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.

Cuero de plena flor ^ Gamuza

Importado

Goma suela

genuino Piel de Granos superior

Suave y transpirable forro de cuero para una comodidad absoluta

No extraíble, ligeramente acolchado, EVA revestido de cuero Plantilla para amortiguar

Flexible entresuela de goma para una mayor comodidad

Durable y flexible moldeado suela de goma con cuero

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

SOREL 13755 Womens B01CHEUVSW Ella Ella Lace-Up Sáhara bb594c1Updated 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.

SOREL 13755 Womens B01CHEUVSW Ella Ella Lace-Up Sáhara bb594c1

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

Sandalia con correa en T para mujer ECCO Polvo / gris cálido Sandalia Patrese de estilo suave para mujer Patente negra Sandalia de cuña WIGG672FLY de mujer FLY London Cool Graphite Sandalia de cuña Karmen de mujer Alegria Peltre Dazzler , CLARKS Lexi Myrtle Sandal para mujer Negro , Sandalia Kinsley de Naturalizer para mujer Papaya Imprimir , Sam Edelman Sandalia Dariel para mujer Estampado de algodón negro / blanco Sandalia plana Nanci de Naturalizer para mujer Mezclilla Sandalia Sharene de Calvin Klein para mujer Blanco suave Sandalias de peonía Easy Spirit para mujer Antik Silver / Antik Silver / Antik Silver , CLARKS Womens Un Reisel Myra Slide Sandal Cuero azul / gris , Sandalia de cambio de mujer Steve Madden Negro J.Renee Womens Mcwayfalls Plata Bomba de cuña Halle Bernie Mev para mujer Negro SAS Mujer, Pampa Sandals Negro , Chaco Mujer Zcloud Sport Sandal Charlie Peach Sandalia LifeStride para mujer Eden Negro , B.O.C. Negro , Sandalias EVA Gizeh de mujer Birkenstock rojo , Sandalia P-Oudry de Yellow Box para mujer Gris pardo , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra Sandalias Galaxy de BZees para mujer Gradiente negro , CLARKS Womens Wynnmere Avah Nobuck gris cálido Sandalia para mujer con correa de tobillo Naot para mujer Púrpura / azul real / cuero de aguamarina Sandalia de cuero Terra Fi Lite de Teva para mujer Negro , Birkenstock Unisex Arizona Essentials Sandalias EVA Khaki Eva Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero marrón , Sandalia de cuña de Easy Street para mujer Champagne Snake Sandalia Huarache Easy Street Allure para mujer Beige / Metálico Imprimir

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 adidas Women s Distancestar W Blanco / Hi-res Verde / Real Purple Zapatillas de running adidas Duramo 8 para mujer Blanco / Cristal Blanco / Gris Claro Heather , Zapatillas para correr Mizuno Women s Wave Paradox 3 Gris-rosa Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Wolf Gray / Black / Cool Gray Sandalia Drew Angela para mujer Dusty Multi Metallic Sandalias Uneek EXO-W de mujer KEEN Negro / gris acero Sandalia de mujer KEEN Rose Aluminio / gris neutro En la ejecución de calzado de mujer en la nube Todo blanco , Nike Hombres Air Zoom Pegasus 33 Crimson Tint / Navy-white Zapato para caminar Ryka Women s Flora Negro / azul / gris , Calzado de mujer de fila Cress Walking Negro, negro, blanco , Trotters Mujer Simmy Mary Jane Flat Negro , Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello , Sandalia Misty Fisherman de Ryka para mujer Turquesa / Amarillo , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Negro, azul, blanco , Sandalia Natalia Ryka para mujer Negro Sandalia Minnie Trotters para mujer Nubuck desnudo Sandalia de vestir Layla de Onex para mujer Cuero marrón , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy Supra Mujer / Hombre Noiz Sneaker Black Oil Slick Zapato de golf de punto híbrido casual para mujer ECCO Blanco Negro / Blanco , Vince Camuto Women s Haydan Slide Sandal Blanco negro Sandalias Poppy de mujer de Eastland Broncearse , Bota de paddock Ovation Women s Sport Rider II Paddock - 469395-Brn marrón TOMS Womens Desert Wedge Bootie Toffee Suede Perforated Leaf Botín de tobillo Cyra Catelyn de Hush Puppies para mujer Leopardo Natural Gamuza Tangas Birkenstock Mocca , Zapatillas OLUKAI Upena - Mujer Negro Botas Nelson de mujer de Matisse Leopardo cazador Pour La Victoire Indy Women s Oxford 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 SOREL 13755 Womens B01CHEUVSW Ella Ella Lace-Up Sáhara bb594c1 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