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.

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Sintético
Shoc Almohadilla en el talón taza transferencias equitativas la energía del impacto durante el plantilla.
Compresión entresuela de EVA moldeada proporciona un superior mezcla de amortiguación y absorción de impactos.
Nylon la caña agrega estabilidad y soporte para el mediopié.
Sintético parte superior de la cinta de nylon, parte superior de neopreno revestimiento, Microban Plantilla de zinc, Plantilla de EVA.
Compresión entresuela de EVA moldeada, espiga de nylon, araña suela de goma.
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


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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 para deportiva/ Tirra para mujer Teva Rasberry/ 11455 Dark 453ac33

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 para deportiva/ Tirra para mujer Teva Rasberry/ 11455 Dark 453ac33

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

Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul Chaco Women s ZX / 2 Yampa Sandal OXW Sandalia deportiva Zvolv X de Chaco para mujer Lavender Liberty Wolky Comfort Jewel Choque de Multi Picasso , Sanda Verra Teva para mujer Granadina Skechers Performance On The Go 400 Vivacity Flip Flop Taupe / Blanco KEEN sandalia de Venecia para mujer Tierra oscura / Mar Caribe , Chancleta Eezay adidas para mujer Blanco / negro / negro , Sandalias Newport H2 de KEEN para mujer Poseidón / Capri Sandalia de mujer Drew Alana Verde / Fucsia Sandalia Savannah de Ryka para mujer Beet Red / Hyper Pink Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7643 Naranja Azul Marino Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul , Sandalia deportiva Nolita para mujer Anne Klein Tela negra , Sandalia Terrell Post II Femenina de Merrell Fucsia , Drew Womens Cascade Nubuck marrón Sandalias Teva Mujer W Terra Fi Lite Atitlan Negro / Blanco Sandalia Terrell Post II Femenina de Merrell Gris pardo Sandalia KEEN Clearwater CNX para mujer Celestial / Vapor CLARKS Sandalia Saylie Jade para mujer Cuero caído negro Sandalia Savannah de Ryka para mujer Azul marino / verde , Sanda Verra Teva para mujer Orquídea púrpura , Sandalia de deslizamiento ultra Comfort NIKE para mujer Mango brillante / negro , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave , Chaco Mujer Zcloud Sport Sandal Negro , Sandalia Savannah de Ryka para mujer Coral / gris , Sandalia Teva Terra-Float Nova para mujeres Lima / Blanco , KEEN Sandalias Newport Eco-W para mujer Negro / Imán Wolky Comfort Jewel Cuero de caviar marino ,
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 Chaco Women s Z2 Classic Athletic Sandal Negro Sandalia original Teva Women s Original Miramar Fade Sage Multi , ASICS Gel-Ziruss para mujer Negro-blanco-plata Zapatillas de running AMPLICA para mujer ASICS Gris Medio / Plata / Aruba Azul , Zapatillas de running NIKE Air Max 2017 para mujer Platino blanco / negro puro , MT-onemix Lovers Calzado deportivo ligero de malla Running Jogging Sneakers Negro / Naranja , Guantes Trail Trail Merrell Women s Glove 4 Negro , Sockwa G-HI Minimal Beach Sneaker Lima Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco / negro / gris oscuro / verde azulado , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Negro JSport by Jambu Women s April Gladiator Sandal Armada Orthofeet 851 Comfort Diabetic Therapeutic Profth Shoes para mujer Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy de arena , Zapatillas deportivas Under Armour Xc Brigade Spike Negro (001) / Blanco Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada , Nike Tech Xtreme Cadet Glove Amarillo de voltaje negro / blanco OLUKAI Womens Eheu Thong Sandal Negro Negro Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Plata blanca Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Carbón Zapatillas de baloncesto Nike para mujer Air Force 1 Hi Premium Black / Black Gum Med Brown Sail Marsopa / Gum Marrón Claro / Cumbre Blanca / Marsopa , Sandalia de la diapositiva de China Girl My Girl Lollipop Red Suede , Zapatillas de volante adidas Bounce W de Crazyflight para mujer, Tinta misteriosa / Blanco / Azul hielo, 9 Medium EE. UU. Negro / Blanco / Rojo de Poder Zapatillas de beisbol de fútbol de nueva generación Fusev1 Metal Fast Pitch de New Balance para mujeres Blanco negro Bomba de vestido Kalila Total Motion Kalport para mujer Warm Taupe Patent , Vionic Women s, Josie Low Heel Pumps Gris Sandalia de vestir Carrson de mujer Steve Madden Rojo / Multi , Zapatillas Iris Easy Spirit para mujer rojo Sandalia de vestir elegante 401 de las mujeres Fabulicious Clr / Gold Chrome , Franco Sarto Womens Garfield Bootie Gamuza de turba Zapatilla deslizante Loki II para mujer BEARPAW Hickory / Champagne ,

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.

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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 para deportiva/ Tirra para mujer Teva Rasberry/ 11455 Dark 453ac33 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 3046410
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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.