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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Cuero
Importado
Sintético suela
Medidas de talón aproximadamente 1 1/2
Plataforma mide aproximadamente 0. pulgadas
Cuero parte superior
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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,

Sandalias Jerusalem Sandalias Deborah Slide Blanco , Sandalia de diapositivas de mujer Kimana Dolce Vita Cuero negro , Vince Camuto Sandalias Ellanna Slide para mujer Hot Berry Pink Sandalia para mujer FRYE Ally Criss Cross Slide Negro , Sandalia Bobbi Slide de Dolce Vita para mujer Orchid Suede , Sandalia de la diapositiva de junio de las mujeres de Propet Plata Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno azul , Sandalias Frankie Slide de mujer NAOT Cuero satinado dorado / Royal Blue Stretch / Bordeaux Patina Elastic Vince Camuto Sandalia para mujer Carran Slide Azul / Natural Sandalia de mujer Hana Slide de Via Spiga Cuero de porcelana , Charles by Charles David Sandalia mujer Kadia Slide Negro UGG Mujer W Joni Niebla de Lavanda , Sandalia de cuña Ollie Wedge Chinese Laundry para mujer Suave Natural , Diapositivas Sola FitFlop para mujer Oro iridiscente , J Slides Sandalia de mujer Naomi Slide Arena , Sandalia Heather Slide de mujer Vía Spiga Cuero negro Sandalia Enroelia Slide de ALDO para mujer Blanco , Birkenstock Arizona Sandalia de ante con suela suave Suela de cuero / blanco de cobre , Calzado de mujer Calvin Klein Patreece Slide Negro , Bernardo Women s Bridget Slide Sandal Guante negro Sandalia Kai Dolce Vita para mujer Cuero Caramel , Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse , Sandalia para mujer Merrell, 1SIX8 Linna AC + Slide Aura astral , Sam Edelman Women s Bay Slide Sandal Cuero metálico yute , BareTraps Women s Rebecca Slide Sandal Negro Negro , Sandalia de cuero Merrell Dwaykair Seaway para mujer Merrell Oak , Sandalia Ryana Marilyn Slide para mujer Blanco negro , Sandalia Kai Dolce Vita para mujer Cuero dorado J Slides Sandalia de mujer Naomi Slide Mezclilla
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 entrenamiento para principiantes Primus Trio Everyday de Vivobarefoot para mujer Negro , Zapatillas de running Flex Ride 630v5 para mujer New Balance Gris oscuro , Sandalia Yucatan ECCO Mujer Titanio , Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco Zapatillas de running NIKE Zoom All Out Low para hombre Obisidian / Paramount Blue-black Salomon Women s X-Mission 3 W-w Aruba Azul / Azul marino Blazer / Verde Flash , Easy Spirit - Zapatillas Faisal2 para mujer Gris Zapatillas Spindrift para mujer Xtratuf Espuma de mar Orthofeet Probado Alivio del dolor Comodidad Serene Womens Orthopaedic Artritis extra ancha Diabetic T-Strap Zapatos Negro , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Arctic Punch / White , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Negro / ondulado / estiramiento Drew Shoe Athena para mujeres Crema / Combo , Artemisa Taupe Soft Buck , Earth Origins Womens Gaven Alpaca Smu Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Rojo-c (talón cónico) Tierra Daylily Cognac Nubuck Zapatillas de deporte Skechers Sport Women s Empire Inside Look Punto negro / blanco , Zapatillas PUMA Tazon 6 Wide Wn para mujer Puma White-island Paradise-puma Silver , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Amapola Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Acuarela / Iridiscente Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Pewter / Calf Under Armour Charged All Around Speedknit Grafito (040) / Negro Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Negro / Negro-medio Olivo-blanco , Sandalia con correas cruzadas Croana adornadas Sloane para mujer Oyster / Gold , ALDO Women s Maxence Slip on Slipper Blanco , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Cielo azul , Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Azul , Sandalia Birkenstock de cuero Florida para mujer Cereza , Nine West Women s Gradskool Suede Oxford Flat Ante negro Botas clásicas de cuero LaRosa para mujer con cordones planos Zapatos Oxford con cordones planos Negro Rojo 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 Sandalia Finn Jamaika de Finn Country para Comfort para mujer Brandy Country 74a26b7 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.