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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Malla
Dinámica Cables Flywire proporcionar un one-to-one en forma.
Diseñado malla proporciona transpirable soporte.
Tri-star patrón de suela mejora la flexibilidad.
Mínimo el diseño del talón proporciona un más natural en forma.
Co-moldeado entresuela usa una inyección más suave Núcleo de phylon (IP) espuma directamente debajo del pie y un poco más firme inyectado unidad única (IU) espuma transportadora a entrega comodidad y durabilidad para un paseo suave
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

Calzado de running NIKE RN 2017 2017 2017 B072Y2NN7R para mujer ligero Carbono ligero/ Carbono ligero e522779

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.


Calzado de running NIKE RN 2017 2017 2017 B072Y2NN7R para mujer ligero Carbono ligero/ Carbono ligero e522779

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 para correr Nike Women s Free Rn Distance 2 Cool Grey / Black-wolfgrey-stealth , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Negro NIKE Men s Flex 2017 RN Negro / Metálico Dorado-blanco Zapatillas de running NIKE Air Max Zero SI para mujer Negro / Ligero Bone-cool Gris-total Crimson , Zapatillas de running NIKE Lunartempo 2 para mujer Aluminio / negro-supremo azul Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Espuma de mar / cielo cubierto , Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo Zapatillas de running adidas Adaptive Adapt para mujer Blanco / Blanco / Gris , Zapatillas de running Brooks Women s Glycerin 16 Road Negro / Rosa / Gris NIKE W Air Huarache City Low Women Ah6804-100 Blanco / Blanco-blanco , Zapato de correr ASICS Gel-Quantum 360 NYC para mujer Run / New / York , Zapatillas de running ASICS Gel-Kahana 8 para mujer Negro / Island Blue / Pink Glow , Zapatillas de running Pearl NZUMi W EM Tri N 2 para mujer Rosa eléctrico / Lima , Zapato de running Saucony Zealot ISO 2 para mujer Coral / púrpura / algodón , Zapatillas de running Nike Air Max Fury para mujer Negro / Blanco-negro , Adidas Women s EQT Support ADV Originals Zapatillas para correr Correr Blanco / Core Black Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco azul NIKE Mujer Air Max 97 Ultra 17 gris / obsidiana 917704-006 - Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird , Short NIKE Tempo para mujer Franja negra / múltiple , HOKA ONE ONE Zapatillas para mujer Arahi Azul Medieval / Lavanda , Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Melón Blanco / Brillante Zapatillas para correr NIKE Women s Air Vapormax Flyknit Gris oscuro / Reflejar órbita azul plateado , Zapatillas de running Nike Kids Air Max Zero Essential GS Olive-violet / Work Blue / Armory Navy Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue Calzado de entrenamiento para correr NIKE Flyknit Lunar2 para mujer NIKE naranja
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 Propet Women s TravelActiv Ss Sandal Negro rojo , Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco Sandalia Unisex Slide de Telic Blanco como la nieve , Zapatillas de running Impulse Altra para mujer Magenta Fade , Zapato deportivo para acolchado 635v2 New Balance para mujer Plata / Amarillo Correpasillos Merrell Women s All Crush Shield Trail Runner Negro / Granito , Zapatillas de running Nike Air Huarache Run Ultra Br para mujer Platino Blanco / Blanco-puro , Zapatillas de running PUMA para mujer Ignite Omphalodes / Coral caliente / Ultramar Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Pink Bloom Bomba de vestido Nika BeautiFeel para mujer rojo Tecnología Vionic Men s Orthaheel White / Navy Walker Equipaje / Negro , Zapatillas de deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Pure Platinum Hyper Violet 001 , Zapatillas de running Adidas Women s Energy Cloud WTC W Negro / naranja / blanco , Vince Camuto Women s Prilla Ballet Flat Recatado , Sandalias de mujer Bandolino Tormenta Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Juanetes caminando Whitney Womens Sneakers Athletic Zapatos Negro Sandalias de cuña de felpa para mujer Aerosoles adelantadas Negro , Bernie Mev Women s Mely Platform Negro , Zapato para caminar Ryka Women s Sky Negro / rosa Entrenador Ryka Eva Nrg Cross para mujeres Amarillo negro , Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow Eva Kate Spade New York Women s Celo Sandal Nappa negro Hush Puppies Mujeres Power Walker Ii Loafer Blanco , APL: Athletic Propulsion Labs Zapatillas de running mujer Techloom Phantom Moteado blanco / negro , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro sin blanco , Sandalia de deslizamiento Rey-Sat de Loeffler Randall para mujer Eclipse KAANAS Sandalias planas de alpargatas tejidas de Martinica para mujer Natural-1 , Bota de vaquero western Heritage Lacer II de Ariat Women s Heritage Black Deertan , Sandalias Heel Glitter de 6 pulgadas para mujer de Ellie Shoes Brillo de plata , Pleaser ILLUSION-1018UV 6 ​​1/2 talón texturado, 2 1/2 textured PF Boot Neon Hot Pink Pu / Hot Pink ,

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.

Calzado de running NIKE RN 2017 2017 2017 B072Y2NN7R para mujer ligero Carbono ligero/ Carbono ligero e522779 - 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 Calzado de running NIKE RN 2017 2017 2017 B072Y2NN7R para mujer ligero Carbono ligero/ Carbono ligero e522779 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 4690612
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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.