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.

Sandalia deportiva Teva B0751PL94Q W deportiva 19072 Hurricane 4 para mujer Hazel Black 6f01296




Textil
Importado
Sintético suela
Nylon caña estabiliza y apoya su pie en desigual terreno
Resistente Durabrasion Rubber la suela es extremadamente durable y suministros gran tracción
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


Sandalia deportiva Teva B0751PL94Q W deportiva 19072 Hurricane 4 para mujer Hazel Black 6f01296 - 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

Sandalia deportiva Teva B0751PL94Q W deportiva 19072 Hurricane 4 para mujer Hazel Black 6f01296

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.


Sandalia deportiva Teva B0751PL94Q W deportiva 19072 Hurricane 4 para mujer Hazel Black 6f01296

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

Sandalias Columbia Sorrento Flip Athletic para mujer Negro, grafito Skechers Performance Women s The Go 400 Radiance Flip Flop Negro , Amalfi de Rangoni Women s Candito Sport Sandal Grafite / Black Nubuck Liz , Sandalia KEEN Clearwater CNX para mujer Negro / Resplandor Sandalia de Hollyleaf de las mujeres de Merrell Negro , Sandalia deportiva Hammond de Spring Step para mujer gris Sandalia para exteriores ECCO Women s Offroad Lite Rainer Sandal Outdoor Blanco / Sombra Blanco , Sandalia NIKE Kawa Slide para mujer Hot Punch / Paramount Blue CLARKS Womens Wave Grip Smokey Brown Chaco Mujer Zvolv 2 Sport Sandalia Celtic Azurite Sandalia deportiva Kankitta de Vince Camuto para mujer Combo puro , Wolky Comfort Jewel Beige , Sandalias Newport H2 de KEEN para mujer Raya Black Sandalia Merrell de mujer con gladiador de poste Dusty Blue Sandalia deportiva ECCO Terra 3S para mujer Arona / Papaya , Earth Origins Sophie Frost Grey Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Helecho Huracán XLT Mujeres Teva - Avellana / Negro Hazel / Black , Wolky Comfort Jewel Estampado de flores negro Sandalia con flecos de cuero original Teva Women s W Olivo oscuro Sandalias de tirantes Sunstone para mujer Merrell Vertiver Sandalia atlética Salomon Women s Crossamphibian Swift W Higo , CLARKS Sandalia de pescador de melón Tuvia para mujer Azul Sandalia atlética de Rka Women s Glance Negro / rosa , Sandalia deportiva Tirra para mujer Teva Negro / Blanco Multi , Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia Azul Sandalias Swiftwater Mesh de Crocs para mujer Gris claro Sandalia NIKE Kawa Slide para mujer Hyper Punch / Negro / Blanco Chaco Women s ZX2 Classic Athletic Sandal Palm aguacate ,
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 KEEN Venice II-W para mujer Aceite de visón / Tierra oscura Flip-flop Teva Women s Mush II Santori Tribal Orange , KEEN Hombres Newport H2 Sandal Negro Zapatillas de running Under Armour Speedform Intake 2 para mujer Negro (001) / Acero Sandalia Yucatan ECCO Mujer Atmósfera / Hielo Blanco / Negro Earth Origins Sophie Escarlata Chaco Zcloud 2 Sport Sandal para hombre Negro , Zapatillas Altra Women s Superior 3.5 Sneaker Gris / morado , New Balance Women s w990v4 Running Shoe Gris / rosa Saucony Women s Ride 10 zapatillas para correr Púrpura Citron , Zapato de trail WT1210 para mujer New Balance Gris / rosa , Zapatillas de trail WT510V2 para mujer New Balance Plata / rosa , Zapatos OLUKAI Pehuea - Mujer Granadina / Granadina Zapatillas para caminar Skechers Performance Go Walk 3 Strike para mujer Negro , La sandalia de gladiador FLEXX Women s Band Together Cherry Nubuck Nike Hombres Air Zoom Pegasus 33 Negro / blanco / antracita / gris fresco , Zapatilla deportiva Kailee de mujer Ryka Gris azulado La sandalia para mujer FLEXX Shoreline Coñac , NIKE Men s in-Season 7 Cross Trainer Gris oscuro / Barely Volt - Aurora Verde , Zapato de entrenamiento cruzado Yui con tratamiento en gel femenino de ASICS Negro / Plata / Blanco , Zapatillas de tenis adidas Barricade Court para mujer Negro / plata mate / rojo destello Clásicos de base Slip-on (Tm) para hombres de Vans Llama Negro Taryn Rose Women s Lana Sheep Nappa Slide Sandalia Negro Sandalia de mujer Lucienne Suede Slide de Nine West Ante negro , Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Denim azul claro Bota Ecuestre Miss141fly FLY London para mujer Alfombra Camel / Suede de Aceite , VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Pewter / Black , 206 Cojín de mocasín de piel de oveja Pearson de mujer Tela escocesa roja , Tempur-Pedic Darla Clog de las mujeres Chocolate , Zapatillas de fútbol MW-Tech RB R TF para césped Diadora para hombre Blanco / Negro / Fluo Amarillo ,

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.

Sandalia deportiva Teva B0751PL94Q W deportiva 19072 Hurricane 4 para mujer Hazel Black 6f01296 - 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 Sandalia deportiva Teva B0751PL94Q W deportiva 19072 Hurricane 4 para mujer Hazel Black 6f01296 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2941571
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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.