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.

PU

Importado

Sintético suela

Plataforma mide aproximadamente 0.5 pulgadas

Slide Sandalia

Talón Tecnología de descanso

Memoria Plantilla de espuma

Diamante Suela flexible

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

Sandalia para de Multi girasol Aerosoles para mujer Bone Aerosoles Multi 8dda617Updated 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.

Sandalia para de Multi girasol Aerosoles para mujer Bone Aerosoles Multi 8dda617

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 Sloan Slide de Naturalizer para mujer Negro , Sandalia Mason para mujer Naturalizer Slide Cuero metalizado lila , Sandalia Chloe Slide de KENDALL + KYLIE menta , Sandalias AS-Mona Slide de mujer Ash Oro negro Sandalia Claire para mujer Dolce Vita Cuero negro , Sandalias Damara Ii Slide de ECCO para mujer Dark Shadow / Moon , Sandalia plana Kip de Trotters para mujer Blanquecino Sandalia de la diapositiva de junio de las mujeres de Propet Negro Zaxi Sandalia con cierre a presión para mujer Zaxy Oro , BareTraps Women s Fergy Slide Sandal Natural , Sandalias Fifi19 Slide de Donald J Pliner para mujer Plata , Sandalia para mujer Castlero Slide de ALDO Armada KAANAS Sandalias planas de alpargatas tejidas de Martinica para mujer Natural-2 Sandalia plana Dionne de Easy Street para mujer Blanco , Lauren by Ralph Lauren Sandalia mujer Davan Slide Deep Saddle Tan Super Soft Leather , Sandalia Rilee Slide Dolce Vita para mujer Cuero Onyx Sandalia Dolce Vita de mujer Kaira Slide Cuero negro Sandalia Cato Dolce Vita para mujer Lt Blue Denim , ALMA NATURAL Sandalia Kiki Slide para mujer Negro Sandalia de la diapositiva Adalea de las mujeres de Dolce Vita Lino negro , Sandalia de mujer Lucienne Suede Slide de Nine West Suede verde claro Sandalia Sloan Slide de Naturalizer para mujer Negro , Sandalia con tiras metálicas Violet Shimmer de Taryn Rose para mujer Oro Sandalia para mujer Ivanka Trump Earin Slide Ante rojo Sandalia Fara Slide de Donald J Pliner para mujer Negro Sandalia para mujer bien descansada Seychelles Gris Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla Sandalia de cuero Merrell Dwaykair Seaway para mujer Cielo azul

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 Teva Women s W Hurricane XLT Infinity Sport de Teva Pimenton , Sandalia deportiva Nolita para mujer Anne Klein Tela negra , Zapatilla Slip-On Hydro-Life W de Teva para mujer Negro , Zapatillas deportivas Spira XLT Camo Limited Edition para mujer Chocolate / Naranja / Blanco , Mujer adidas Originals Ultraboost St Indigo crudo / tinta noble / naranja de alta resolución Zapatillas para correr WRL247WM New Balance para mujer Estratos , Zapatillas Columbia Trail Trail WMNS de Plains Ridge para mujer Silla de montar / pluma , Zapatillas adidas Outdoor Terrex Agravic Speed Shoe de mujer Rosa táctil, negro, naranja fácil , Zapatillas de agua Merrell para mujer All Out Blaze Sieve Blue Crush Zapato de running Saucony Zealot ISO 3 para mujer Azul marino Púrpura Zapatillas para correr Saucony Freedom Iso Unisex Oro blanco Me Too OLYMPIA1 Zapato plano de cuero para mujer Rose Gold Metallic Ryka Heather Cross Trainer para mujeres Blanco azul Under Armour Threadborne Push para mujer Negro (004) / Negro Zapatillas Superga 2750 Cotu para mujer Violeta total Easy Spirit Twist2 Sneaker para mujer Natural Drew Shoe Athena para mujeres Crema / Combo nativo AP Mercury Liteknit Naranja Sunset / Blanco Shell / Caucho Natural , Zapatillas de tenis Babolat Propulse Fury All Court para mujer Princess Blue / Fandango Pink , Zapato de skate casual Heathrow DC para hombre Azul marino / caqui , Sam Edelman Women s Bay Slide Sandal Cactus floreciente multi brillante UGG Mujer W Joni Negro Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / real Sandalia con tiras metálicas Violet Shimmer de Taryn Rose para mujer Oro , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA7036 Tacón de 3 pulgadas + plegable Paquete de cepillo Patente roja , Bomba de vestido Pointe para mujer Easy Street Patente desnuda Corral Boots Mujeres de 5 pulgadas Círculo G de cuero de vaca Cuero punta redonda Recorte Tan Shortie Western Boot marrón , OTBT Womens Khora Sneakers Azul eléctrico Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Patente negra Propet Women s Tour Walker Strap Sneaker Blanco / Berry ,

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 Sandalia para de Multi girasol Aerosoles para mujer Bone Aerosoles Multi 8dda617 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