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.

Zapatillas para caminar ProFit TravelFit negro Slide Slide para 19962 mujer Azul negro 4650a31




Malla
Importada
Sintética único
Eje mide aproximadamente Parte baja del arco
Ancho anchos
Flexible suela
Transpirable malla superior
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


Zapatillas para caminar ProFit TravelFit negro Slide Slide para 19962 mujer Azul negro 4650a31 - 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

Zapatillas para caminar ProFit TravelFit negro Slide Slide para 19962 mujer Azul negro 4650a31

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.


Zapatillas para caminar ProFit TravelFit negro Slide Slide para 19962 mujer Azul negro 4650a31

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

Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Negro Zapatillas de deporte Converse Chuck Taylor All Star Coral Ox para mujer Vapor rosa / blanco Zapatos de mujer de neopreno clásico Pedors Negro Drew Shoe Athena para mujeres Crema / Combo , Trotters Mujer Simmy Mary Jane Flat Sabio , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Hueso , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Estaño , Zapatillas deportivas para caminar Spira CloudWalker para mujer con resortes Blanco negro , Toni Pons Womens Noa Piedra Cordelia Zapato para caminar Skechers Women s You Negro Calzado para caminar Persad de la Mujer del Dr. Scholl s Shoes Monument Action Leather Linea Paolo Felicia | Vestido de gamuza estampado de camuflaje marino oscuro / verde Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Plata blanca , Zapatillas para caminar RYKA Women s Devotion Plus Frost Grey / Steel Grey / Rose Violet , Zapatillas de deporte Skechers Performance Go Walk 4 Exceed con cordones para mujer Blanco negro , Zapato para caminar Skechers Go Joy para mujer Gris pardo , Prop & eacute; t Olivia Oxford para mujeres Hueso , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Juanetes caminando Whitney Womens Sneakers Athletic Zapatos Blanco Zapatillas Skechers Go Walk Lite-15360 para mujer gris Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Blanco , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Negro , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Estaño , Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Blanco negro Skechers Women-14960 Sneaker melocotón I-RUNNER Pro Series Mujer - Slip, Oil, Skid Resistance Extra Depth Zapato de cuero con cordones Negro Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Merlot Suede , SAS Womens Sunny Mandarina Naranja Drew Shoes Cologne Diabético terapéutico con cremallera extra para mujer Cremallera de cuero Marron oscuro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco 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 Carro de mujer Brooks Negro Earth Origins Sophie Women s Sandal Negro , adidas MUJERES ORIGINALES CLIMACOOL 02/17 ZAPATOS # BY9289 Gris / gris Calzado casual tejido NIKE NIKE para hombre Team Royal / Off Blanco-negro NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Gimnasio Red / Black-Noble Red Zapato de agua negro V-Aqua Vibram para mujer Negro Zapatillas de montaña de montaña Akyra La Sportiva para mujer Esmeralda / menta , Brooks Womens Adrenaline GTS 18 Rosa / Plum / Champagne , Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Dragon Fruit / Sea Salt / Phantom , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Azul Púrpura / Violeta / Airy Blue Zapatillas de mujer Reebok DMX Lite Slip Walking Gable Gray / Brave Blue / Vitamin C / White / Matte Silver Inov-8 Unisex F-Lite 195-U Zapato de entrenamiento cruzado Neón Rosa / Negro / Blanco Zapato para caminar WW577 para mujer New Balance Negro adidas Athletics 24/7 Tr M Cross Trainer Core Black / Core Black / Hi-res Blue , Vans Unisex Old Skool Zapatillas clásicas de skate High-rise / Pewter Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / blanco , Zapatillas de golf Minimus WP para mujer New Balance Gris / menta , Sandalia para mujer Unterman de ALDO Blanco , Sandalia Aerosoles Midday Women Slide Tela de mezclilla , Sandalia tobogán elástica bebe para mujer Negro Sandalia Ryana Marilyn Slide para mujer Armada , Así que Danca CH50 Character Shoe con 1.5 Heel Blanco Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Azul marino / blanco Mocasines casuales Alpargata Deconstructed de TOMS para mujer Mustard Slub Chambray Zapatillas de ciclismo All-Road v4 Pearl iZUMi para mujer Blanco / gris oscuro , Mujer adidas Sprintstar W Blanco / Negro / Choque Rosa Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Red Patent / Nude Trim / Gold Studs Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Cuero de Trigo TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Dark Tortoise Shell Brown Nubuck / Rope

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.

Zapatillas para caminar ProFit TravelFit negro Slide Slide para 19962 mujer Azul negro 4650a31 - 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 Zapatillas para caminar ProFit TravelFit negro Slide Slide para 19962 mujer Azul negro 4650a31 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

Thirteen elementary straightedge 1923478
3491259
4291053
5033660
5578936
9246796
10188429
11039336
11380678
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.