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.

Trotters Jane Mujer Simmy Mary 19994 Jane Flat B01MYR3NG0 Sabio 72a5c5e




Cuero
Importado
Goma suela
Medidas de talón aproximadamente 0.75
Cuero plano ofreciendo contraste costura y asimétrica Correa Mary Jane
Elástico insertar y gancho y lazo cierre en la correa
Flexible Suela anti-impacto
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


Trotters Jane Mujer Simmy Mary 19994 Jane Flat B01MYR3NG0 Sabio 72a5c5e - 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

Trotters Jane Mujer Simmy Mary 19994 Jane Flat B01MYR3NG0 Sabio 72a5c5e

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.


Trotters Jane Mujer Simmy Mary 19994 Jane Flat B01MYR3NG0 Sabio 72a5c5e

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 San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Blanco Zapatilla deportiva Kailee de mujer Ryka Frost Grey / Cotton Candy / Summer Grey , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Borgoña / ondulado / estiramiento , Klogs Footwear Astoria para mujer Perdiz , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Negro Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Plata , Zapato de agua Easy Spirit para mujer First Walker Terciopelo-amanecer-azul Zapatillas Skechers Women You-14966 Azul marino Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Moca , Zapato para caminar WW411v2 New Balance para mujer Blanco / rosa Dr. Comfort Annie-X Mujer diabética terapéutica Doble Profundidad Zapato Lycra Velcro Negro Zapatillas Skechers Performance para mujer You-BCA 2017 Azul marino / blanco Zapatillas para caminar Go Walk 4 Achiever de Skechers Performance para mujer Negro , Easy Spirit Deiny Women s Walking Negro Easy Spirit Deiny Women s Walking Azul oscuro Easy Spirit Women s Motion Lace up Oxford Cuero de concha , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Denim Lea / Nubuck Zapato para caminar Ryka Women s Sky Negro / rosa , Hush Puppies Men s Gus Oxford Broncearse , Xelero Visalia Comfort Comfort Terapéutico Extra Profundo Casual Shoe Leather Velcro Negro Zapatillas para caminar Brooks Women s Addiction Walker Blanco Earth Origins Tova Polvo , Zapato clásico de mujer Spira para caminar con correa EZ Blanco , Zapatillas para caminar Skechers Go Step Lite-14750 para mujer Negro Calzado de espuma fresca New Balance para mujer 1165v1 Púrpura , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Blanco , SAS Womens Sunny Teal , Bota Cami Alegria para mujer Tectónico OLUKAI Nohea Nubuck - Mujeres Dark Java / Tan ,
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 adidas Performance Ultraboost X para mujer Carbono / Cristal Blanco / Núcleo Negro , Sandalia Shaye Sport mujer Donald J Pliner Almendra / Salvia / Limón Zapatillas de deporte de moda Kanye West Desert Rat 500 Zapatillas de correr 3M para hombres Mujeres DB2908 DB2966 Sonrojo ONEMIX Zapatillas para correr Air Cushiong para mujer, Zapatillas deportivas deportivas ligeras Azul rojo , Piel Kulapa Kai de mujer OLUKAI Negro Negro Sandalias de encaje Ros Hommerson para mujer Oro , Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Negro , Soulsfeng para hombres Moda femenina Zapatos deportivos de encaje up Amortiguación transpirable Diseño de flores de tela Zapatillas de deporte para parejas Blanco Prop & eacute; t Zapatillas para caminar TravelFit para mujer Púrpura Zapato deportivo ASICS Gel-Excite 4 para mujer Aruba Azul / Plata / Acuario Zapatillas de running adidas Cloudfoam QT Racer para mujer Negro / Negro / Blanco Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata Zapatillas de senderismo QOMOLANGMA para mujer Zapatillas de senderismo a prueba de deslizamiento para correr Trekking Outdoor Training Azul / naranja , Entrenador cruzado Fresh Foam Crush V1 de New Balance para mujer Dusted Peach / Sunrise Glo Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Mushroom Nubuck , Zapatillas de skate DC Midway W para mujer Dark Shadow / White Vans Old Skool Unisex Adults Low-Top Trainers Mono Bumber / Red Vans Unisex Old Skool Zapatillas clásicas de skate lona negra / negra Sandalias FRYE Carla Criss Cross Open Toe Casual Slide Ceniza , Sandalia de diapositivas para mujer LifeStride Remix Negro , Bota de moda Emma Tall BEARPAW para mujer Burdeos Dr. Martens Gunby Steel Toe Shoe Negro , FRYE Sacha Moto Shootie para mujeres Cuero vintage de corte de carbón Zapatilla de cuero Zoili de Nine West para mujer Cuero natural claro Zapatillas Zeppa en suela francesa para mujer Ola negro Zapatillas de fútbol adidas X.2.2 Fg W, para mujer Negro / Platino Core Red S , Zapatillas de fútbol adidas Women s Nemeziz, 17,3 cm Fg W Tinta Blanca / Misteriosa / Coral Fácil Sandalia de tacón Daniel de Yosi Samra para mujer Cuero texturizado metálico de oro oscuro , Propet Women s Tour Walker Strap Sneaker Blanco CLARKS Women s CloudSteppers Sillian Jetay Flat Black Perf Textile ,

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.

Trotters Jane Mujer Simmy Mary 19994 Jane Flat B01MYR3NG0 Sabio 72a5c5e - 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 Trotters Jane Mujer Simmy Mary 19994 Jane Flat B01MYR3NG0 Sabio 72a5c5e 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 3542866
9786455
13496139
14980872
15946553
16807454
17196381
18019405
18539901
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.


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