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.

SAS SAS Womens Womens Sunny Oro Oro 62dccfc




Cuero
Hecho en EE. UU. o importados
eva suela
Medidas de talón aproximadamente 0.5 pulgadas
Inicio tu mañana libre correcto con el Mocasín soleado sin cordones de SAS & reg;.
Soft piel genuina abrazos superiores tu pie y características un mocasín atado a mano dedo del pie.
SAS Casual El último está diseñado para envolver por completo alrededor de tu pie para un acogedor próximo para sentir la piel.
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


SAS SAS Womens Womens Sunny Oro Oro 62dccfc - 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

SAS SAS Womens Womens Sunny Oro Oro 62dccfc

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.


SAS SAS Womens Womens Sunny Oro Oro 62dccfc

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

Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Negro Negro , Zapato de caminar WW1300v1 para mujer New Balance Gris Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Negro , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco , Zapatillas para caminar para caminar al aire libre Go Skechers Performance para mujer Negro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Blanco negro Vivien Flat de Bella Vita para mujeres Almond Kid Suede Leather , Drew Shoe La alegría de las mujeres Cuero blanco / malla , Prop & eacute; t Olivia Oxford para mujeres Bronco Brown , Earth Origins Tova Azul cobalto Adidas Sport Performance Mujeres Terrex CC Voyager Sleek Sneakers Negro / Negro / Blanco Tiza Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Negro Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy , Zapato para caminar Skechers Go Joy para mujer Coral , Zapato para caminar Eden Propet para mujer Blanco , Trotters Mujer Simmy Mary Jane Flat Sabio Zapatillas Skechers Women You-14966 Azul marino Zapatillas para caminar antideslizantes Spira WaveWalker para hombre Naval Blanca , Easy Spirit - Zapatillas Faisal2 para mujer Negro , Skechers Women s Define Sneaker Azul marino , Orthofeet Probado Alivio del dolor Comodidad Serene Womens Orthopaedic Artritis extra ancha Diabetic T-Strap Zapatos Broncearse , Drew Shoe Athena para mujeres Blanco azul Zapatos de mujer RYKA Sky Walking Cool Mist Gray / Orchid Bouquet , Vivien Flat de Bella Vita para mujeres Cuero de gamuza de cabrito negro , Ballet de niña de flores Aerosoles femenino Serpiente de coral Zapatillas de senderismo Merrell Moab 2 GTX para mujer Sedona Sage , JBU by Jambu JBU Womens Aberdeen Low Top Slip On Running Sneaker Negro ,
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 Converse Chuck Taylor All Star Seasonal Color Hola Rojizo-Marron Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Blanco negro Zapato de running ASICS para mujer Torrance Negro / Gris piedra Zapato de running ASICS Gel-Quantum 360 cm para mujer Melocotón negro / Onyx New Balance Vazee Running Shoe-Breathe Pack Zapatillas de deporte de moda para mujer Arrecife Zapato de agua negro V-Aqua Vibram para mujer Negro Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / Campanula / Carbono Zapatillas de running adidas Ultraboost X Parley para mujer Gris crudo / Carbono / Azul Espíritu , NIKE Men s Benassi Just Do It Athletic Sandal Negro Negro Bomba de cuña Lansing para mujer Trotters Gamuza marrón oscuro , Zapatillas para caminar Skechers Performance Go 4 Inspire para mujer Negro Flip Flop de Hari Mari Fields - Mujeres Negro / Seafoam / Negro Zapatillas Superga 2750 Cotu para mujer Oro rosa , Adidas Originals Adidas Athletics 24/7 W Cross Trainer para mujer Negro / Negro / Blanco , Entrenador cruzado New Foam Crush V1 de New Balance para mujer Pigmento / amarillo solar , Trotters Mujer Simmy Mary Jane Flat Sage Veg Tumbled Leather Zapatillas de golf Skechers Women s Go Golf Birdie Azul marino / rosa , Sandalia UGG Royale Slide para mujer Blanco , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Cuero Sahara Bota de trabajo Alisa Roper para mujer Negro Zapatillas de beisbol Softball Velo V1 para mujer New Balance Azul marino / blanco Bomba de vestido Daisie de mujer Steve Madden Gris claro gamuza , Alegria Nanook Solid Negro acolchado , TOMS - Slip-On de lienzo clásico para hombre Cayenne Canvas Tile Sandalia Birkenstock de cuero Florida para mujer Cuero engrasado , Sandalia Birkenstock de cuero Florida para mujer Sfb Black Birko-flor Sanuk Women s Vee K Shawn Chukka Boot Natural L & rsquo; Artiste de Spring Step Women s Gabriel Oxford Negro , Dr. Comfort Vigor Mujer diabético terapéutico Extra profundidad senderismo botas de cuero de encaje Negro , Rockport Mujeres Ayva Oxford Negro

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.

SAS SAS Womens Womens Sunny Oro Oro 62dccfc - 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 SAS SAS Womens Womens Sunny Oro Oro 62dccfc 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.

Para caminar
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.