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.

Gamuza

Hecho en EE. UU. o importados

Caucho único

tradicional zapatilla mocasín en piel de oveja genuina.

Suave lana natural ayuda absorber la humedad de tu piel.

Fácil estilo sin cordones.

Suave forro de piel de oveja.

Acolchado espuma para extra confort.Sierra zapatilla interior / exterior suela.Este producto contiene pieles de verdad de Sheep.Fur Origen: China.Real La piel ha sido artificialmente teñido y tratado.Medidas: Peso: 8 oz Mediciones del producto fueron tomadas usando tamaño 8, ancho M. Tenga en cuenta que mediciones pueden variar por tamaño

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

Mocasines con suela de 3341 piel de suela oveja Minnetonka B01GEZUOD2 para mujer - 3341 Gris 415c624Updated 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.

Mocasines con suela de 3341 piel de suela oveja Minnetonka B01GEZUOD2 para mujer - 3341 Gris 415c624

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

Zapatilla de piel sintética Cally Minnetonka para mujer Canela , Pantuflas Mahabis Classic para hombre gris Gris oscuro / Gotland Green JOE N JOYCE Mujer London Soft-footbed Metallic SynSoft Sandalias Zapatillas Estrechas Negro , Zapatos OLUKAI Pehuea - Mujer Trench Blue / Trench Blue , Bomba de vestido de consuelo para mujer LifeStride Negro , Zapatillas de deporte de verano Breezy Dualyz Fit Breezy con suela extraíble Gris claro / amarillo Glerups Unisex G-01 - Botas de fieltro Naturebrown Alpargatas antideslizantes de lona clásicas para mujer de TOMS Armada Zapatos de mujer VANELi Allure Pumps Ante / elástico negro Bomba de vestido Kalila Total Motion Kalport para mujer Deep Ocean Nappa Zapatillas Vionic Adilyn Mujer punta redonda Negro Ellie Shoes Women s 711 Coco Platform Sandal Claro / Claro , Propet Cush N Foot Slipper & Oxy Cleaner Bundle Armada , UGG Mujer Chickaree Ante negro , Zapatilla de cuero ferocidad de las mujeres del oeste nueve Negro , Zapatilla Rs80016 Giuseppe Zanotti para mujer Nerón , Vionic con Orthaheel Indulge Gracie Zapatillas para mujer Gris claro Dualyz Fit Unisex 100% lana Zapato con suela extraíble Gris oscuro / negro , Zapatillas de casa de lana de invierno U-lite cálido cálido Shearling de las mujeres, zapatillas de lana caliente para las mujeres Gris (Gamuza de cuero de vaca) Bomba de vestido Henrika de tela Anne Klein para mujer Cuero dorado claro , Haflinger Women s GZ Zig Zag Flat Picea CLARKS Bomba de cuña Flores tulipán para mujer Cuero negro , Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento rojo Bomba Saldana LifeStride para mujer Gris pardo , Ropa para mujer Patricia Green Rosa Azul Bomba de vestido de pasión Easy Street para mujer Patente negra Pantuflas de seda de mujer color verde de Patricia Green Azul 1 TOMS Clásicos de mujer Black Cherry Poly Canvas Zapatillas VI-S para fitness y yoga Vibram para mujer Gris ,

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 KEEN Women s Uneek Slide-w Negro Negro , Calzado casual tejido NIKE NIKE para hombre Sequoia / Pale Gray-black , Sandalia Terrell Post II Femenina de Merrell Atlantis Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / Navy , NIKE Women s WMNS Zoom Fly, negro / blanco Gran gris / antracita-atmósfera gris Zapatillas de running Pearl NZUMi W Em Road N 2 V3 para mujer Ibis Rose / White , NIKE Women s Tanjun Antracita Wht Black Platinum , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 negro Sandalia deportiva Zcloud Chaco para mujer Marquise Pink , Sandalias Columbia Columbia Sunlight Vent II 2 Correa Sorbete Bluff OluKai Ohana Pokeberry / Dark Java Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Blanco negro Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Vivid Purple Black Fucsia Glow 500 , Zapatillas de running Gobiv2 para mujer New Balance Berenjena / Vintage Indigo , Zapato de caminar ECHONON LE2 para mujer Saucony Negro , Dr. Comfort Victory Profundidad extra terapéutica para mujer Athletic Shoe leather / mesh cord-up Blanco , Easy Spirit Women s Traveltime Mule Natural , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada Zapatillas para caminar Brooks Women s Addiction Walker Blanco Vans Unisex Old Skool Zapatillas clásicas de skate Boom Boom Black / True White Vans Auténtico Salmón fresco Sam Edelman Sandalia Bernice Slide para mujer Saddle Vaquero Saddle Leather Zapatillas de volante adidas Bounce 2 de Crazyflight para mujer Blanco / Plateado Metálico / Gris , 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) 1662- Flor blanca y azul Zapatillas de beisbol PowerAlley 4 W TPU de adidas Performance para mujer Poder rojo / blanco / blanco Bota de tobillo con puntera en punta de cuero concho de Dingo para mujer - Di8940 Negro Aquatalia de Marvin K. Botín de ante perforado mujer Shari para mujer Negro , CLARKS Keesha Rosa Dress Pump para mujer Oro / Metálico Sandalias de vestir Patti de Sam Edelman para mujer Yute Glitzy Fabric Zapatillas Zeppa en suela francesa para mujer Impresión en lámina roja

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 Mocasines con suela de 3341 piel de suela oveja Minnetonka B01GEZUOD2 para mujer - 3341 Gris 415c624 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