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
Rápido Tecnología de rebote
FantomFit
Ligero entresuela acolchada
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


Zapato Estratos para caminar 17516 WW1865v1 New caminar Balance para mujer Estratos 1d6db71 - 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

Zapato Estratos para caminar 17516 WW1865v1 New caminar Balance para mujer Estratos 1d6db71

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.


Zapato Estratos para caminar 17516 WW1865v1 New caminar Balance para mujer Estratos 1d6db71

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

Hush Puppies Walker clásico de las mujeres Off White Multi , Bernie Mev Women s Gummies Gem Flat Estaño Hush Puppies Power Walker Sneaker para mujer Blanco Sandalias de mujer Ros Hommerson Mellow Armada Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 Negro / Lava Glow / Blanco , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Púrpura , Skechers Women s Go Walk Revolution Ultra Sneaker Negro / rosa caliente Zapato para caminar WW577 para mujer New Balance Negro , Zapato para caminar WW577 para mujer New Balance Negro Reebok Walk Ahead Action RS Walking Shoe para mujer Blanco negro Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Negro , Drew Shoe La alegría de las mujeres Negro , Hush Puppies Power Walker Sneaker para mujer Blanco Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Azul marino / gris / blanco Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On Cuero Vino Zapatillas para caminar Ryka Women s Devo Plus 2 Negro / púrpura Hush Puppies Men s Gus Oxford Marron oscuro , Easy Spirit Twist2 Sneaker para mujer Negro , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Azul marino / ondulado / estiramiento Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Hueso , Zapatillas de viaje Proather para mujer II Azul claro Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Blanco invierno Zapatillas Slip-on de mujer Skechers You Inspire Blanco negro Zapatillas para caminar WW669V1 New Balance para mujer Gris , Hush Puppies Mujeres Power Walker Ii Loafer Cuero Taupe , Apex Emmy Slip para mujer Negro , New Balance WW813 Hook and Loop Walking para mujer Gris , Drew Shoe Women s Tulip Comfort Ternero 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 Zapatillas de running Mizuno Wave Viper para mujer Capri-plata , Sandalia original Teva Women s Original Campo Negro / Blanco PUMA Men s Roma Basic Sneaker Blanco / gris claro Zapatillas Drew Shoe para mujer Paige Ternero blanco , NIKE Women s Tanjun Igloo / Blanco-negro NIKE Women s Revolution 4 Running Shoe Gris oscuro Pure Platinum Cool Gray White , Camiseta adidas Swift Run W para mujer Núcleo Negro / Blanco / Núcleo Negro , Zapato de trail running ASICS GT-2000 4 para mujer Azalea / Melon / Perfect Plum , Vivobarefoot Ultra 3 Bloom para mujer , Zapato de caminar WW928HW2 para mujer New Balance Blanco , Xelero Visalia Comfort Comfort Terapéutico Extra Profundo Casual Shoe Leather Velcro Negro Sandalia Jeana Mujer Yellow Box Multi , Easy Spirit Women s Traveltime Mule Blanco / Verde oscuro / Cuero Multi Kate Spade New York Women s Celo Sandal Nappa negro Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black , Skechers Womens Be-Light - Florescente Gris pardo Supra Vaider LC Sneaker Negro / Oro / Oro / Blanco , Converse Chuck Taylor All Star Core Ox Entintado / entintado Sandalia Sabani Sabani Slide de Franco Sarto Plata 50 tonos de oro 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) 2829- Oro Sparkle & Snake Bomba Hazel para mujeres Sam Edelman Multi Jacquard azul marino / jardín secreto Zapatilla Slip-On Hydro-Life W de Teva para mujer Granito 50 tonos de oro 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) S92307 Gold Flower & Gold , Botas de suela de cuero antideslizante de 6 \ Ls60 negro Bomba de vestido Kalila Total Motion Kalport para mujer Deep Ocean Nappa , Belleville 700V WP Black Combat Flight Boots para hombre Negro Zapatillas para mujer Del Rey TOMS Desert Taupe Poly Canvas TOMS Kids 10009918 Alpargata-K Alaskan Blue Heather Jersey / Faux Shearling , UGG Womens Mini Bailey Petal Negro , Mocasín con flecos Minnetonka para mujer marrón

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 Zapato Estratos para caminar 17516 WW1865v1 New caminar Balance para mujer Estratos 1d6db71 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.

Para caminar
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2925589
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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.