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 Edge de running adidas Women s Edge gris Rc Rc W Blanco/ gris claro/ blanco 8656467




Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Runner tipo: neutral
BOUNCE proporciona energizado comodidad para todos deportes, todo el día
Ligero malla técnica superior
STRETCHWEB se adapta a cada Picada del corredor
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 Edge de running adidas Women s Edge gris Rc Rc W Blanco/ gris claro/ blanco 8656467 - 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 Edge de running adidas Women s Edge gris Rc Rc W Blanco/ gris claro/ blanco 8656467

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 Edge de running adidas Women s Edge gris Rc Rc W Blanco/ gris claro/ blanco 8656467

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 correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , Zapatillas de running adidas Ultraboost W para mujer Unity Ink / Unity Ink / Dark Navy , Zapatillas de running NIKE Air Huarache para hombre Luz de hueso / piedra pómez , NIKE Men s Flex 2017 RN Negro Gris oscuro Gris oscuro , Zapato de running Saucony Zealot Iso 2-w para mujer Plata / Menta / Coral , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris oscuro , Zapatillas de running NIKE RN 2017 para mujer gratis MULTI-COLOR / MULTI-COLOR Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Blanco negro , Adicción a Brooks 12 Antracita / Bluefish / Silver , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul , Zapatillas de running 430v1 New Balance para mujer Negro / cardo Zapatillas de running ASICS GT-2000 6 para mujer Negro / Blanco / Carbono , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Gris medio / Aruba / gris medio Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower , Brooks Women s PureFlow 6 Heather / Marañón anacardo / cocodrilo Zapato de running de punto fuzeX para mujer ASICS Carbono / negro / blanco Zapatillas de running Nike Air Huarache Run Ultra para hombre Negro / Negro / Negro , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Gris medio / Aruba azul / gris medio , Zapatillas de running HOKA ONE ONE Clayton 2 para mujer Azul Medieval / Fucsia , Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-blanco-negro Zapatillas de correr Saucony Liberty ISO para mujer S10410 Niebla / azul , Skechers Performance Women s Go Run 400 Obstruct Negro / Aqua NIKE Women s Revolution 4 Running Shoe Platino Blanco / Blanco-puro , Zapatillas de running ASICS Gel-Contend 4 para mujer Diva Azul / Plata / Orquídea Saucony Women s Ride 10 zapatillas para correr Gris , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Corona Azul / Teaberry / Stonewash , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Puma White
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 para correr ASICS Gel-DS Trainer 22 para mujer Azul / morado / negro / coral Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento CLARKS Womens Wave Grip Cuero blanco 2 Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco KEEN Sandalias con correa de mujer Maya Negro Zapatillas de running NIKE Air Max 2017 para mujer Platino blanco / negro puro Merrell Siren Edge Hiker para mujer Azul Zapatillas sin cordones Flex Merit para mujer Merrell Armada Zapatillas de trail running Pearl Trail i3 para mujeres Pearl iZUMi Vino Púrpura / Clementina , Zapatillas de running La Sportiva Helios 2.0 para mujer - SS18 Verde ASICS GT-2000 5 Trail Runner para mujeres Aluminio / Plata / Orquídea , Calzado casual NIKE Air Max 90 Ultra 2.0 Flyknit para mujer Platino Blanco / Blanco-puro , Sandalia Skechers Women s on-Go 600 Negro CLARKS Danelly Millie Sneaker para mujer Cuero negro Sandalia Tallis de mujer VANELi Marina de verano , Steve Madden Daphnee para mujer Cuero Cognac Naturalizer Samantha Pointed Toe Flat para mujer Oro negro Movimiento fácil de las mujeres del alcohol Cuero de concha Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Atlético femenino Zapatos - 910 Zapatillas Superga 2750 Cotu para mujer Gray Seashell Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor rojo , Easy Spirit - Zapatillas Faisal2 para mujer Rosado Vans Unisex Old Skool Zapatillas clásicas de skate Negro CLARKS Womens Rosilla Durham Cuero metalizado dorado claro , Sandalia ecológica ECCO Women s Offroad Lite Slide Negro Negro Zapato de bailarina Swing Tic-Tac-Toes para mujer Estaño Birkenstock Mules para mujeres Azul Sandalias de vestir Patti de Sam Edelman para mujer Greige Suede Zapatilla Nakiska SOREL para mujer Negro Zapatillas sin cordones Slip-On Sport Sport Roper para mujer, punta redonda, marrón marrón ,

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 Edge de running adidas Women s Edge gris Rc Rc W Blanco/ gris claro/ blanco 8656467 - 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 Edge de running adidas Women s Edge gris Rc Rc W Blanco/ gris claro/ blanco 8656467 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 731057
3423501
8449694
10907293
13276286
14824231
15252905
17977902
18014650
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.