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.

Calzado de running ancho NIKE para running mujer - Experience 7 Blanco de NIKE Negro/ Blanco - Blanco 1ee0a7e




Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Hexagonal ranuras de flexión en la suela promueve flexibilidad natural.
Heathered tejer ofrece peso ligero comodidad y transpirabilidad
Redondeado talón promueve un rango natural de movimiento
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


Calzado de running ancho NIKE para running mujer - Experience 7 Blanco de NIKE Negro/ Blanco - Blanco 1ee0a7e - 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

Calzado de running ancho NIKE para running mujer - Experience 7 Blanco de NIKE Negro/ Blanco - Blanco 1ee0a7e

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.


Calzado de running ancho NIKE para running mujer - Experience 7 Blanco de NIKE Negro/ Blanco - Blanco 1ee0a7e

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 Huarache Run SE para mujer Azul , Zapatillas para correr WRL247WM New Balance para mujer Estratos Zapatillas de running Reebok para mujer Zprint Her Elle MTM Gris Ceniza / Carbón / Vitamina C / Blanco Zapatillas de running NIKE Roshe Two para mujer Negro / negro-cool gris , Zapatillas de running Questr Cc W de mujer adidas Aero Pink S, Aero Pink S, Carbono S , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue Zapatillas de running W1540V2 New Balance para mujer Blanco azul Zapatillas de running NIKE Air Max Flair para hombre Blanco / puro platino , Zapatillas de running New Balance para mujer 940v3 Azul negro , Zapato de tenis ASICS Gel-Dedicate 5 para mujer Plata / Rojo Rojo / Blanco , Zapatillas Reebok Women s Runner 2.0 MT Track Negro / violeta violeta / peltre Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello , Zapato de running ASICS Gel-Quantum 360 cm para mujer Melocotón negro / Onyx , New Balance Women s 555 Lifestyle Fashion Sneaker-Suede / Mesh Zapatillas para correr Silver Mink / Gráfico Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Negro / sal marina ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Phantom / Black / White , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosado , Brooks Womens Hyperion Peacoat / azul marino / cacatúa ASICS Kanmei Shoe Casual para mujer Cosmo rosa / negro / ciruela , Brooks Womens Adrenaline GTS 18 Rosa / Plum / Champagne , Zapatillas de running NIKE Lunar Skyelux para mujer Wolf Gray / Black / Cool Gray / White , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Negro , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-negro Zapatillas de running Vibram para mujer V ostra , Zapatillas para correr Saucony Women s Cohesion 10 Armada , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Primer gris / verde azulado victoria / plata , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ecru Tint / Ash Pearl / Ash Pearl , Zapatillas de running adidas Women s Edge Lux W Ash Pearl / Chalk White / Chalk Coral Zapatillas de running NIKE Air Zoom Vomero 12 para mujer Platino blanco / negro puro ,
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 NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Negro / Blanco / Volt / Cool Gray , Topo ST-2 para mujer, zapatos para correr y ejercicio, banda para la cabeza, paquete Azul blanco , Sandalia universal W Sanborn de Teva para mujer Nuez , Zapatillas de running adidas Adizero Boston 7 para mujer Hi-res Aqua / Clear Mint / Mystery Ink adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Negro , Mizuno Women s & rsquo; s Wave Hayate 3 (W) Trail Running Shoes Negro (Dark Shadow / Zinnia / Black) Zapato acuático de fitness Speedo Hydroforce Xt para mujer Cuero gris , Sandalia Terra FI Lite de Teva para mujer Agua NIKE Men s Flex 2017 RN Sequoia / Black-medium Olive-dust , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Plano extra ancho Feet Breeze Womens Zapatos para caminar , Sandalias de cuero de Viv Viv de TOMS Multi Tribal , Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Cuero negro Kenneth Cole New York Mujer Kingcro Triple Hook y Loop Sneaker Blanco , Splendid Feodora Sandalia para mujeres Sonrojo Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco-negro-corredor Azul , Brooks Womens Adrenaline GTS 18 Azul / Menta / Rosa , Skechers Women s Walk Walk Lite-Daisy Loafer Negro , Sandalia de cuña Daytona SoftWalk para mujer Negro Sandalia Dionay Calvin Klein para mujer Negro 1.STATE Womens Celvin Black Mexico , Onitsuka Tiger Asics Unisex Lawnship 2.0 Borgoña / Borgoña adidas Originals Mujer Superstar Blanco / negro / blanco , Zapatillas de golf adidas W Adipure DC de adidas para mujer Ftwr Blanco / Real Coral / Silver Met. , Zapatillas de golf Skechers Go Golf Elite Ace Jacquard para mujer Carbón de leña / azul Birkenstock Mules para mujeres Azul Sandalia Cleo Birkenstock para mujer Cuero plateado metalizado , Calzado de ballet Freeform FF01 Femenino de Capezio Caramelo Zapatilla de deporte Blair Fashion de Naturalizer para mujer Negro , Zapato Folclórico FLT101 para Mujer Dance Class Negro Clásicos de base Slip-on (Tm) para hombres de Vans (Bum de goma) True White / True White

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.

Calzado de running ancho NIKE para running mujer - Experience 7 Blanco de NIKE Negro/ Blanco - Blanco 1ee0a7e - 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 Calzado de running ancho NIKE para running mujer - Experience 7 Blanco de NIKE Negro/ Blanco - Blanco 1ee0a7e 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 1864216
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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.