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

Importado

Sintético suela

Medidas de talón aproximadamente 1

Medidas de la plataforma aproximadamente 0.5 pulgadas

Radicalmente liviano y flexible

Goga energía máxima de retorno plantilla

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

Skechers Flip Performance Mujeres H2 GOGA Laguna Flip Flop Coral Carbón Flop de leña/ Coral a65dd68Updated 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.

Skechers Flip Performance Mujeres H2 GOGA Laguna Flip Flop Coral Carbón Flop de leña/ Coral a65dd68

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 deportiva Terrell Ari Lattice de Merrell para mujer Resquicio de esperanza , OluKai Kulapa Kai Negro Negro , Skechers Performance On The Go 400 Vivacity Flip Flop Blanco / azul claro , Sandalia Merrell para hombre All Out Blaze Aero Sport Vapor OluKai Kulapa Kai Pokeberry / Black Sandalia de vestir Tillman de SoftWalk para mujer Cuero Nappa Suave Negro , Sandalias KEEN Mujer Ana Cortez Flip-W Concha de tortuga Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Sodalita , Sandalias Newport H2 de KEEN para mujer Cuervo / Capri , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Negro , Sandalia de deporte Merrell, para mujer Terran Ari Backstrap Sodalita Wolky Comfort Jewel 30340 Cuero de lino , Sandalia Teva Terra Stella Lux de mujer Teva Negro , Sandalia de mujer Drew Alana Blanco negro , OluKai Ohana Platino / Negro , OluKai Ohana Punch / Ice , Chaco Mujer Zcloud X2 Sandalia Deportiva Plata metálica , Sandalia deportiva Zvolv X de Chaco para mujer Arco caliente , Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia gris Chaco Women s ZX2 Classic Athletic Sandal Palm aguacate KEEN Hombres Newport H2 Sandal Vestido Azul / Azul Clásico Revs New Classic Masaje de reflexología Sandalias para hombres & Mujer. Mira. Diseño. Terapia. Cojín Comfort Con plantilla de masaje & Absorción de impactos Único. Broncearse Chaco Women s Zx3 Classic Athletic Sandal Aqua Mint , Chaco Zcloud 2 Sport Sandal para hombre Varsity Sun , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Amarillo Sandalia Savannah de Ryka para mujer Gris oscuro Chaco Mujer Zcloud X Sandalia Deportiva Dahlia Eclipse OluKai Kulapa Kai Pokeberry / Black Chaco Mega Z / Cloud Blazer Navy

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 Sandalias de mujer con tirantes Peep Toe de Bordello by Pleaser para mujer Negro , KEEN Hombres Newport H2 Sandal Dark Shadow / Bombay Brown , Converse Chuck Taylor All Star Seasonal Color Hola Sunblush Zapatillas deportivas 574S Sport New Balance para hombre Nimbus Cloud / Navy Sandalia deportiva NIKE Kawa Slide para hombre Hyper Cobalt / White / Wolf Gray , Sandalia Drew Angela para mujer Dusty Multi Metallic Sandalia deportiva MBT Jasira 6 Trail Sandal para mujer Negro / gris carbón de malla , PUMA Men s Roma Basic Sneaker Blanco / Limoges Zapatillas de trail running W620v2 para mujer New Balance Negro / Plata Zapatillas de deporte con cordones Sneakers de hombre Carson Runner de PUMA Gris / Blanco / Plasma Brillante CLARKS Sonar Pilot Sandalia para mujer Cuero blanco New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Armada , Zapato de entrenamiento Cross-Trainer Vazee Wx77v1 para entrenamiento New Balance para mujer Deep Porcelain Blue / Bleached Sunrise , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Negro Negro , ASICS Tiger Unisex Gel-Lyte V Sanze Bosque oscuro / mazapán , Zapatillas de tenis adidas Adizero Defiant Bounce W, de mujer, adidas Tiza Azul / Blanco / Tiza Coral , Zapato de golf perforado Biom Hybrid 2 ECCO para mujer Blanco / Granito Verde , Sandalias Fifi19 Slide de Donald J Pliner para mujer Oro Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro Sandalia Beau Slide Kate Spade New York 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) 1692- Cuero dorado claro / malla de carne 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2702- Silver Stardust & Silver Leathe 50 sombras de blanco 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) 1606- Satén de marfil VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Armada Zapatillas MTB SD15 SD-15 Mountain Bike de Sidi para hombre marrón , Bota caila mujer Musse & Cloud Gris pardo , TOMS Womens Katalina Espadrille Sandal Cordón floral negro , FRYE Phillip Harness Tall Boot (Niño pequeño / Niño pequeño / Big Kid) Whisky , TOMS Kids 10009918 Alpargata-K (Marina, microfibra) , Botas cortas de cuero Mordenmiss para mujer Zapatos nuevos Estilo 1-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 Skechers Flip Performance Mujeres H2 GOGA Laguna Flip Flop Coral Carbón Flop de leña/ Coral a65dd68 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