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.

Zapato para de running Pink ASICS Gt-2000 5/ para mujer Negro/ Bay/ Diva Pink 010acdb




Malla / sintético
Importado
Caucho suela
I.G.S (Orientación de impacto) Sistema) Tecnología: Filosofía de diseño ASICS que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.
Dinámico Soporte DuoMax Sistema: esta evolución del sistema DuoMax mejora la estabilidad y apoyo, con peso reducido y plataforma aumentada apoyo.
Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.
FluidRide Entresuela: FluidRide proporciona lo último combinación de recuperarse y propiedades de amortiguación con peso reducido y excepcional durabilidad.
Orientación Sistema Trusstic Tecnología: Esto Sistema Trusstic integra orientación Construcción de línea para una marcha mejorada eficiencia mientras proporcionar el mediopié integridad estructural.
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


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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

Zapato para de running Pink ASICS Gt-2000 5/ para mujer Negro/ Bay/ Diva Pink 010acdb

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.


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

Calzado deportivo New Balance para mujer Vazee Rush V3 Vivid Coral / Fiji Short NIKE Tempo para mujer Negro / Rojo WMNS Zoom Fly Apenas gris / Sequoia , Zapatillas de running Under Armour Speedform Intake 2 para mujer Puerto marítimo (300) / Marfil Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Peacoat Zapatillas Reebok para mujer Trilux Run PNT Sneaker Prnt - Negro / Blanco Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Grey / Rose Gold , Zapatillas de running Kinvara 8 Saucony para mujer Azul Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / hierro blanco púrpura Zapatillas de running AMPLICA para mujer ASICS Gris Medio / Plata / Aruba Azul , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Utility Gray / Utility Gray / Maroon Zapatillas de running adidas NMD_CS2 PK Originals para mujer Lino / Correr Blanco Calzado de levantamiento de pesas Fastlift 335 para mujeres Inov-8 Rosa / Púrpura / Azul Zapatillas de running NIKE Huarache Run SE para mujer Negro / Negro-blanco , Zapatillas para correr Nike Women s Free Rn Distance 2 Carbón de leña blanco / Sangria-light , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco Zapatillas de running NIKE Mens Air Zoom Structure 19 Blanco / negro / verde eléctrico , adidas Yeezy Boost 700 Gris sólido / Tiza Blanco con núcleo negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Binario Azul / Blanco Amortiguación 460v2 New Balance para mujeres Azul plateado Zapatillas de running New Balance 520v3 para mujer Negro , Amortiguación 635v2 New Balance para mujer Plata EN Zapatilla Cloudster para mujer Magenta / Berry Zapato para correr ASICS Conviction X 2 para mujer Negro / Carbón / Flash Coral , Zapatillas de running Reebok Zpump Fusion 2.0 para mujer Aleación / Gris Estaño / Rosa Solar / Carbón / Blanco Zapatillas de running Laguna Zoot para mujer Punch / gris , Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Negro / Negro / Blanco Zapatillas de running ISO para mujeres de la guía Saucony Gris / menta , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul amarillo
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 Sandalia Terrell Ivy Wrap Sport Merrell para mujer Negro Zapatillas de running NIKE Unisex Flyknit Racer Negro / Blanco-blanco Botas de agua y botiquín para mujeres Salomon Women s Crossamphibian Swift Deep Peacock Blue / Ceramic / Living Coral Sandalia Deportiva Jordan Spires de Via Spiga Albaricoque gamuza Earth Origins Sophie Negro , Converse Chuck Taylor All Star Seasonal Color Hola rojo , Wolky Comfort Jewel Patente metálica vieja de cobre Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , Zapatillas para correr Mizuno Women s Wave Rider 17 Blanco Alpargata Chambray Clara denim de mujer TOMS Chambray / Cuero texturizado negro J.Renee Womens Soncino Blanco / Kidskin Easy Spirit Solana Oxford para mujeres Gris , Sandalias Deynee3 Easy Spirit para mujer Luna de oro / Luna de oro WHITE MOUNTAIN Moon Sandalia para mujer Cuero blanco Zapato de malla Nohea de OLUKAI - Hombre marrón , Drew Shoe Women s Paradise II Velco Slip-on Gris pardo , Comfortiva Mujer, Faye Sandalia de tacón medio Whiskey Otago Nubuck , Zapatillas de running Golden Spike de Altra para hombre Lima / Azul Under Armour Women s Charged 24/7 Low Negro (001) / Blanco , Sandalias Jerusalem Sandalia David Slide marrón La sandalia plana Naomi con diseño de diapositivas extra grandes para mujer de Fix Denim Textile , Sandalias planas Nori de Easy Street para mujer rojo , Diapositivas de Sam Edelman Gertrude para mujeres entrecruzadas Red Kid Suede Leather Sansha Salsette 1 Zapatillas de baile unisex Gris carbón , Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Negro / púrpura Zapatillas de casa de gamuza con forro de piel sintética de mujer de HomeIdeas, mocasines de exterior de interior transpirables marrón , Minnetonka Womens Chrissy Slipper Bootie Negro / Tostado , Bomba de vestido animado de LifeStride para mujer Tierno Taupe , Chinese Laundry Women s Cecilia Oxford Patente negra , Zapatos antideslizantes clásicos de lona de mujer de TOMS 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.

Zapato para de running Pink ASICS Gt-2000 5/ para mujer Negro/ Bay/ Diva Pink 010acdb - 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 Zapato para de running Pink ASICS Gt-2000 5/ para mujer Negro/ Bay/ Diva Pink 010acdb 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 1290570
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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.