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.

Pantalón largo clásico para NIKE Golf para Gray mujer Tech Essentials B072MFRN87 Negro/ Mtlc Cool Gray fd92975




Nike Hombres NIKE AIR MAX SECUESTRO 2, NEGRO / MTLC FRESCO GRIS ANTRACITA, 6.5
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


Pantalón largo clásico para NIKE Golf para Gray mujer Tech Essentials B072MFRN87 Negro/ Mtlc Cool Gray fd92975 - 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

Pantalón largo clásico para NIKE Golf para Gray mujer Tech Essentials B072MFRN87 Negro/ Mtlc Cool Gray fd92975

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.


Pantalón largo clásico para NIKE Golf para Gray mujer Tech Essentials B072MFRN87 Negro/ Mtlc Cool Gray fd92975

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

Zapatillas de running adidas Women s Edge Lux 2 Energy Aqua / Trace Blue / Gray One , Zapatillas de running NIKE Air Huarache para hombre Hueso de Goma Ligera / Orquídea Amarillo Zapatillas de running NIKE Airstream Sequent 3 para mujer Antracita negra Zapatillas para correr NIKE Women s Air Zoom Pegasus 34 Provence Violeta / Violeta Persa Zapatillas de running ASICS Gel-Contend 4 para mujer Azul índigo / Rosa Cosmo / Negro , Zapatillas de entrenamiento Primus Lite para entrenamiento de mujer Vivobarefoot Todo negro Zapatillas de running Salomon Women s Sense Link Lucite Verde / Negro / Granny Green , Zapatillas de running Reebok Print Smooth Ultk para mujer Fuego Coral / Rosa Estelar / Blanco / Stonewash / Peltre Zapatillas de entrenamiento Nike Womens Metcon 3 Gris oscuro / verde fantasma Zapatillas de running Nike Zoom para mujer Hyper Pink / White / Pink Blast , Zapatillas para correr Saucony Women s Triumph 11 Gris / verde azulado / cidra , Zapatillas de running NIKE Air Max Flair para hombre Piedra Pómez / Legión Verde Zapatillas de running New Balance 790v6 para mujer Espacio exterior / guayaba Nike Air Max Zero QS femenino Under Armour Speedform Velociti para mujeres Rhino Gray (076) / Glacier Grey Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , NIKE Mujeres Downshifter 7 Blanco / puro platino , Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink NIKE Women s Tanjun Paramount Blue White Lava Glow Zapatillas para caminar Reebok Women s Cloudride LS DMX Polvo de asteroide / nube gris / blanco / negro , Zapatillas de deporte Nike Air Max Mercurial 98 para hombre Rojo (Unvrsty Rd / Unvrsty Rd-rflct Sl 600 Zapatillas de running NIKE Lunarglide 9 para mujer Negro / Blanco / Antracita , Zapatillas de running ASICS Gel-Contend 4 para mujer Gris piedra / plata / rosa caliente Short NIKE Tempo para mujer Verde lima / negro / blanco Zapatillas para correr New Balance W840V3 para mujer Trueno , Zapatillas Nike SF Air Force 1 Mid para mujer Marfil / Marfil-Mars Stone AA3966-100 Piedra Ivory / Ivory-mars , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo Zapatillas de running Nike Zoom All Out Low para mujer Pure Platinum / Chrome-glacier Blue ,
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 Simplemente Taupe , KEEN Sandalias Newport Eco-W para mujer Rose Garden / Tawny Port , Adicción a Brooks 12 Ombre Azul / Obsidiana / Púrpura Fucsia , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Asfalto , adidas Adizero Defiant Bounce para mujeres Flash rojo / blanco / escarlata Sandalia Drew Angela para mujer Bone Smooth Leather Huracán XLT Mujeres Teva - Avellana / Negro Hazel / Black , NIKE Kids gratis Rn (grande) Gym Red / Off White Track Red Sandalias OLKAKAI Ohana - Hombre Carbón de leña / carbón Zapatillas de running ASICS Gel-Contend 4 para mujer Gris Medio / Mar Glaciar / Plata Zapatillas de paseo Slip-on Go Step Lite de Skechers Performance para mujer Azul marino / gris Sneachers Sport Premium-Premix Slip-On Sneaker para mujer Naval Blanca , Zapatillas de running New Balance 990v4 para mujer Faded Rose / Komen Pink , XYD Prom Dancing Shoes Elegant Open Toe Strappy Sandalias de tacón de tobillo Wrap vestido de bombas para las mujeres Desnudo Zapatillas de running adidas Women s Edge Lux W Negro / Blanco / Dgh Gris Sólido Easy Spirit Solana Oxford para mujeres Nuez Sandalias de Progreso Mujer NAOT Cuarzo, Silv Mirror, Beige Micro , Melissa Women s Diane + Jason Wu Slides Marrón / negro Zapatillas de entrenamiento CUSH + WX577V4 New Balance para mujer Negro / Heather , Sandalia Uneek-W de mujer KEEN Ágata gris / abedul plateado , Zapatillas de skate Vans Unisex Authentic Platform (Retro Sport) Sunflower 7294 , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco , Supra Flow Run Evo Skate Shoe Negro Negro Padders Women s Cherish Soft Microfiber Velcro Cierre Bootie Slipper (Púrpura) Extra Wide - Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Azul morado Mocasines de conducción en punta D con cordones en piel para mujer Twisted X - Marrón / Flores combinadas Flores marrones / molidas , Bota sobre la rodilla Kantha para mujer de Vince Camuto Granite Peak , Bota Electra-2020 para mujeres Pleaser Patente blanca Bota Deia para mujer de TOMS Canela Suede / Tribal ENMAYER Mujeres Moda retro negra Mocasines con punta redonda Resbalón en piso al aire libre Zapatillas con peludo y hebilla Brown (vaca dividida) ,

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.

Pantalón largo clásico para NIKE Golf para Gray mujer Tech Essentials B072MFRN87 Negro/ Mtlc Cool Gray fd92975 - 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 Pantalón largo clásico para NIKE Golf para Gray mujer Tech Essentials B072MFRN87 Negro/ Mtlc Cool Gray fd92975 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 271474
2288350
4197310
5866226
6515560
8942390
13391101
14273191
18702666
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.