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.

Zapatillas de Blue running 19158 ASICS Gel-Nimbus B0010TORW2 19 para mujer Indigo Blue/ Paradise Green/ Pink Glow 2054c00




Sintético
Importado
Caucho suela
FlyteFoam Tecnología de entresuela. 3 mm de adicional altura en las mujeres los modelos ayudan a aliviar tensión de Aquiles.
FluidRide Entresuela: FluidRide proporciona lo último combinación de recuperarse y propiedades de amortiguación con peso reducido y excepcional durabilidad.
Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.
FluidFit Parte superior: ASICS FluidFit tecnología superior combina multi-direccional malla elástica con refuerzos de estiramiento que se adaptan a la pie de atleta, creando un verdadero personalizado como un guante ajuste.
Clutch de talón Tecnología de sistema: Talón exoesqueleto contador proporciona soporte mejorado y crea mejoras entorno de ajuste del talón.
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


Zapatillas de Blue running 19158 ASICS Gel-Nimbus B0010TORW2 19 para mujer Indigo Blue/ Paradise Green/ Pink Glow 2054c00 - 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

Zapatillas de Blue running 19158 ASICS Gel-Nimbus B0010TORW2 19 para mujer Indigo Blue/ Paradise Green/ Pink Glow 2054c00

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.


Zapatillas de Blue running 19158 ASICS Gel-Nimbus B0010TORW2 19 para mujer Indigo Blue/ Paradise Green/ Pink Glow 2054c00

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

NIKE Women s Tanjun Sport Fuchsia White Tart Zapatillas de running New Balance 990v4 para mujer Dark Mulberry Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Azul marino / rosa / verde azulado , Zapatillas de running adidas Women s Edge Lux w Misterio Ruby / Bahia Magenta / Calzado Blanco Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello Under Armour Women s Drift 2 Negro (001) / Blanco , NIKE Tanjun ENG Sneaker para mujer Negro Negro Blanco Racer Pink Zapato de running ASICS Gt-2000 5 para mujer Diva Blue / White / Aqua Splash Zapatillas de running NIKE para hombre Air Max BW OG Negro / persa Violet-white Zapatillas de running Nike Air Huarache Run Ultra para hombre Blanco , Zapatillas para correr ASICS Gel-DS Trainer 22 para mujer Acuario / Aqua Splash / Flash Coral , Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Azul Zapatillas de running mujer HOKA ONE ONE Clifton 4 Rosado Zapatillas de running NIKE Lunartempo 2 para mujer Pink Blast / Clr Jd / Black / Brly Grn Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris oscuro , Zapatillas para mujer Reebok Zprint Her MTM para mujer Negro / Gris Ceniza / Blanco Zapatillas de running Under Armour Charged Bandit 3 para mujer Purple Rave (959) / Penta Pink , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul humo / azul oscuro / Cantalupo , Zapatillas de running New Balance para mujer Fresh Foam Zante V3 Breathe Pack Vivid Ozone Blue / White , Zapatillas de running NIKE Lunartempo 2 para mujer Hyper Violet / Black / Concord / Gamma Blue , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover Zapatillas de running NIKE Air Max 2017 para mujer Blanco negro , Zapatillas de entrenamiento Nike Womens Metcon 3 Gris oscuro / verde fantasma Zapato de running de punto fuzeX para mujer ASICS Rosa Glo / Flash Coral / Negro , Zapatillas de running adidas Originals Women s Tubular Dawn Originals Utiblk / Utiblk / Crywht Zapatillas de running adidas Adizero Boston 6 W para mujer Utilidad Negro / Platino / Negro Zapatillas de running adidas Supernova Glide 8 para mujer Lgh Solid Gray / Lgh Solid Gray / Silver Metallic ,
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 Sandalia deportiva Zvolv X de Chaco para mujer Alloy Dancer , Zapatillas para correr Mizuno Women s Wave Daichi 2 True Blue / Micro Chip / Blue Topaz Zapatillas de running ASICS Gel-Kayano 23 para mujer Blanco / Plata / Acuario NIKE Men s Benassi Just Do It Athletic Sandal Oro negro / metálico , Salomon Women s Speedcross 4 W Trail Runner Bluebird / Acid Lime / Black Calzado de running Trail para mujer New Balance para amortiguar 620v2 Granada / Outerspace , Zapatillas para correr Nike Women s Free Rn Distance 2 Gris blanco Zapatillas de running NIKE Air Max Zero Premium para hombre Gimnasio Rojo / Gimnasio Rojo / Gris Lobo Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy de arena , OLUKAI Alohi - Mujeres Fray Brown / Fray Brown , Sandalia Savannah de Ryka para mujer Moonrock / Snowline Ecru , Nina Womens Contesa Suedette reflectante de plata verdadera Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Multi , Zapatillas para caminar Skechers Go s Lite-Blooming Walking para mujer Gris pardo , CLARKS Sillian Stork Fisherman Sandal para mujer Nubuck rojo sintético Zapato de entrenamiento cruzado HIIT XT de Altra para mujer Gris / Lima , Zapatos de mujer RYKA Sky Walking Negro / rosa , Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Blanco Converse Chuck Taylor All Star Core Ox Rosado Zapatilla de deporte KAANAS Santa Fe Fashion Skate Slip-on Casual Sneaker Fumar , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / de peltre , FootJoy Embody BOA Golf Shoes 2016 CLOSEOUT Mujeres Ártico , Stuart Weitzman Landslid femenino Ante rojo BareTraps Women s Rebecca Slide Sandal Blanco , Sandalia Sisleysu Slide de Musse & Cloud para mujer rojo , Sandalia Dolce Vita Jene Slide para mujer Denim de Ceniza Zapato Capezio Mary Jane Tap para mujer Negro Mocasín holgazán Pandora para mujer de Dr. Martens Rojo cereza , Timberland Flyroam Go Lace Chukka para mujer Pure Cashmere Escape / Natural Sandune , Blondo Womens Ellie Waterproof Cuero 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.

Zapatillas de Blue running 19158 ASICS Gel-Nimbus B0010TORW2 19 para mujer Indigo Blue/ Paradise Green/ Pink Glow 2054c00 - 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 Zapatillas de Blue running 19158 ASICS Gel-Nimbus B0010TORW2 19 para mujer Indigo Blue/ Paradise Green/ Pink Glow 2054c00 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 2619907
5354215
6294239
6809872
7758431
12289053
13318008
15790137
18381003
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.