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 19643 de running running NIKE Blanco-gris Zoom para mujer Negro/ Blanco-gris oscuro ae9a240




Malla
Importada
Zoom Unidad de aire en el el talón proporciona respuesta amortiguación.
Dinámico Tecnología de ajuste bloquea tu pie.
Co-moldeado entresuela es liviana y receptivo.
Single-layer malla proporciona transpirable apoyo.
Waffle la suela es durable y proporciona una gran tracción multi-superficie. En forma de Pentágono los terminales tienen 5 ángulos para un agarre excelente a altas velocidades.
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 19643 de running running NIKE Blanco-gris Zoom para mujer Negro/ Blanco-gris oscuro ae9a240 - 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 19643 de running running NIKE Blanco-gris Zoom para mujer Negro/ Blanco-gris oscuro ae9a240

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 19643 de running running NIKE Blanco-gris Zoom para mujer Negro/ Blanco-gris oscuro ae9a240

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 de running NIKE Air Max Thea Ultra PRM para mujer Summit White 100 Nike Men s Air Presto Essential Azul marino / difuso , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Negro / Mtllc Plata-volt-blanco Zapatillas de running NIKE Air Max Thea Jacquard para mujer Negro / Metalic Plata / Blanco Zapatillas de running New Balance 990V3 para mujer Gris Zapatillas de running New Balance 520v3 para mujer Bronce / Blanqueado Amanecer / Trueno , Brooks Womens Ravenna 9 Rosa / Plum / Champagne Zapatillas de running NIKE Mens Lunarglide 8 Blue Glow Zapatillas de running NIKE Women s Air Zoom Structure 20 Negro / Blanco-lava Glow-chlorine Blue , Zapatillas para caminar Reebok Women s Cloudride LS DMX Blanco negro Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Verde Ceniza , Amortiguación 635v2 New Balance para mujer Negro , New Balance Women s w990v4 Running Shoe Negro / Plata , Calzado de running NIKE Kids Roshe One (GS) Rosado , Zapatillas de deporte NIKE Womens Air Zoom Pegasus 32 Running 749344 Sneakers Shoes Azul Zapato de running ASICS Gel-Quantum 360 cm para mujer Flash Coral / Negro / Plata , Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Choque Verde / Negro , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running adidas Women s Edge Lux W Colegial azul marino / azul claro / azul claro Zapatillas de running Reebok para mujer Pump Plus Vortex Seaside Grey / Mist / Silver Met Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosa Fuego / Negro / Crema de Melocotón Zapatillas Reebok para mujer Trilux Run PNT Sneaker Prnt - Negro / Blanco Nike Men s Air Presto Essential Universidad / Rojo / Negro / Negro New Balance Women s w940v3 Running Shoe Plata Lanzamiento de Brooks Womens 4 Negro / Fucsia Púrpura / Azul deslumbrante Zapatillas de running NIKE Mens Lunarglide 8 Multicolor , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / desierto Moss-desierto musgo
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 deporte NIKE para mujer Air Max Jewell PRM TXT 917672 Zapatillas de deporte Black Sail Dark Gray 002 , Sandalia Omnium de mujer Teva Pizarra Zapatillas náuticas para mujer Saucony Originals Carbón de leña / rosa Chaco Women s Z2 Classic Athletic Sandal Eclipse total de Yellowstone , Zapatillas Drew Shoe para mujer Paige Ternero negro , Zapato de running ASICS Gel-Quantum 360 cm para mujer Melocotón negro / Onyx , HOKA ONE ONE Recuperación ORA para mujeres Blanco negro Zapatillas de running La Sportiva Helios 2.0 para mujer - SS18 Verde Zapato acuático NRS para hombre Armada Zapato para caminar WW411v2 New Balance para mujer Blanco / rosa , Zapato de entrenamiento cruzado Ryka Women s Influence 2 Black / Purple Ash / Mint Ice CLARKS Mujer Gracelin Art Mary Jane Flat Cuero Tan oscuro , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Arena Skechers Sport Skech Air 2.0 City Love Fashion Sneaker Azul marino / azul claro , Skechers Women s Go Walk Joy-Paradise Sneaker Azul marino / blanco Zapatillas de tenis Yonex Power Cushion Eclipsion para mujer, blancas / grises , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Blanco sin blanquear , Sandalia de cuña ECCO Women s Shape 35 Wedge, 2 correas, mujer Negro , Sandalias planas de mujer de Easy Street Coñac , Sandalia para mujer Kate Spade New York Coby para mujer Oro , Zapatillas de tenis adidas Ligra 5 W para mujer Tinta noble / oro metálico / tiza púrpura , Pantuflas Kris de Katy Perry para mujer Rojo verde , Botas altas kensie para mujer Negro Zapatilla de UGG Women s Lane Bronceado , TOMS Kids 10009918 Alpargata-K (Marina, microfibra) , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue rojo , FRYE Botas de mujer Danica Peep Bootie Sand Soft Oiled Suede Allrounder de Mephisto Womens Lets Go Lace up Shoes Liquen Blue Mesh / Teal Suede , Zapato femenino Keli de Alegria Estaño , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Caoba roja

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 19643 de running running NIKE Blanco-gris Zoom para mujer Negro/ Blanco-gris oscuro ae9a240 - 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 19643 de running running NIKE Blanco-gris Zoom para mujer Negro/ Blanco-gris oscuro ae9a240 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.

Carretera en funcionamiento
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1703160
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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.