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 Nike Air Max/ Essential para hombre Negro hombre/ Negro rojo carmesí d951c20




Cuero
Cuero y malla superior
Aire burbuja máxima en el talón
Caucho suela para agregar tracción
Waffle suela exterior
Light peso y seguro
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 Nike Air Max/ Essential para hombre Negro hombre/ Negro rojo carmesí d951c20 - 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 Nike Air Max/ Essential para hombre Negro hombre/ Negro rojo carmesí d951c20

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 Nike Air Max/ Essential para hombre Negro hombre/ Negro rojo carmesí d951c20

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

NIKE Men s Benassi Just Do It Athletic Sandal Negro / Rojo de desafío , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Violeta profundo / Amanecer New Balance Women s w990v4 Running Shoe Negro / Plata , Zapatillas para correr Zante V3 para mujer New Balance Poisonberry / Dark Mulberry Zapatillas para correr Saucony Women s Cohesion 10 Gris azul , Zapatillas de running adidas Adaptive Adapt para mujer Negro / Carbono / Blanco Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Púrpura Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Mica Blue / White / Racer Pink / Sport Fuchsia NIKE Women s Revolution 4 Running Shoe Negro Negro , Calzado de running Nike Flyknit Lunar2 para mujer Rojo azul Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Violet / Gamma Blue / Concord / Black , Zapatillas de running MBT MBT para hombres MBT Amarillo / Lima , Zapatillas de running Brooks Women s Glycerin 16 Road Ébano / verde / negro , Zapatillas de running NIKE RN 2017 para mujer gratis Binario azul / oscuro cielo azul-obsidiana Zapatillas de running Nike Kids Air Max Zero Essential GS Blanco / lobo blanco gris , Zapatillas de running Under Armour Charged Bandit 4 Team para mujer Acero (100) / Acero NIKE Air Max Zero QS para mujeres , NIKE Men s Flex 2017 RN Antracita / Volt / Cool Gray / Black Zapatillas de running Nalu para mujer Ryka Azul / plata , Zapatillas de running Alphabounce W para mujer adidas Performance Negro / blanco / negro de utilidad , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Negro / sal marina Zapatillas de running New Balance 1500v3 para mujer Porcelana / Alpha Pink Zapatillas de running Topo Athletic Terraventure - Mujer Teal / Mint , Zapatillas de running Flex Ride 630v5 para mujer New Balance Gris New Balance Women s 555 Lifestyle Fashion Sneaker-Suede / Mesh Zapatillas para correr Negro Negro Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla oceánica / Resplandor azul-blanco , Zapatillas para correr Mizuno Women s Wave Creation 16 Mulberry Purple / Silver Calzado de running ft. Perpetua ft de memoria Fila Women s Castlerock / Aruba azul / blanco Zapatillas de running New Balance para mujer 490v4 Droplet / Silver ,
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 New Balance Foam Cruz V1 para mujer New Balance Negro / sal marina / goma de mascar Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Lino Verde 6367 Teva Terra Sandal para hombre Madang Olive Webbing Teva Terra Sandal para hombre Madang Blue , Zapatillas de entrenamiento de vuelo Jordan Nike para hombre Antracita / Negro / Blanco , Zapatillas de running Cavu para mujer HOKA ONE ONE Negro / Bluebird NIKE Kids gratis Rn (grande) Plata negra / metálica Wolky Comfort Jewel Piel Metálica Suave Antracita Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Negro / Estaño / Plata Metálica Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / Carbón Calzado de senderismo Skechers Go s 15601 para mujer Negro Sandalias planas para mujer Naturalizer Juniper Strappy con punta abierta Lady Blue Leather , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral Birkenstock Kid s Arizona Sandal Moca Newton Running El destino de las mujeres Rosado New Balance WX77 Cross Trainer para mujeres Negro / Tie Dye Gray / Gráfico Skechers Sport Women s Burst City Scene Zapatillas de moda ciruela , BOC B.O.C - Mujeres - Alisha Negro Zapatillas para caminar WW1400v1 New Balance para mujer Marron oscuro Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Black-paradise Pink Zapatillas de entrenamiento cruzado Realflex Train 4.0 de Reebok para mujer Goma de goma negra / blanca / Rbk / plateada metálica , Converse Chuck Taylor All Star Core Ox Blanco negro , Zapatillas de ciclismo Pearl XZUMi W X-Alp Canyon para mujer Ahumado Perla / Niebla Verde Sandalia de vestir Taimi de Naturalizer para mujer Cuero guijarroso negro iridiscente Zapatilla de joya de mujer Foamtreads Negro , Bota de campo Timberland para mujer Camdale Nubuck de trigo FRYE Mujeres Erica Oxford Blanco , Zapatillas para mujer Del Rey TOMS Lámina natural Bobcat / Gold Cloud Aline rojo Orthofeet Probado Pain Relief Tivoli Cómodo apartamento Pies Fascitis Plantar Diabetic Orthopaedic Mujeres extra anchas Botas de velcro ,

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 Nike Air Max/ Essential para hombre Negro hombre/ Negro rojo carmesí d951c20 - 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 Nike Air Max/ Essential para hombre Negro hombre/ Negro rojo carmesí d951c20 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.

Carretera en funcionamiento
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1106408
7480744
7694688
9213551
13113743
15652249
15749246
16672306
18814393
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.