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 de running NIKE B019DKDB44 Roshe Two 18445/ para mujer Caqui/ Negro/ Negro/ Blanco e3e4c21




Malla
Caucho único
acanalado el diseño cambia entre 2 colores como tú mover
Entresuela características 3 espuma densidades para superior amortiguación
Bootie la construcción entrega un ajuste cómodo ajuste
acolchado collar proporciona amortiguación de felpa
Sockliner hecho de lenta recuperación espuma aumenta la comodidad y amortiguación
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 de running NIKE B019DKDB44 Roshe Two 18445/ para mujer Caqui/ Negro/ Negro/ Blanco e3e4c21 - 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 de running NIKE B019DKDB44 Roshe Two 18445/ para mujer Caqui/ Negro/ Negro/ Blanco e3e4c21

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 de running NIKE B019DKDB44 Roshe Two 18445/ para mujer Caqui/ Negro/ Negro/ Blanco e3e4c21

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 deportivas para hombre NIKE Lunar Apparent Índigo ligero de carbono / obsidiana neutral Camiseta adidas Swift Run W para mujer Núcleo Negro / Blanco / Núcleo Negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Binario Azul / Blanco Zapatillas de running New Balance Zante v4 Fresh Foam para mujer Rosado Calzado de running casual para mujer New Balance WL555, para mujeres Crema , NIKE Air Max 90 Womens Style: 325213 Mujeres 325213-048 Lobo Gris / Blanco / Gris Oscuro , Zapatillas de running Brooks Women s Glycerin 16 Road Negro / Rosa / Gris , Zapatillas de running New Balance Urgev2 para mujer Alpha Pink / Vintage Indigo , Zapatillas de running adidas Cloudfoam Super Flex W para mujer Misterio Azul / Blanco / Naranja Brillante , Zapatillas de running Kinvara 6 Saucony para mujer Blanco / azul / coral New Balance Women s w990v4 Running Shoe Armada , Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower , Zapatillas de entrenamiento Primus Lite para entrenamiento de mujer Vivobarefoot Todo negro , Brooks Womens Levitate Negro / Ébano / Plata , Zapatillas de running NIKE Roshe Two para mujer Midnight Navy / Midnight Navy , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Azul índigo / negro / amarillo de seguridad Adidas Women s EQT Support ADV Originals Zapatillas para correr Correr Blanco / Core Black , Zapatillas de running NIKE Roshe Two Flyknit V2 para mujer Blanco / Lobo Gris Blanco Blanco Calzado de running New Balance para mujer 530v2 de New Balance Ártico Fox / amanecer blanqueado Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Pedernal , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco Zapatillas de running Kinvara 8 Saucony para mujer Gris , Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Burdeos / Desert Moss Zapato de tenis ASICS Gel-Dedicate 5 para mujer Blanco / Porcelana azul / azul índigo , WMNS Zoom Fly Equator Blue / Summit White-lagoon Pulse , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Brooks Women s Ravenna 8 Plata / Clematis Azul / Muy Berry , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Violeta profundo / Amanecer
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 Sandalias Therafit Summer Women s Slingback Sport - para fascitis plantar / dolor en el pie Negro , Zapatillas de running NIKE Juvenate para mujer Hueso Ligero / Hueso Ligero / Negro / Blanco Zapatillas de running La Sportiva para mujer Trailido Bushido Gris / menta , Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Easy Coral, Negro, Blanco , Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Azul marino / azul claro , Sandalia Jenley de Lucky Brand para mujer marrón SJP de Sarah Jessica Parker Mujer Meteor Sneaker Pistacho Gamuza Zapatillas de running con amortiguación W560v7 para mujer New Balance Gris negro , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina , Sandalia de vestir de plataforma Preya de Naturalizer para mujer Color de malva , Sandalias FitFlop Mujer Lulu Cross Slide Shimmer Print Negro , New Balance WW577 Hook and Loop Walking para mujer Hueso , Diadora Unisex I.C. Negro / Plata Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Monocromo negro Sandalia plana Sofie de Badgley Mischka para mujer Plata , Sam Edelman Sandalia Glenn para mujer Cuero negro , Sandalias de mujer Amalia de Donald J Pliner Adular Vince Camuto Sandalias Ellanna Slide para mujer Brillante plata Sandalia Ryana Marilyn Slide para mujer Blanco negro Sandalia FitFlop Glitzie Slide para mujer Desnudo , Sandalias planas para piscina Walking Cradles para mujer Abrigo blanco de cachemira / corcho , Zapatillas de salón de baile femenino de Very Fine CD6017 con HP 2.75 \ Satén negro , Bota Ecuestre Lanesha de Lucky Brand para mujer Ivy Green , Botín Wagner para tobillo de mujer Steve Madden Ante negro Rieker Womens L.Slipper Black Negro , Nine West de mujer Garroy Leather Oxford Flat Cuero negro Cole Loan de Conductor de borla de rodeo de mujer de Cole Haan Brillo de Platino , Propet Women s Tour Walker Strap Sneaker Blanco azul , Mocasines de vestir de punta estrecha para mujer de Me Too Women Negro , KR Strikeforce L-053-085 Mist Zapatos de bolos, blanco / negro / azul, talla 8.5 -

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 de running NIKE B019DKDB44 Roshe Two 18445/ para mujer Caqui/ Negro/ Negro/ Blanco e3e4c21 - 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 de running NIKE B019DKDB44 Roshe Two 18445/ para mujer Caqui/ Negro/ Negro/ Blanco e3e4c21 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 173866
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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.