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.

Converse Chuck Taylor All Core Star Core Converse Ox 19925 Ratón/ Ratón/ Ratón b6785ba




Cuero
Suela de goma
Converse
Hombre
Zapatos informales
Blanco
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


Converse Chuck Taylor All Core Star Core Converse Ox 19925 Ratón/ Ratón/ Ratón b6785ba - 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

Converse Chuck Taylor All Core Star Core Converse Ox 19925 Ratón/ Ratón/ Ratón b6785ba

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.


Converse Chuck Taylor All Core Star Core Converse Ox 19925 Ratón/ Ratón/ Ratón b6785ba

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 para mujer Del Rey TOMS Desierto de color topo , adidas Originals Mujer Superstar Blanco / Onix / Metálico / Dorado , Botas de tobillo Pallaphoenix K Palladium para mujer Verde ASICS Tiger Womens Gel-Lyte V Mysterioso / Mysterioso 1 Diadora N.9000 Arrowhead Azul , Onitsuka Tiger Asics Delegación Unisex de México Borgoña / Borgoña , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco Converse Chuck Taylor All Star Core Ox Armada , Zapatillas de deporte especializadas Gola para mujer Indian Teal / blanco roto Zapatillas para mujer Del Rey TOMS Luz Marina , Diadora Unisex N9000 III Blanco / Princesa Azul Converse Chuck Taylor All Star Core Ox Negro Negro Zapatillas de deporte de hombre Gel-Lyte MT de ASICS India Ink / India Ink , Z-CoiL Pain Relief Footwear Legend Legend Slip Resistant Black Tennis Tennis Shoe Negro JBU by Jambu Women s Gwen Rain Shoe marrón JBU by Jambu Women s Gwen Rain Shoe Armada , Zapatillas de tenis NIKE Women s Court Lite Blanco / Lava Glow-vast Grey , APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Plata / Blanco hueso , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Pewter / Calf Zapatillas de deporte cuadradas Harrier de Gola para mujer Gris pálido Ryka Womens Divya Blanco / plata cromada , Zapatillas de tenis Wilson Rush Pro 2.5 para mujer Very Berry / White / Pink Glow Ryka Womens Divya Gris / azul / blanco , Zapatillas de tenis 696v3 New Balance para mujer Pigmento / coral vivo , Zapatillas de tenis New Court 896v2 Hard Court para mujer Naval Blanca Calzado de tenis de bádminton Yonex Aerus 2 LX para mujer gris , Zapatillas Skechers Burst TR para mujer Negro Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco
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 Fresh Foam 1080v6 de New Balance para mujer Gris azulado , Sandalia KEEN Women s Uneek Slide-w Negro Negro Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Fila Navy / Wedgewood / Deep Pink Correpasillos Merrell para mujer 3 Trail Runner Negro / Plata Zapatillas de correr Mesh Lone Peak 3.5 de Altra para mujer Púrpura / Naranja , Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Menta , Zapatillas de running New Balance 990v4 para mujer Verde militar / verde militar , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Púrpura / plateado Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro , Earth Origins Selby Azul marroquí Ros Hommerson Calypso Slip para mujer Azul / Denim Sandalia plana Halle de Walking Cradles para mujer Blanco , Revere Miami sandalia de confort para la mujer w. Bronce / Serpiente , NIKE Men s Benassi Just Do It Athletic Sandal Dark Obsidian / Electric Yellow , Ryka Womens Divya Blanco / plata cromada Clásicos de base Slip-on (Tm) para hombres de Vans PANEL NEGRO NEGRO Vans Unisex Old Skool Zapatillas clásicas de skate Negro / Blanco verdadero Zapatillas de skate Vans Unisex Old Skool Silver Lining True White Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Armada , Birkenstock Arizona Sandalia de ante con suela suave Cuero engrasado , Sandalia Emilia High Slide de Cole Haan para mujer Ocelote Sandalia Cato Dolce Vita para mujer Cuero negro Zapato de césped de la dinastía Ringor Azul marino / blanco Zapatilla Glamour de mujer verde Daniel Gold Kidskin , UGG Womens Selarra Slipper castaña , Zapatillas Dr. Mary Comfort Flauta Lycra Elasticables Diamante Mary Jane Licra , Bota Deia para mujer de TOMS Broncearse , Zapatillas Nike Women s Flex Trainer 5 Blanco / Elemental Rose / Barely Rose Calou Stockholm Women s Hanna borgoña , Aquatalia de Marvin K. Botines de gamuza para mujer Tobillera Arena

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.

Converse Chuck Taylor All Core Star Core Converse Ox 19925 Ratón/ Ratón/ Ratón b6785ba - 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 Converse Chuck Taylor All Core Star Core Converse Ox 19925 Ratón/ Ratón/ Ratón b6785ba 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.

Tenis y deportes de raqueta
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

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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.