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.

KEEN H2 marino Hombres Newport H2 Sandal Azul marino/ gris medio medio fe2ac18




Cinta de poliéster
Hecho en los EE. UU. o importados
Caucho único
Rendimiento sandalia deportiva con sistema de cordones bungee y tracción trasera lazo
Soporte arco contorneado ; sandalias de agua son lavable en la lavadora. Usa una pequeña cantidad de detergente, lavado en ciclo suave y secar al aire. Consejo: este estilo está ejecutando un 1/2 talla pequeña. sugerir ordenar un 1/2 de tamaño más grande que tu usual tamaño
metatarsiano soporte de cresta
Cuidado - Lavable en la lavadora, usa una pequeña cantidad de detergente, lavado en ciclo suave y secar al aire
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


KEEN H2 marino Hombres Newport H2 Sandal Azul marino/ gris medio medio fe2ac18 - 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

KEEN H2 marino Hombres Newport H2 Sandal Azul marino/ gris medio medio fe2ac18

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.


KEEN H2 marino Hombres Newport H2 Sandal Azul marino/ gris medio medio fe2ac18

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

Calzado casual Slip-On de Bernie Mev para mujer Sapphire-multi Drew Womens Cascade Denim Blue Nubuck , Sandalias planas Eileen Fisher para mujer Sport-Nu Armada Sandalia Drew Angela para mujer Bone Smooth Leather , Sandalia Drew Element para mujer Denim Blue Nubuck , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Cristal marino , Sandalia Yucatan ECCO Mujer Polvo blanco Wolky Comfort Jewel Cuero de lino , Earth Origins Sophie Platino , Sandalia deportiva Tirra para mujer Teva Simplemente Taupe , Sandalia deportiva Chaco Zvolv X2 para mujer Dash Black Earth Origins Sophie Negro Tacones Pleaser 766 High Feels de Pleaser Negro Negro , Chaco Mujer ZX1 Classic Sport Sandalia Negro Chaco Mujer Zcloud X Sandalia Deportiva Durazno estándar Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Púrpura Sandalia Teva Women s W Hurricane Xlt2 Sport Negro , Chaco Women s ZX2 Classic Athletic Sandal Credo Golden , Sandalia de mujer Drew Alana Pewter-bronce , ALMA NATURAL Sandalia de pescador Beatrice para mujer Rosa KEEN Hombres Newport H2 Sandal Azul marino / gris medio , Chaco Mujer Zcloud X2 Sandalia Deportiva Scuba Eclipse , Sandalias planas Eileen Fisher para mujer Sport-Nu Caqui OluKai Ohana Pale Lime / Dk Java , Chaco Women s Z2 Classic Athletic Sandal Cañón , Sanda Verra Teva para mujer Surf Negro / Multi Sanda Verra Teva para mujer Manta Rojo Rayas Rojo , Sandalias de Gladiador Mujeres Fretwork Sandalias de Tacón Extrañas Mujeres Zapatos de Cuña de Verano para Mujer Negro , Chaco Mujer ZX1 Classic Sport Sandalia Dart Peach
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 Skechers Performance Women s Go Run 400 Obstruct Gris / azul , Zapatillas de running con amortiguación W560v7 para mujer New Balance Silver Mink , Calzado de running Filaity Memory Finity de Fila Negro, blanco, plata metalizado Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco verde , Zapatillas de running NIKE Mens Lunarglide 8 Rio Teal / Volt-clear Jade-medianoche Turquesa Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro Sandalia atlética del Caprizee de la mujer de Columbia Miami / Jade Lime La Sportiva Akasha Running Shoe - Mujer Stone Blue / Jade Green , Helly Hansen 11302 Bandera de sal para mujer F-1 Sneaker Vintage Indigo / Plum / O Zapatillas MBT Simba 17 W para mujer Cian / Blanco / Rosa , Zapatillas para caminar Brooks Women s Addiction Walker Negro , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Artic Fox / Poisonberry Calzado de running Nike Flyknit Lunar2 para mujer Pure Platinium / Atomic Purple / Corte Púrpura / Blanco Brooks Women s PureGrit 5 Azul pavo real / Rosa virtual / Azul patriota Nike Hombres Air Zoom Pegasus 33 Pure Platinum Black White 002 Amortiguación 635v2 New Balance para mujer Negro , Easy Spirit Twist2 Sneaker para mujer Porcelana blanca CLARKS Sandalia Arla Kaydin para mujer Arena / Blanco elástico jaspeado , Under Armour UA Team Ultimate Speed Rojo blanco , Zapatillas de deporte Reebok para mujer CL Nylon Mesh M Niebla / tiza Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Ciruela / Plateado / Rojo Rouge , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Crema / Combo , Supra Vaider LC Sneaker Cool gris-blanco , Zapatillas de golf Minimus WP para mujer New Balance Gris / menta , Sandalia para mujer Merrell, 1SIX8 Linna AC + Slide Azalea , UGG Mujer Alena mocasín Anochecer , Bota western para mujer FRYE Sacha Moto Shootie Black-78001 , TOMS Clásicos de mujer Plumas de hoja de taupe desierto Sandalia Birkenstock de cuero Florida para mujer Sfb Blue Birko-flor Bed | Stu Botines para mujer estridentes Tinte Di Moro Dip Dye

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.

KEEN H2 marino Hombres Newport H2 Sandal Azul marino/ gris medio medio fe2ac18 - 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 KEEN H2 marino Hombres Newport H2 Sandal Azul marino/ gris medio medio fe2ac18 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 2405423
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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.