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 de running ASICS Alpine XT para de mujer Azul Fucsia Morado Morado/ Fucsia Morado/ Azul índigo 3f6f2f0




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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


Zapato de running ASICS Alpine XT para de mujer Azul Fucsia Morado Morado/ Fucsia Morado/ Azul índigo 3f6f2f0 - 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

Zapato de running ASICS Alpine XT para de mujer Azul Fucsia Morado Morado/ Fucsia Morado/ Azul índigo 3f6f2f0

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.


Zapato de running ASICS Alpine XT para de mujer Azul Fucsia Morado Morado/ Fucsia Morado/ Azul índigo 3f6f2f0

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 Topo Athletic Ultrafly - Mujer Púrpura / Lila , Zapatillas de deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Vino de Oporto Blanco 602 Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Blk / azul Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / gris antracita-cool , adidas Performance Ultraboost X para mujer Misterio Ruby / Core Black / Trace Pink Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Añada Indigo / Sal Marina Amortiguación 635v2 New Balance para mujer Gris , Under Armour Women s HOVR Sonic NC Blanco (102) / Elemental Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Ash Gray / Black / Sour Melon / Pewter / White / Alloy NIKE W Air Vapormax Flyknit C para mujer Ah8448-001 Atmósfera Gris / Blanco-blanco-caliente Punch Zapatos para caminar Altra Provision para mujer Blanco Zapatillas de running Adidas Women s Energy Cloud WTC W Fácil menta / blanco / negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hyper Turquoise / Pure Platinum-Igloo Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Negro-blanco-gris oscuro , Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul Zapato de running ASICS GT-2000 5 para mujer Regatta Blue / Flash Coral / Indigo Blue Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Volt / Blanco Zapato deportivo ASICS Gel-Cumulus 18 para mujer Blanco / azul índigo / lavanda , Zapatillas para correr Saucony Progrid Echelon 3 para mujer Blanco / verde / cidra NIKE Men s Flex 2017 RN Negro / Metálico Cool Gray / Light Carbon , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Wolf Gray / Volt-black , Brooks Women s Ravenna 8 Negro / Plata / Acuario Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Binario Azul / Blanco Zapato de running ASICS GT-2000 4 para mujer Rosa Brillo / Calmante Mar / Azul Índigo , Short NIKE Tempo para mujer Pizarra Púrpura / Pizarra Púrpura / Gris Lobo , Zapatillas de running Nike Kaishi para hombre Negro / Metálico Plateado-blanco ,
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 Kids gratis Rn (grande) Negro / antracita / gris oscuro Calzado de running rápido para mujer Reebok, tejido flexible Carbón / Blanco / Moonglow / Ácido rosa Zapatillas de running NIKE Unisex Flyknit Racer Volt / Black-secoya Zapatillas de running Mizuno para mujer Synchro MX Blue Atoll / White Zapatillas de running PUMA para mujer Ignite Omphalodes / Coral caliente / Ultramar Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar _Blanco negro , Sandalia de cuña de Easy Street para mujer Champagne Snake Drew Shoe Women s Tulip Comfort Croc negro Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina , Easy Spirit Solana Oxford para mujeres Nuez , Sandalia plana Halle de Walking Cradles para mujer Serpiente negra / plateada Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / Begonia rosa , Sandalia estampada Benikei JDI de mujer Nike Blanco Blanco , Tierra Orígenes Shane Negro , Easy Spirit Women s Traveltime Mule Azul marino / ante blanco , Adidas Mujeres Aleki X Cross Tainer Amarillo / Amarillo / Aqua Zapato de tenis K-SWISS Bigshot Light para mujer Blanco / Océano Hawaiano , Zapatos de entrenamiento Nike Free TR 6 para mujer Negro / Carmesí total - Blanco Sandalia de cuero Nirveli de Nine West para mujer Cuero negro , Sam Edelman Women s Bay Slide Sandal Sugar Pink , Sandalia de plataforma de playa Coast de Seychelles Blanco Sandalias planas Nori de Easy Street para mujer Pewter Metallic CORRAL Señoras Marrón Crater Bone Bordado Western Boot marrón 3N2 Mofo Turf Trainer Negro Zapatilla de hombre viejo amigo castaña , Zapatillas Zeppa en suela francesa para mujer Nappa negro , Bomba Parigi de LifeStride para mujer Negro brillante , ED Ellen DeGeneres Womens Garance Cuero blanco , TOMS Kids 10009918 Alpargata-K Marina, microfibra CLARKS Women s CloudSteppers Sillian Jetay Flat Microfibra perfilada 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.

Zapato de running ASICS Alpine XT para de mujer Azul Fucsia Morado Morado/ Fucsia Morado/ Azul índigo 3f6f2f0 - 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 de running ASICS Alpine XT para de mujer Azul Fucsia Morado Morado/ Fucsia Morado/ Azul índigo 3f6f2f0 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 945754
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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.