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 y Textil
Importado
Sintético único
Eje mide aproximadamente parte baja del arco
ACTEVA entresuela
ABZORB tacón
Respuesta NB 2.0 Performance insertar
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

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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.


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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 running W1540V2 New Balance para mujer Blanco azul , Zapatillas de running Nike Air Zoom Vomero 11 para hombre Squadron Blue / Blue Grey / Total Crimson / White , En Running Womens Cloudflow Shoe Atlantis / Llama , Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Plata / gris / rosa , Brooks Womens Adrenaline GTS 18 Negro Negro Zapatillas de deporte para correr Nike Air Max Ivo Fashion Running Gym para mujer Pure Platinum / Blanco Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-negro Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Binario Azul / Blanco Camiseta adidas Swift Run W para mujer Core Black / Core Black / White , adidas Women s Edge Lux 2 Sol / Plata / Naranja Zapatillas de running ASICS Women s Roadhawk FF SP Cereza / flor / abedul , Zapatillas de running NIKE para mujer Lunarepic Flyknit Shield Ocean Fog Blue 400 Zapatillas de running adidas Ultraboost W para mujer Ice Blue / Ice Blue / Blue Night Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer HERON / Blanco / Agua Zapato de running ASICS Gel-Quantum 360 cm para mujer Punto medio gris / glaciar / coral NIKE Womens Air Zoom Wildhorse 4 Running Shoe 880566-007 Negro / Volt Negro / antracita-antracita Zapatillas Reebok Women s Runner 2.0 MT Track Blanco / Poison Pink / Silver / Pink Ribbon Zapatillas de running adidas Ultraboost W para mujer Blanco / Blanco-2 / Blanco , Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout Camiseta adidas Swift Run W para mujer Gray Two / Ice Pink / White , Zapatillas para correr New Balance Rushv3 para mujer Poisonberry / Black Camiseta adidas Swift Run W para mujer Traza púrpura / Traza púrpura / Traza púrpura , Zapatillas de running Nike Kaishi para hombre Negro , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Dragon Fruit / Sea Salt , Zapatillas de running Flex Ride 630v5 para mujer New Balance Azul marino / plateado , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul Mediano / Negro-aluminio-electro Verde , Calzado de running NIKE Kids Roshe One (GS) Gimnasio Rojo / Blanco / Gris Oscuro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Wolf Gray / Cool Gray / Black , NIKE Mujeres Air Max 90
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 ASICS Gel-Scram 4 para mujer Negro / gris oscuro , Amortiguación 635v2 New Balance para mujer Negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco negro , Zapatillas para trail running Mutant Backcountry de La Sportiva para mujer Green Bay Zapato de running Mission XR para mujer Salomon Magnent / negro / púrpura Zapatillas para caminar RYKA Women s Devotion Plus Frost Grey / Steel Grey / Rose Violet Sanita Sanena Women s Deena Turquesa , Zapatillas de running New Balance para mujer 940v3 Azul negro , New Balance WW813 Hook and Loop Walking para mujer Blanco Walking Cradles Womens Henna Nubuck estampado rojo , Rockport Cobb Hill Collection Mujeres Cobb Hill Abbott Curvy T-Strap Cuero caqui Easy Spirit Women s Traveltime Mule Plata , Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Multi , Zapatillas para mujer Del Rey TOMS Dril de algodón gris / lavado Zapatillas de skate DC Midway W para mujer Azul oscuro , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Baya , Easy Spirit - Zapatillas Faisal2 para mujer Blanco Zapatillas de tenis Adidas Adizero Club w, para mujer Tiza Coral / Blanco / Azul Tinte , JiYe Performance Sports Shoes Zapatillas de deporte de baloncesto para hombre Verde negro Sandalia Heather Slide de mujer Vía Spiga Ladrillo de gamuza , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ UGG Mujer Alena mocasín Negro Mordenmiss Nuevo verano de primavera y verano de flores hechas a mano zapatos planos de cuero Mocasines Estilo 3 Rojo Zapatilla de deporte de alta moda para mujer Soft VII de ECCO Negro CLARKS Amberlee ROSI Tobillera para mujeres Cuero negro Bota Dafne Thierry Rabotin para mujer Negro FRYE Mujeres Anna Oxford Negro Timberland Flyroam Go Knit Chukka para mujer Negro , B.O.C - Mujeres - Peggy Peggy Gray Tooled Pu TOMS - Slip-On de lienzo clásico para hombre Blue Marl

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 Zapato para deportivo para acolchado 680v5 New Silver Balance B01GIPLC8E para mujer Silver Mink fff97bb 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

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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.


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