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.

Sam Edelman Sequins Small Women s Gigi Pewter Pewter Small Sequins 57acd48




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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


Sam Edelman Sequins Small Women s Gigi Pewter Pewter Small Sequins 57acd48 - 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

Sam Edelman Sequins Small Women s Gigi Pewter Pewter Small Sequins 57acd48

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.


Sam Edelman Sequins Small Women s Gigi Pewter Pewter Small Sequins 57acd48

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

Naot Footwear Mujer Verbena Beige Nubuck / Black Madras Leather Salvatore Ferragamo Mujer Varina Nero 1 Zapatilla OLUKAI Kahiko - Para mujeres Pizarra / Tan FLY London - Bomba YIPI831FLY para mujer Plata / Blanco Borgogna / Alfombra Sandalias de peonía Easy Spirit para mujer Gris pardo , NIKE Men s Benassi Just Do It Athletic Sandal Equipo oscuro Rojo / Solar Rojo / Negro , Sandalia Tonya de mujer Trotters rojo , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Candy Pink Suede , Sandalia de gladiador sintético Xema de Nine West para mujer Rosa / Multi , BZees Women s Cruise Slip-on Gris negro Bota Deia para mujer de TOMS Caramelo , Taipán de mujer de David Tate Verona Cordero negro Sandalias de mujer Havaianas Slim Sandalia Tropical Blue Splash , Sam Edelman Women s Laney Pump Dusty Rose Kid Suede Leather Taipán de mujer de David Tate Verona Champagne Lamb , Vionic con Orthaheel Tide Sport Sandalia para mujer Charol de cuero verde azulado , Drew Womens Ariana Dusty Gold SJP de Sarah Jessica Parker Mujer Meteor Sneaker Candy Pink Suede MONTAÑA BLANCA Miso Mule cuero de mujer Arena clara L`Amour Des Pieds Womens Amadour Sandal Gris pardo Chaco Mujer Fallon Sandal Negro , KEEN Women s Uneek 8MM Sandal Neutral Gray / Dahlia Mauve , Sandalia Tonya de mujer Trotters Negro Tecnología Vionic Men s Orthaheel White / Navy Walker ciruela Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra , Sanda Verra Teva para mujer Surf Taupe Multi , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja CLARKS Lexi Myrtle Sandal para mujer Cuero azul / gris , Bomba Nikole para mujer Via Spiga Midnight Suede
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 Tirra para mujer Teva Trufa de ciruela Zapatillas para correr NIKE Women s Air Zoom Structure 21 Platimum puro / Antracita Skechers Performance Women s The Go 400 Radiance Flip Flop Negro , Sandalia deportiva Tirra para mujer Teva Simplemente Taupe , Sandalia para mujer Teva Terra-Float Travel Knit Paloma Salvaje , Calzado deportivo NIKE CK Racer para mujer Cerúleo / cerúleo / vela Skechers Women s Go Trail 2 - Zapatillas de senderismo con agarre Negro Zapatillas Slip-on de mujer Skechers You Inspire gris , Zapatillas Isla de las mujeres Softinos marrón , Sandalias de cuero de Viv Viv de TOMS Tribal negro , Sandalia Sunkiss Cara de Sperry Top-Sider para mujer Broncearse Splendid Women s Brittani Sandal Plata , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro Sandalia del vestido de Flaunt de las mujeres de LifeStride Plata Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Azul / iridiscente Earth Origins Tova Hibiscus Red , Comfortiva Mujer, Faye Sandalia de tacón medio Antracita adidas Women s Adilette Cloudfoam + Sandalia deslizante Negro / Blanco / Negro , Zapatillas Skechers Go s Lite Lite-Mystic para caminar Negro Vans Unisex Old Skool Zapatillas clásicas de skate (Gamuza / Lona) Frost Grey / True White Calzado de baloncesto NIKE Women s Court Borough Low Negro / Vivid Pink-Vivid Pink Zapatillas de golf Skechers Go Golf Elite Ace Jacquard para mujer Azul marino / rosa Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Oro / Metálico Bota de campo soberano de Mountain Horse Negro , Wonderheel 7 \ - Bota de moto Fulton para mujer Roxy Broncearse Splendid Midi Perf Vionic para mujer Slip-On Nubuck Perfume Negro , Sam Edelman Loraine Loaine para mujer Navy multi guinga TOMS - Slip-On de lienzo clásico para hombre Negro sobre negro , CLARKS Women s CloudSteppers Sillian Paz Slip-on Loafer Nubuck sintético de arena

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.

Sam Edelman Sequins Small Women s Gigi Pewter Pewter Small Sequins 57acd48 - 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 Sam Edelman Sequins Small Women s Gigi Pewter Pewter Small Sequins 57acd48 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
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 3857953
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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.