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.

TOMS Clásicos Clásicos TOMS de mujer 19092 Gris a46e278




Caucho
Importado
100% Auténtico, cuando comprado de - Tienda ilimitada
Clásico Diseño Alpargata
Elástico v para encendido / apagado fácil
Gamuza plantilla
Con cada par de zapatos usted compra, TOMS dará un nuevo par de zapatos para un niño necesitado Uno por uno
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


TOMS Clásicos Clásicos TOMS de mujer 19092 Gris a46e278 - 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

TOMS Clásicos Clásicos TOMS de mujer 19092 Gris a46e278

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.


TOMS Clásicos Clásicos TOMS de mujer 19092 Gris a46e278

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

Ballet Tali de mujer Cole Haan Bow Bow Maple Sugar Leather , U-lite para mujeres Zapatilla de lana de piel Casual plana interior y exterior Mocasín Slip-on, Zapatillas de mocasín para mujeres Negro (piel de cerdo de gamuza) TOMS - Slip-On de lienzo clásico para hombre Cable de hierro gris forjado de punto con Shearling , B.O.C - Mujeres - Peggy Azul marino grano completo , TOMS - Slip-On de lienzo clásico para hombre Plata , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Microfibra negra , CLARKS Jocolin Vista Boat Shoe para mujer Off White White Perforated Textile TOMS - Slip-On de lienzo clásico para hombre Acero gris microfibra monocromo , Drew Shoe Women s London II Slip On Textil Casual Flats Armada , Drew Shoe Women s London II Slip On Textil Casual Flats Armada , Zapatillas Slip-On Roamer SAS para Mujer Santolina , Kmeioo Slip On Mocasines, mujeres Bordado mula Zapatos con felpa Terciopelo de piel de cordero Sin respaldo Toe Mule Slides Zapatilla de satén negra Betunia Loafer de mujer Aerosoles Floral negro Zapatillas Skechers Women s Microburst One hasta Fashion Sneaker Negro TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Ceniza Trotters Liz Loafer para mujer Blanco Sanuk Mujer Donna Hemp Flat Natural Zapato femenino Keli de Alegria Jardín de invierno Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / de peltre , Clásicos de base Slip-on (Tm) para hombres de Vans Black Off White Checker , Zapatillas Skechers Women s Microburst One hasta Fashion Sneaker Negro / Plata , Skechers You de Women s One Slip-on Shoe gris , Easy Spirit Women s Traveltime Mule Dk Brown Easy Spirit Women s Traveltime Mule rojo Mocasines de vestir de punta estrecha para mujer de Me Too Women Blue Champagne Denim Easy Spirit Women s Traveltime Mule Plata Bota de lluvia Tinsley para mujer Sam Edelman Moss Green / Black TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Lino negro , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Chocolate ,
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 Zante V3 para mujer New Balance Poisonberry / Dark Mulberry , Saucony Women s Ride 10 zapatillas para correr Coral negro Skechers Performance Women s Go Run 400 Obstruct Negro Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Brillante Crimson / Wolf Gray-black-white , Zapatilla de deporte ECM Biom Fjuel Tie Sneaker para caminar Negro Negro , Zapatillas para caminar ProFit TravelFit Slide para mujer Azul negro , Zapatos de mujer de neopreno clásico Pedors Negro , OOFOS Unisex Ooriginal Sport Thong Flip Flop Negro / Aqua Zapatillas de running adidas Aerobounce ST w para mujer Aero Blue / Chalk Blue / Trace Blue , Skechers Women s Go 4-14918 Calzado para caminar Carbón , Zapato para caminar con cordones Go Step de Skechers Performance para mujer Negro Blanco Multi , Easy Spirit Women s Traveltime Mule Blanco Multi , Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Azul Indigo / Plata / Azul Regatta Plataforma de satén mono de ante PUMA para mujer Silver Pink-whisper , Vionic con sandalia Orthaheel Technology Rio Mujer Blanco , Zapatillas de tenis adidas Barricade Club para mujer Haze Coral / Blanco / Core Pink Empuje cargado de Under Armour para mujer Rhino Gray (076) / Negro , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede Zapatillas de tenis adidas Barricade Court para mujer Negro / plata mate / rojo destello , Zapatillas altas para mujer Converse Chuck Taylor All Star Hi Top Azul marino / blanco Plataforma de campo Leisa para mujeres de CLARKS Cuero rojo Sandalia de la diapositiva Adalea de las mujeres de Dolce Vita Ante amarillo , La Plume Jen de las mujeres Denim Tool Zapatillas Ivy de mujer TOMS Lana tribal gris / fucsia Tangas Birkenstock rojo TOMS - Slip-On de lienzo clásico para hombre Armada Bomba de Angel II para mujeres estilo suave Patente negra , Bota de montar Riviera de Ivanka Trump para mujer Gris oscuro Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 14 A.98 Botas de moto Isha 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.

TOMS Clásicos Clásicos TOMS de mujer 19092 Gris a46e278 - 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 TOMS Clásicos Clásicos TOMS de mujer 19092 Gris a46e278 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.

Mocasines y Slip-Ons
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1548014
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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.