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.

Texto de poliéster Plástico Sintético

Suave, entresuela de doble densidad se ajusta a su pie.

Diseñado las vainas ayudan a absorber impacto para mejorar Amortiguación.

Transpirable la parte superior de malla crea un ajuste, como un guante en forma.

Flywire cables se integran con los cordones para proporcionar una dinámica soporte y bloqueo en el mediopié.

Interno, botín de una pieza envuelve el pie proporcionando un ajuste, en forma de calcetín.

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

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 running NIKE/ Lunar Skyelux para mujer Aluminio Mediano Binario/ Binario Azul/ Azul Mediano a6fcc34Updated 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:

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 running NIKE/ Lunar Skyelux para mujer Aluminio Mediano Binario/ Binario Azul/ Azul Mediano a6fcc34

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 ASICS Gel-Kayano 25 para mujer, blanco / blanco, 6.5 B (M) US - , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Zapatillas para correr NIKE Kaishi 2.0 para mujer Wolf Gray / Pink Black / White Skechers Sport Skech Air Run High Fashion Sneaker Adornos de malla negra / rosa fuerte , Amortiguación 635v2 New Balance para mujer Gris Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink Black Bright Mango 600 , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , NIKE Men s Roshe Two Running Shoe University Red / White / Track Red , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel Zapato de running ASICS Alpine XT para mujer Gris Pizarra / Gris Pizarra Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver Skechers Performance Women s Go Run 400 Obstruct Azul marino / Aqua Zapatillas de running adidas Ultraboost W para mujer Misterio Rojo / Misterio Rojo / Rosa Táctil Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento NIKE Women s Tanjun Iced Jade Aurora verde blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo NIKE Men s Benassi Just Do It Athletic Sandal Midnight Navy / Windchill , Nike Men s Air Presto Essential Universidad / Rojo / Negro / Negro , Calzado de carretera Inov-8 Road-X-Treme 250 Gris / Negro / Rojo Pantalón largo clásico NIKE Golf para mujer Tech Essentials Binario Azul / Blanco-azul Luna , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Negro , NIKE Women s Revolution 4 Running Shoe Gunsmoke / Ocean Bliss / Dark Grey Adicción a Brooks 12 Ombre Azul / Obsidiana / Púrpura Fucsia Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Peacoat Zapatillas de running adidas Ultraboost W para mujer Aero Blue S, Aero Blue S, Noble Indigo S Zapatillas de running NIKE para mujer Epic React Flyknit Perla rosa / rosa perla rosa apenas rosa

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 Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico Sandalias OLKAKAI Ohana - Hombre Dark Java / Ray Zapatillas Reebok Women s Runner 2.0 MT Track Cráneo gris / Verde azulado / Blanco / Negro / Plateado Metálico NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Obsidiana / obsidiana Wolky Comfort Jewel 12750 Cal y Nubuckleather Brooks Womens Adrenaline GTS 18 Púrpura / negro / champán , Zapatillas de running New Balance 520v3 para mujer Artic Fox / Black , Columbia Montrail Mountain Masochist IV Outdry Trail Running Shoe para mujer Grafito, camelia roja , CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Chaqueta amarilla , Zapatos OLUKAI Pehuea - Mujer Paprika / Paprika , Helly Hansen 11302 Bandera de sal para mujer F-1 Sneaker OFF BLANCO / SHELL PINK , Sandalia de vestir SB-Raven azul de Betsey Johnson para mujer Tela plateada , Bomba de vestido Nika BeautiFeel para mujer Esmeralda , Vionic Grace Samar - Sandalia Ajustable Mujer Serpiente natural Chaco Zcloud 2 Sport Sandal para hombre Tri Navy , Entrenador campeón Gusto Cross Trainer Malla blanca negra Zapatillas de tenis adidas Adizero Defiant Bounce W, de mujer, adidas Tiza Azul / Blanco / Tiza Coral Zapatillas Cross-Trainer Ryce para mujer Enhance 3 Gris oscuro , Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Blanco , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / de peltre , Sandalia Dolce Vita Celita para mujer Cuero negro CLARKS Leisa Cacti Slide Sandalia para mujer Cuero de peltre Vince Camuto Sandalias Ellanna Slide para mujer Negro 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) S9282 Trenza dorada y plateada Trotters para mujer Dea Slingback Pump GOLDWASH / NEGRO Bota Khola de mujer de Penny Loves Kenny Negro / Multi La bota de moda Fix para mujer Skylar Mid-Shaft Material-Block Cuero en relieve de la serpiente de pitón Bota de mujer L19 de Arcopedico Reptil negro , Tierra Pegaso Ladrar , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma 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.

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 running NIKE/ Lunar Skyelux para mujer Aluminio Mediano Binario/ Binario Azul/ Azul Mediano a6fcc34 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.

Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge
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6612629

7542024

8130160

10158536

11689149

12191834

14324016

17185399

and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

For email to reach me, the word