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.

Sandalia Paula Fisherman Mocha de Haflinger para mujer Haflinger Cuero Nubuck Mocha Nubuck 3043401




Cuero
Goma suela
Altura del talón: 1
Origen: Importado
Ajuste: Fiel al tamaño
Plantilla: Gamuza
Suela: Caucho
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


Sandalia Paula Fisherman Mocha de Haflinger para mujer Haflinger Cuero Nubuck Mocha Nubuck 3043401 - 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

Sandalia Paula Fisherman Mocha de Haflinger para mujer Haflinger Cuero Nubuck Mocha Nubuck 3043401

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.


Sandalia Paula Fisherman Mocha de Haflinger para mujer Haflinger Cuero Nubuck Mocha Nubuck 3043401

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

Earth Origins Sophie Platino Columbia Women s Barraca & shy; Sunlight Athletic Sandal Barro / Guijarro , Chaco Mujer Zcloud Sport Sandal Noche Emplumada , OluKai Kulapa Kai Burbujeante / negro Sandalia de mujer Drew Alana Verde / Fucsia BZees Sandalia Deportiva Jive para Mujer marrón , Sanda Kayenta para mujer Teva Negro KEEN Hombres Newport H2 Sandal Negro Drew Womens Cascade Nubuck azul marino , Chaco Mujer Z Eddy X1 Sport Sandalia Cipher Gold Sandalia deportiva Nolita para mujer Anne Klein Luz Natural / Oro Metálico , Propet Women s TravelActiv Ss Sandal Oro / negro , Chaco Women s ZX2 Classic Athletic Sandal Cascada Bosque Teva Terra Sandal para hombre Rocio Olive Fabulicious Women s Caress 401Dm sandalias de diapositivas Claro Rhinestone negro / peltre , Sandalia Terrell II de Merrell para mujer Gris pardo Wolky Women s Rio Sandalia Black Crash Suede Sandalia Teva Mujer Terra-Float Livia Lux Negro Sandalias Swiftwater Mesh de Crocs para mujer Gris claro , Drew Womens Ariana Cuero de peltre , OluKai Ohana Blanco negro Xero Shoes Sandalias deportivas con diseño descalzo - Z-Trek - Mujer Carbón de leña / carbón negro Zapatilla de deporte Sina 11 de Josef Seibel para mujer Pantalones Chaco Mujer Zcloud X Athletic Sandalia Island Tango Earth Origins Sophie Platino , Wolky Comfort Jewel Gray Croco Sandalias planas Eileen Fisher para mujer Sport-Nu Armada OluKai Kulapa Kai Violeta profundo / negro , Sandalia KEEN Women s Uneek Slide-w Dapple Gray / Vapor ,
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 para correr Under Armour UA Speedform Fortis 2.1 para mujer Negro / blanco / azul veneciano Zapatillas de running New Balance Fresh Foam Lazr V1 Blanco negro , Zapatillas de running New Balance Fresh Foam Lazr V1 Blanco negro , Sandalia Teva Mujer Terra-Float Livia Lux Armada Sandalias de cordones Joanie Tobillera para mujer de SOREL Negro Flip-Flops de The North Face para hombre Base Camp Lite Zinc Grey y Bossa Nova Red Linea Paolo Felicia | Suede perforado rosa claro Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory Sandalia de cuña Karmen de mujer Alegria Wrapture Camiseta adidas Swift Run W para mujer Carga nocturna / Carga nocturna / Blanco Sandalia de vestir Layla de Onex para mujer Azul Zapatillas adidas Performance Crazytrain Pro-Mid Cross-Trainer-Shoes, para mujer Core Red / Ftwr White Hush Puppies Walker clásico de las mujeres Piedra Easy Spirit Mujer Ezcool Mule Azul marino Combo de gamuza Clásicos de base Slip-on (Tm) para hombres de Vans Blanco (lienzo) adidas Originals Adi-Ease Fashion Sneaker Escarlata / negro / blanco Zapatillas de golf Adidas Golf Adidas, para mujer, de adidas Golf Mujer Gris / rojo noche Sandalia para mujer FRYE Ally Criss Cross Slide Secoya , Sandalia Vince Westcoast para mujer Vince Caucho magenta Zapatilla N275 de grano suave Ocean-Grafito / El Viajero para mujer El Naturalista Océano / Grafito Glerups Unisex G-01 - Botas de fieltro Carbón Pantuflas de mujer TOMS Classic Desierto de color topo , Mocasín Amity Australia mujer EMU Carbón Glerups Unisex G-01 - Botas de fieltro Púrpura , BareTraps BT Ricarda tobillera para mujer Marron oscuro , Dakota femenina de UGG Negro , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Azul Trotters Liz Loafer para mujer Light Tan Tonal , Easy Spirit Women s Traveltime Mule Cuero negro / negro Zapato femenino Keli de Alegria Nuez dura

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.

Sandalia Paula Fisherman Mocha de Haflinger para mujer Haflinger Cuero Nubuck Mocha Nubuck 3043401 - 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 Sandalia Paula Fisherman Mocha de Haflinger para mujer Haflinger Cuero Nubuck Mocha Nubuck 3043401 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 y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1760584
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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.


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