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.

Textil y sintético

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

Para todas las necesidades de entrenamiento.

Grip para un movimiento rápido: Caucho pegajoso suela en clave áreas que incluyen en la parte delantera del pie para un pie seguro colocación en rápido movimientos y giros.

Potencia y estabilidad: Una plataforma más firme con Heel-lock para el poder y el equilibrio al levantar pesas o haciendo sentadillas.

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

Inov-8 Women s All 19993 Train 215 (W) Gris Cross (W) Trainer Gris/ rosa 56c0954Updated 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.

Inov-8 Women s All 19993 Train 215 (W) Gris Cross (W) Trainer Gris/ rosa 56c0954

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

Entrenador cruzado de trabajo 627v2 New Balance para mujeres Gris / rosa , Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Negro , Adidas Mujeres Aleki X Cross Tainer Black / Bliss Coral / Intense Blue , Zapatillas de deporte WN Slip-on Farece para mujer de PUMA Puma Black-puma Black Zapatillas de entrenamiento Reebok Crossfit Lite Lo TR para mujer China Rojo / Electro Rosa / Acero Zapatos de entrenamiento Nike Free TR 6 para mujer Multicolor / Multicolor-negro , Skechers Sport Women s Burst City Scene Zapatillas de moda Natural NIKE Entrenador flexible para mujeres 6 Negro / Lava Glow / Antracita OLUKAI Eleu Trainer - Mujeres Dark Shadow / Deep Red Entrenador cruzado 20v7 de New Balance para mujeres Vivid Coral / Fiji , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Silver Mink / Dusted Peach NIKE Entrenador flexible para mujeres 6 Blanco / gris oscuro Zapatos de entrenamiento Nike Free TR 6 para mujer Blanco / Juego Royal Pure Platinum , Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Negro , Inov-8 Mujer F-Lite 195 V2 (W) Cross Trainer Gris oscuro adidas Athletics 24/7 Tr M Cross Trainer Core Black / Core Black / Hi-res Blue , Zapatos de entrenamiento Nike Cheer Scorpion Cross para mujer Blanco / Blanco / Platino puro Entrenador cruzado New Foam Crush V1 de New Balance para mujer Dusted Peach / Sunrise Glo Entrenador cruzado Fresh Foam Crush V1 de New Balance para mujer Negro , Zapato de entrenamiento WX1267 para mujer New Balance Negro / rosa , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Gris claro Zapatillas Superga 2750 Cotu para mujer Azur Entrenador cruzado de trabajo 627v2 New Balance para mujeres Negro / púrpura New Balance Women s Wx715cw3 Carbón Zapatos de entrenamiento Nike Free TR 6 para mujer Blanco / medianoche azul marino / puro platino , Under Armour Charged All-Around Neutral Nordic Green (903) / Nori Green PUMA Mujer Ignite Flash Evoknit Satin En Pointe Wn Sneaker Melocotón Beige-perla-puma Blanco , Zapatillas Cross-Trainer Gel-Fit Tempo 3 para mujer de ASICS Negro / Plata / Blanco Plataforma de satén mono de ante PUMA para mujer Silver Pink-whisper ,

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 Zapato de running ASICS GT-1000 6 para mujer Insignia Azul / Plateado / Rojo Rouge Brooks Women s PureGrit 5 Peacoat / Passion Flower / Ceramic Under Armour Women s Charged Rebel Apollo Gray (100) / Blue Infinity , Sandalia Terrell Ivy Post Sport Merrell para mujer Sodalita MBT Shoes Mujeres GT 17 Athletic Shoe Cuero / Malla Con cordones Blanco negro , Sandalia de vestir Tillman de SoftWalk para mujer Broncearse Sandalia de mujer KEEN Rose Negro / Gris neutro , nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Mar oscuro metálico / turquesa de medianoche , CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Chaqueta amarilla , Zapatillas para correr Saucony Xodus 4.0 GTX para mujer Negro Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple Comfortiva Mujer, Faye Sandalia de tacón medio Negro , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Dragon Fruit / Sea Salt Sandalia de cuña Drama BZees para mujer Terciopelo negro Sandalias casual Debra 19 2 Strap de Josef Seibel para mujer Castagne , Zapatillas para caminar Skechers Performance Go 4 Inspire para mujer Gris pardo Birkenstock Men s Arizona Taupe Suede Soft Footbed Hush Puppies Mujeres Power Walker Ii Loafer Cuero blanco , New Balance Wrl247hk para mujeres Chocolate , Vans Unisex Old Skool Zapatillas clásicas de skate Negro / Blanco verdadero Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Oro negro 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 \ 1620s- Cuero plateado Bomba Lloret para mujer J.Renee Naranja / azul / verde Zapatillas de running Adidas Performance Adizero Prime SP con púas Blanco / Infrarrojo / Plata Metálica Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Negro , Mocasín de Kentucky de Old Friend Women castaña , Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Morado Patentes / Nude Trim / Gold Studs , Zapatos holgazanes Joleen de mujer Alegria Tar Tooled Mocasines casuales Alpargata Deconstructed de TOMS para mujer Toffee Suede Cupsole TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Azul marino Batik Stripe ,

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 Inov-8 Women s All 19993 Train 215 (W) Gris Cross (W) Trainer Gris/ rosa 56c0954 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