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 Zapatillas running Fresh Life Life Arishi V1 para 19805 mujer New Balance Estratos 466f7b7




Sintético / malla superior
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Fresco Entresuela de espuma
sin costura aplicación material
NB Rendimiento de la respuesta 1.0 insertar
No-Sew aplicación material
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 Zapatillas running Fresh Life Life Arishi V1 para 19805 mujer New Balance Estratos 466f7b7 - 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 Zapatillas running Fresh Life Life Arishi V1 para 19805 mujer New Balance Estratos 466f7b7

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 Zapatillas running Fresh Life Life Arishi V1 para 19805 mujer New Balance Estratos 466f7b7

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 ASICS GT-2000 6 para mujer Gris medio / plata / carbono , Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Volt / Blue Lagoon / Blanco , Zapatillas NIKE Air Max 90 Ultra 2.0 Flyknit para mujer White Dark Raisin 102 Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello , Zapato para correr Saucony Women s Cohesion 9 Plata / azul marino / verde azulado Zapatillas de running ISO para mujeres de la guía Saucony Pizarra / Melocotón , Zapatillas de running adidas Originals Women s Tubular Dawn Originals Trace Cargo , Zapatillas adidas Ultraboost X All Terrain LTD Shoe para mujer Negro Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Blanqueado Lila / Rosa Blast-white , Calzado de running Filaity Memory Finity de Fila Negro, blanco, plata metalizado Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Negro / rosa pavo real / ciruela Caspia Calzado de running Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle Zapato de running ASICS para mujer Torrance Negro / Gris piedra Zapatillas de running Nike Zoom para mujer Cool Grey / Black / Fire Pink / Dk Grey Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Púrpura / rosa / verde azulado Zapatillas de running con amortiguación W560v7 para mujer New Balance Gris negro , Calzado deportivo de punto Chelsea con memoria de Fila para mujer Blanco / Blanco / Highrise , Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / Blanco / Rosa , Zapatillas de running Reebok para mujer Negro / Rosa ardiente / Blanco NIKE Women s Season TR 5 Cross Trainer Stealth / Pink Pow / Cool Gray / White , En running para hombre Cloudflow Malibu / Neon Sneaker - 12 D (M) US - Zapatillas de deporte clásicas para uso general Summer Utility WL696 para mujer de New Balance Playa Arena / Libélula Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Racer Rosa / Antracita / Cool Gray Zapato deportivo para acolchado 680v5 New Balance para mujer Silver Mink Under Armour Women s Dash 2 Apollo Gray (101) / Blanco Zapatillas de running NIKE Mens Lunarglide 8 Racer azul / negro , Zapatillas de running ASICS Gel-Kayano 23 para mujer Púrpura / Plata / Menta Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Negro / Blanco / Antracita , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Negro / Blanco / Antracita
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 Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Recordar / Obsidiana azul Sandalia de mujer KEEN Rose Aluminio / gris neutro Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Negro Chaco Zcloud 2 Sport Sandal para hombre Tri Navy Zapatillas de running Nike Kids Air Max Zero Essential GS Light Bone Black White 001 Sandalias KEEN Mujer Whisper-w Sterling Blue / Vestido Azul Chaco Women s ZX2 Classic Athletic Sandal Haz de luz , Calzado de running Vitality V de Fila para mujeres Fila Navy / Aruba Blue / Fuchsia , Zapatillas Altra AFW1755F Mujeres Lone Peak 3.5 Púrpura / Naranja Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Atlético femenino Zapatos - 910 Sandalia Alegria Mujer Anah Strap Negro Bomba natural Nora para mujeres champán , Zapatillas de deporte Reebok Print Run Dist para hombre Negro / Aleación , Hush Puppies Walker clásico de las mujeres Cuero blanco Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Blanco , Zapatillas de running ASICS Gel-Contend 4 para mujer Rosa Brillo / Plata / Negro , Easy Spirit Women s Traveltime Mule Azul , Zapatillas de entrenamiento PUMA Women s Pulse XT V2 Prisim WNS Cross-Trainer Puma White / Puma Silver , Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Armada , Zapato para caminar Skechers Go Joy para mujer Blanco negro , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre (Cheque de dos tonos) Azul imperial / Blanco verdadero , Sandalia Haviva Slide Dolce Vita para mujer Tela de marfil Sandalia plana Kip de Trotters para mujer Equipaje , Plataforma CLARKS Women s Parram Waltz Cuero negro , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Partícula Rose / Flash Crimson-thunder Gray , Sandalia Birkenstock de cuero Florida para mujer Cuero engrasado Zapatillas de casa de lana de invierno U-lite cálido cálido Shearling de las mujeres, zapatillas de lana caliente para las mujeres Castaño (Gamuza de piel de oveja) , SoftWalk Womens Relax Negro / Gris Lycra / Microfibra , Bota de mujer Rieker Felicitas 84 Ozean / Mogano / Multi , Trotters Liz Loafer para mujer Estaño

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 Zapatillas running Fresh Life Life Arishi V1 para 19805 mujer New Balance Estratos 466f7b7 - 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 Zapatillas running Fresh Life Life Arishi V1 para 19805 mujer New Balance Estratos 466f7b7 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 2719393
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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.


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