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.

Sintético

Hecho en EE. UU. o importados

Suela: Caucho sin marca suela con un arrecife de coral diseño para añadido agarre.

Plantilla: Anatómico moldeado por compresión; Entresuela de EVA con ICEVA plantilla sin cita para personalizado en forma.

Vegano Friendly: Nuestro vegano calzado amistoso ofrece la durabilidad, tracción y soporte esperas de OluKai mientras usas 100% de alta calidad materiales sintéticos, 0% productos animales.

Agua Listo: Ligero, de secado rápido, transpirable y materiales duraderos construido para adentro y alrededor del agua

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

Sandalias OLKAKAI Ohana/ - B005SKJY8Q Hombre Stormy Blue Hombre/ Black a153bdeUpdated 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.

Sandalias OLKAKAI Ohana/ - B005SKJY8Q Hombre Stormy Blue Hombre/ Black a153bde

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 deporte Onitsuka Tiger Mexico 66 Oro negro , Converse Chuck Taylor All Star Seasonal Color Hola Armada Carro de mujer Brooks Blue Atoll / Black , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Ture White Zapatillas de running Saucony Women s Triumph ISO 4 Negro / Aqua / Violeta , Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado NIKE Men s Roshe Run Negro Negro Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Negro NIKE Men s Lunar Fingertrap Cross Trainer Negro / Blanco - Negro Converse Chuck Taylor All Star Seasonal Color Hola Zafiro claro Zapatillas de running NIKE para hombre con versatilidad de tren gratis Blanco negro , Zapatillas de senderismo adidas AX2 CP para mujer Ch Gris Sólido / Vapor Azul / Gris Cinco Under Armour Speedform Slingwrap para mujer Negro (001) / Glacier Grey Zapatillas de trail running para mujer Asics Gel Fujiattack 5 - SS17 Aqua Splash Diva Pink 6720 NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Wei / Schwarz / Grau , Correa de plataforma PUMA para mujer Satin En Pointe WN Sneaker Puma Black-puma Black , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Negro Zapatillas de deporte casual Gazelle de adidas para hombre Blanco apagado / blanco Zapatillas para correr Nike Women s Flex Experience Rn 5 Palest Purple / Clear Jade / Fierce Purple Zapatillas de running Saucony Women s Triumph ISO 4 Azul marino / menta , Zapatillas NIKE Air Huarache Run SE (GS) Girls Fashion 904538 Amarillo / Blanco / Platino NIKE Kids gratis Rn (grande) Burdeos / Plata Metálica Zapatillas de deporte casual Gazelle de adidas para hombre Negro / Blanco / Oro Met. NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Gimnasio Red / Black-Noble Red , Zapatillas Nike Women s Flex Trainer 5 Antracita / Ghost Green / Chlorine Blue NIKE Women s Tanjun Blanco / Plata Metálica Converse Chuck Taylor All Star Seasonal Color Hola Sunblush Zapatillas de deporte Onitsuka Tiger Mexico 66 Birch / India Ink / Latte , I-RUNNER Elite Terapia atlética atlética extra Profundo Cuero / Velcro de malla Negro ,

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 NIKE Air Huarache Pink para mujer 40,5 (EU) -9 (US) Rosa - Camiseta adidas Swift Run W para mujer Carga nocturna / Carga nocturna / Blanco , Zapatillas de running W775V2 para mujer New Balance Blanco negro , Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Guayaba / Negro Zapatillas de trail running Merrell para mujer Pace Glove 3 Corona azul , Zapatillas de agua Merrell para mujer All Out Blaze Sieve Brown Sugar / Blue Heaven , Zapatillas de senderismo Pro Pro 3D Pro de mujer Ryka Carbón de leña / negro CLARKS Lexi Myrtle Sandal para mujer Blanco Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Piedra , Sandalias Paulie T-Strap de mujer Vionic rojo Zapatillas de running NIKE Lunartempo 2 para mujer Negro , J.Renee Womens Soncino Cuero metalizado Taupe Zapato de entrenamiento cruzado 09v1 New Balance para mujer Gris blanco Zapato deportivo Cross-Trainer 711V3 de New Balance para mujer Gris / Rojo Energia Sandalias Casual Slide de Michael Kors MK Negro / Blanco Óptico DC Youth Rebound Skate Shoes Negro Easy Spirit Mujer Ezcool Mule Negro / negro gamuza , Zapatillas de golf NIKE Explorer 2 de mujer AA1846-101 Pulso blanco / blanco / al atardecer Sandalia de mujer Hana Slide de Via Spiga Cuero naranja caliente , Sandalia de cuero Lucili de Nine West para mujer Cuero negro , Birkenstock Mules para mujeres Blanco Sandalia Mila Slide de mujer Naturalizer Ante negro , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7013- Red Velvet Cobb Hill Rockport - Aubrey Dress Pump para mujer Azul / Multi Bota de mujer Merrell Ashland Chukka Ocre rojo , Womens Fly London Yama plataforma cuña talón botín aceite gamuza medio talón Diesel , CLARKS Willow Jade tobillera para mujer Nubuck marrón , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Zapatos de mujer - Wichita Negro , Ros Hommerson Nadia Oxfords para mujer Patentado de color marrón oscuro, gamuza / marrón Croc Patente Sam Edelman Loraine Loaine para mujer Arena

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 Sandalias OLKAKAI Ohana/ - B005SKJY8Q Hombre Stormy Blue Hombre/ Black a153bde 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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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