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.

Zapatillas adidas Irana 2 Cross-Trainer Zapatillas de White B073WRCQKD adidas para mujer Pop/ Shock Red/ White 740cbdb




Sintético
Importado
Caucho único
Ligero sintético y textil superior proporciona durabilidad; oferta de subcapas transpirabilidad
Perforado cuarto panel da ventilación
Adidas Etiquetas deportivas Stella en la lengua y el talón
Talón tirar para fácil en y fuera; ofrece soporte para el tobillo
Pivot-point cupsole de goma para un agarre excelente
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


Zapatillas adidas Irana 2 Cross-Trainer Zapatillas de White B073WRCQKD adidas para mujer Pop/ Shock Red/ White 740cbdb - 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

Zapatillas adidas Irana 2 Cross-Trainer Zapatillas de White B073WRCQKD adidas para mujer Pop/ Shock Red/ White 740cbdb

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.


Zapatillas adidas Irana 2 Cross-Trainer Zapatillas de White B073WRCQKD adidas para mujer Pop/ Shock Red/ White 740cbdb

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 de Cross-Trainer de las mujeres de Ryka Gris oscuro New Balance WX77 Cross Trainer para mujeres Blanco / gris gráfico , Zapatillas Superga 2750 Cotu para mujer Lt amarillo Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Peacoat-island Paradise Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Negro / blanco / puro platino Calzado de trabajo resistente a los resbalones de Fila, Memory Comfort Trainer Blanco / Blanco / Plata Entrenador cruzado 611v1 de New Balance para mujeres Artic Fox / Metallic Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Nrgy Peach-puma Black NIKE Entrenador flexible para mujeres 6 Negro / Lava Glow / Antracita Zapatillas de deporte cruzadas WX40V1 de New Balance para mujer Negro / Trueno / Amanecer , Nike Tech Xtreme Cadet Glove Gimnasio Rojo / Negro / Brillante Carmesí / Blanco K-Swiss Tubos de mujer Infinity CMF Cross Trainer Patriot Blue / Raspberry Wine , Zapatillas Superga 2750 Cotu para mujer Grey Sage Under Armour Charged All-Around Neutral True Ink (918) / Marfil , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Seafoam / Blanco , Zapatillas de cross-trainer Fierce Bright Mesh para mujer de PUMA Prisma rosa , PUMA Women s Fierce Rope Pleats Wn Sneaker Paradise Pink-paradise Pink Inov-8 Mujer F-Lite 235 V2 (W) Cross Trainer Coral rojo , Reebok Crossfit Nano 7.0 Track Shoe para mujer Rosa solar / negro / plomo / blanco , Entrenador cruzado New Foam Crush V1 de New Balance para mujer Pigmento / amarillo solar Under Armour Women s Precision Mid Purple Rave (959) / Premier Purple , Under Armour Women s Street Precision Low Lux Bayou Blue (953) / Marfil Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Negro , Nike Tech Xtreme Cadet Glove Amarillo Zapato de entrenamiento cruzado Yui con tratamiento en gel femenino de ASICS Negro / Plata / Blanco Zapatillas Cross-Trainer Ryce para mujer Enhance 3 Coral rosa púrpura K-Swiss St329 CMF Sneaker para mujer Angel Wing / Blanco Zapatillas de agua RYKA Hydro Sport 2 Cross-Training para mujer Negro / Gris hierro / Bluebird PUMA Tazon 6 IRI Wn Sneaker para mujer Puma White-cantera ,
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 Sandalias de tirantes Sunstone para mujer Merrell Negro , Skechers D Lites-Retro Glam-Rhinestone River Style Deportivo Sport Sport Sandalia Rosa claro NIKE Men s Lunar Fingertrap Cross Trainer Wolf Gray / Midnight Navy , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Lino Verde 6367 , Under Armour Charged CoolSwitch de mujer Glacier Grey (002) / Overcast Grey , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Añada Indigo / Sal Marina , Skechers Performance On The Go 400 Vivacity Flip Flop Taupe / Blanco Corredor SCARPA Neutron Wmn Running Shoe Trail Runner para mujer Atolón Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Trueno Zapatillas de running Wstro para mujer New Balance Alpha Pink / Black / Lime Glow Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Fiyi , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Piso de mujeres de Coral Pies extra anchos Diabético ortopédico Zapatos atléticos Sneakers para caminar Turquesa Sandalia de vestir de tela Leisa de nueve mujeres del oeste Combo blanco ASICS Gel-Nimbus 20 Platinum femenino Carbono / Plata , Mocasín de mujer RIALTO Vette Tejido negro Zapatillas de running Under Armour Speedform Intake 2 para mujer Negro (001) / Acero , Cobb Hill Rockport Women s Angelina Dress Pump Almendra Easy Spirit Women s Traveltime Mule Blanco / Verde oscuro / Cuero Multi Sandalia Uneek o2-w de mujer KEEN Puerto / blanco , Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer Blanco negro adidas Athletics 24/7 Tr M Cross Trainer Noble Rojo / Noble Rojo / Cristal Blanco MBT Zapatos Mujer Tunisha Mary Jane Casual cuero de zapatos / malla hebilla Negro / Nappa , Zapatillas Mujer adidas adizero Ubersonic 3 JADE Tactile Green / Forest / Fairway , Sam Edelman Women s Bay Slide Sandal Cuero Vaquero Negro Vaquero Sandalias Jerusalem Sandalia Nuri Slide rojo , Birkenstock Arizona Sandalia de ante con suela suave Negro 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 5008mirage- Party Sparkle & Ultra Silver Bota del oeste del pájaro azul del vintage de las mujeres de Dan Post Sanded Chocolate / Teal Blue Bird , Bota de mujer H de Hudson para mujer Armada , Cuero de Windup para mujer Caterpillar Lace Up Fashion Oxford rojo ,

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.

Zapatillas adidas Irana 2 Cross-Trainer Zapatillas de White B073WRCQKD adidas para mujer Pop/ Shock Red/ White 740cbdb - 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 Zapatillas adidas Irana 2 Cross-Trainer Zapatillas de White B073WRCQKD adidas para mujer Pop/ Shock Red/ White 740cbdb 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 1470831
3464754
6005197
6184605
9054955
12996648
15305573
15830078
19029416
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.