurro de mujer Keen Sombra Oscura / Cerámica Sandalia Yucatan ECCO Mujer Blanco / estampado de flores , BZees Sandalia con cuña de beso para mujer Blue Multi Print KEEN Hombres Newport H2 Sandal Negro / Gris piedra Sandalia Shaye Sport mujer Donald J Pliner Platino Sandalia Terrell Ivy Post Sport Merrell para mujer Negro Sandalia deportiva Tirra para mujer Teva Negro / Blanco Multi , Sandalias Bali KEEN para mujer Nórdico azul / azul ópalo Chaco Women s Z2 Classic Athletic Sandal Negro Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia gris , Chaco Women s ZX2 Classic Athletic Sandal Dolman Pine Sandalias Newport H2 de KEEN para mujer Rosa negra / brillante Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose , CLARKS Sandalia Saylie Jade para mujer Armada Sandalia KEEN Women s Uneek Slide-w Dapple Gray / Vapor Earth Origins Sophie Frost Grey , Sandalia KEEN Women s Uneek Slide-w Aqua Sea / Turquesa Pastel Sandalia atlética al aire libre del cordón Vulc N Vent de Columbia para mujer Tiburón, Cool Gray Chaco Women s Z2 Classic Athletic Sandal Glaciar negro , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth Sandalia de susurro de mujer Keen Negro / Gris neutro , Sandalia de susurro de mujer Keen Imán / Sangría Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth Sandalia KEEN Women s Uneek Slide-w Blanco negro Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad
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 de running Mizuno Wave para mujer con sombra Fiery Coral / Blanco Drew Womens Ariana Cuero de peltre , Zapatillas de deporte Onitsuka Tiger Mexico 66 Gris vaporoso / Peacoat Sanda Verra Teva para mujer Orquídea púrpura , Chaco Mujer ZX1 Classic Sport Sandalia Dart Peach Mujer adidas Originals Ultraboost St Negro / Plata Metálico / Carbono , Zapatillas de running Gobiv2 para mujer New Balance Semillas con Grove Zapatillas de running New Balance para mujer 590v3 Reflection / Vintage Indio , Zapatillas de running Reebok para mujer All Terrain Super O Berry / Violet , Wolky Comfort Jewel 12390 Beige Nubuck , Zapatillas de deporte trail running Hiero V2 para mujer New Balance Typhoon / Supercell / Alpha Pink , Sandalia casual destello de las mujeres de ECCO Negro , Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / Blanco-gris oscuro , Ros Hommerson Calypso Slip para mujer Mezclilla , En la ejecución de calzado de mujer en la nube menta Zapatillas para caminar Nell Ryka para mujer Negro / Menta Sandalia Pikolinos Antillas para Mujer Negro New Balance WW813 Hook and Loop Walking para mujer Negro , Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Azul / Microdot Zapatillas de deporte Fenty x Trainer Mid de PUMA para mujer Puma Black / Puma White , DC Youth Rebound Skate Shoes Negro / Aqua , Zapatillas de golf Lexington Marc Joseph New York para mujer Patente de la marina de guerra Calvin Klein Dalana Slide Sandalia para mujer Plata / Oro rosa / Estaño Sandalias Hush Puppies para mujer Dachshund Slide Cuero Ivory 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5008- Cuero blanco , Zapatillas Skechers Sport Synergy 2.0 High Spirits para mujer Gris oscuro , Me Too Womens zena14 Suede Almond Toe Tobillera Botas de moda Dark Ruby Suede , Mujeres muy finas Ballroom Latin Tango Salsa Dance Zapatos 2707 2.5 \ Talón y plegable Paquete de cepillo para zapatos Satén blanco PIC / PAY Hazel | Cuero desnudo Toms Womens Classics Black / Black Canvas 10002472 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 Northside Womens B00VHVXUQM Savannah Sport Closed/ Toe Sandal Closed Tan/ Coral f7c1f60 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 301700
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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.