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.

New Balance Women s w990v4 Running B00LBZ05BW Shoe w990v4 Gris/ Running rosa 5b18e61




Cuero
Hecho en los EE. UU.
Caucho único
ENCAP Anillo de PU con EVA entresuela central
ENCAP
3M Reflectividad
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


New Balance Women s w990v4 Running B00LBZ05BW Shoe w990v4 Gris/ Running rosa 5b18e61 - 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

New Balance Women s w990v4 Running B00LBZ05BW Shoe w990v4 Gris/ Running rosa 5b18e61

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.


New Balance Women s w990v4 Running B00LBZ05BW Shoe w990v4 Gris/ Running rosa 5b18e61

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 adidas Questar Ride para mujer Carbono / Negro / Gris Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul , Zapatilla para mujer Saucony tipo A8 Azul / Citron , Zapatillas de running ASICS Gel-Nimbus 18 para mujer Melocotón Melba / rosa fuerte / guayaba , Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Glaciar Gris / Blanco / Carbono , Zapatillas de running NIKE Air Zoom Mariah Flyknit Racer para mujer Cargo Khaki / Cargo Khaki aa0521-301 , Calzado informal NIKE Darwin de mujer Bright Mango Bright Crimson 881 , Zapatillas de running NIKE Air Max 90 LX para mujer Gris , Zapatillas para correr W630V5 para mujer New Balance Plata , Zapatillas de running adidas Aerobounce Pr W para mujer Núcleo Negro, Plata Met., Ftwr Blanco , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris claro / blanco Under Armour Women s Drift 2 Negro (001) / Blanco Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover , Short NIKE Tempo para mujer Fresh Mint / Black / Wolf Gre Zapatillas de running Brooks Women s Glycerin 16 Road Blanco / azul / rosa , Zapatillas de running mujer HOKA ONE ONE Clifton 4 Blanco negro Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Negro , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Lobo gris / negro-antracita , Zapatillas para caminar Reebok Women s Cloudride LS DMX Cráneo gris / gris plano / blanco / rosa venenoso Zapatillas de running ASICS Gel-Cumulus 19 para mujer Gris medio / Carbono / Amarillo de seguridad Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hyper Turquoise / Pure Platinum-Igloo , Zapatillas de running Saucony Women s Breakthru 3 Berry Coral , Zapatillas de running ASICS Gel-Kahana 8 para mujer Negro / Island Blue / Pink Glow , Zapatillas de running NIKE Lunartempo 2 para mujer Dinastía púrpura / Mango brillante Calzado de running NIKE RN 2017 2017 para mujer Black / Hot Punch / Aurora Green Zapatillas para correr Mizuno Women s Wave Rider 21 Trébol / blanco , Zapatillas de calle YKTM Performance Women Boost 4.0 Street Running Shoe Road Negro , Zapatillas de running NIKE Air Zoom Mariah Flyknit Racer para mujer Cargo Khaki / Cargo Khaki aa0521-301 adidas Performance Ultraboost X para mujer Trace Scarlet / Crayon White / Trace Orange
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 Borgoña / negro BZees Sandalia con cuña de beso para mujer Negro , Sandalia Drew Element para mujer Denim Blue Nubuck Sandalia Jacardia Merrell para mujer Armada Merrell Siren Edge Hiker para mujer Azul , NIKE Air Huarache Run Ultra Blanco / Blanco-Blanco-Negro (Mujer) Merrell Women s Mqm Flex Gris negro , Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Easy Coral, Negro, Blanco , Sandalias EVA Gizeh de mujer Birkenstock Khaki Eva Sandalia de cuero universal Arvabelle de Teva para mujer Negro , Prop & eacute; t Zapatillas para caminar TravelFit para mujer naranja SOREL Womens Joanie Wrap Sandal Negro Sandalias KEEN Rose City T Strap para mujer Embarcadero Earth Orígenes Mujeres Lorena Alpaca , Converse Chuck Taylor All Star Core Hola Blanco óptico , Zapatos de golf AsJei para mujer FootJoy Negro Propet Women s Mary Lou Mary Jane Sneaker Naval Blanca , Sandalia UGG Royale Slide para mujer Negro , Sandalias Birkenstock Unisex Arizona Tobacco Oiled Leather - 43 N EU / 10-10.5 2A (N) US Men - Sandalia Otto Slide de mujer Yellow Box Sonrojo 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2707- Satén blanco y adornos de plata # 119 , Bota vaquera Laredo para acceso de mujeres Negro , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Cáñamo azafrán , J. Rene233; Gris pardo , LifeStride Women s Lively 2 Pump Sonrojo , Rockport Mujeres Truwalk Zero Touring T-Toe Oxford Cálido roble La jerga de las mujeres chinas de lavandería Cuero Vainilla , Botines de cuero negros Toulouse de Rieker para mujer 00 negro , Walking Cradles Hardy Oxford para mujeres marrón TOMS - Slip-On de lienzo clásico para hombre rojo ,

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.

New Balance Women s w990v4 Running B00LBZ05BW Shoe w990v4 Gris/ Running rosa 5b18e61 - 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 New Balance Women s w990v4 Running B00LBZ05BW Shoe w990v4 Gris/ Running rosa 5b18e61 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 62629
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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.