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.

Mordenmiss - Sandalias Huarache de cuero, para mujer, Huarache B06ZY6WZXC de de verano, zapatos planos de tacón Estilo 1-negro be20a1b




El estilo 1 está hecho de piel de vaca, Estilo 2 es de cuero artificial
Único, lindo, bien construido zapatos, puede usar en primavera, verano y otoño
Para cualquier ocasión: hogar, trabajo, ropa de calle, fiesta, playa, travl, vacaciones......
Esto los zapatos no son anchos forma, si tienes pie gordo, sugerir para elegir un tamaño Arriba
Limpiar con limpiador de cuero
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


Mordenmiss - Sandalias Huarache de cuero, para mujer, Huarache B06ZY6WZXC de de verano, zapatos planos de tacón Estilo 1-negro be20a1b - 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

Mordenmiss - Sandalias Huarache de cuero, para mujer, Huarache B06ZY6WZXC de de verano, zapatos planos de tacón Estilo 1-negro be20a1b

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.


Mordenmiss - Sandalias Huarache de cuero, para mujer, Huarache B06ZY6WZXC de de verano, zapatos planos de tacón Estilo 1-negro be20a1b

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

Wolky Comfort Jewel 80200 gris metalizado de cuero metálico Sandalias de encaje Ros Hommerson para mujer Black Snake Print OluKai Ohana Pewter / Black Sandalia atlética de Rka Women s Glance Azul / Menta / Coral , Chaco Women s ZX2 Classic Athletic Sandal Barras de orquídea Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7643 negro azul Sandalia deportiva Tirra para mujer Teva Taupe Multi Sandalia Drew Angela para mujer Bone Smooth Leather Sandalia deportiva ECCO Terra 3S para mujer Dark Shadow / Emerald , Columbia Women s Barraca & shy; Sunlight Athletic Sandal Tin Grey Steel / Stream Sandalia deportiva Zvolv X de Chaco para mujer Arco caliente OluKai Ohana Burbujeante / negro , Sandalia de mujer W Verra Sport de mujer Teva Atlántico Norte Skechers Performance On The Go 400 Vivacity Flip Flop Blanco negro Chaco Women s ZX2 Classic Athletic Sandal Crest Citrus Sandalia KEEN Women s Venice H2 Gruta Celeste / Azul Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Negro Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Lago Azul / Gris , Encaje de quimera de Columbia para mujer Ballena / Dark Mirage KEEN Hombres Newport H2 Sandal Azul marino / gris medio , Sanda Verra Teva para mujer Deep Teal , Sandalia Dusty Sport mujer muy volátil Negro , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave Sandalias de encaje Ros Hommerson para mujer Impresión de serpiente blanca Skechers sandalia ancha de deporte 600-brilliancy para mujer Armada Sandalias KEEN Venice II H2-W para mujer Skydiver / Blue Opal Clorts Women s Whisper Sandal Outdoor Seaside Senderismo Zapatillas de agua SD207 Rosa , OluKai Ohana Blanco / oscuro Java Skechers Women s on-The-Go 600-15315 Wide Sport Sandal Gris pardo
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 Fresh Life Arishi V1 para mujer New Balance Gris oscuro Zapatillas de running Saucony Women s Triumph ISO 4 Niebla / gris , Zapatillas de running NIKE Air Max Zero Premium para hombre Gimnasio Rojo / Gimnasio Rojo / Gris Lobo , NIKE Men s Flex 2017 RN Negro Gris oscuro Gris oscuro , Sandalia Merrell Tetrex Crest Wrap Sport para mujer Legion Blue Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo , Calzado casual Slip-On de Bernie Mev para mujer Bronce , Zapato de running ASICS Alpine XT para mujer Gris Pizarra / Gris Pizarra Short NIKE Tempo para mujer Negro / Negro / Brillante Magenta / Plata mate Camiseta adidas Swift Run W para mujer Maroon / Maroon / Blanco , Sandalia de susurro de mujer Keen Negro / Gris neutro Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Gris tres, negro, tiza coral Zapatillas de running NIKE Air Huarache para hombre Hueso de Goma Ligera / Orquídea Amarillo , Yellow Box Mujeres Janalee Armada MAIERNISI JESSI Clásico de mujeres para hombres Estiramiento de dos correas Sandalias de tacón bajo (Las mujeres necesitan 1.5-2 Tamaño abajo) Negro Zapatillas de deporte Romy de Easy Spirit para mujer Blanco negro LifeStride Womens Corinne Blanco Calzado de senderismo Go Joy 15602 de Skechers para mujer Negro / rosa caliente Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Blanco apagado / blanco Vans Auténtico New York Yankees / Navy Sandalia Cato Dolce Vita para mujer Ante azul marino Supra Vaider LC Sneaker Verde oliva / blanco Zapatilla Saffron2 Easy Spirit para mujer Gris , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2702- Silver Stardust & Silver Leathe Ariat Womens Devon Pro VX Paddock Negro , VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Blanco negro , Regimiento de mujeres de Lugz Hi Fleece Water Resistant Rust / Cream / Gum Thermabuck , Bota Cwen Gwen de Hush Puppies para mujer Ante negro , Sperry Womens Sky Sail Suede Driftwood , Bota de moda Pascal de Dr. Mortens Black Military Canvas Negro ,

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.

Mordenmiss - Sandalias Huarache de cuero, para mujer, Huarache B06ZY6WZXC de de verano, zapatos planos de tacón Estilo 1-negro be20a1b - 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 Mordenmiss - Sandalias Huarache de cuero, para mujer, Huarache B06ZY6WZXC de de verano, zapatos planos de tacón Estilo 1-negro be20a1b 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 206838
1917505
2323017
3831104
10120103
10532695
13400609
15601667
16915257
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.