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.

Charol

Importado

Talón mide aproximadamente 0.5 pulgadas

Por favor Nota: COACH & trade; los artículos no pueden ser enviado a militares direcciones (APO o FPO) y direcciones en Hawaii, el Islas Virgenes, Guam o cualquier otro lugares fuera del continental EE. UU.

No niegues usted mismo la elegancia de la Caterine sandalia.

Cuero superior.

Ajustable cierre de hebilla en el tobillo.

Hermoso logo de hardware en vamp. diseño. revestimiento. plantilla acolchada. Suela hecha por el hombre. Importado. Altura del talón: 1 & frasl; 2 en peso: 6 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

Sandalia Caterine de de cuero de charol Caterine para Patent mujer Chalk Patent 38ae5d1Updated 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 Caterine de de cuero de charol Caterine para Patent mujer Chalk Patent 38ae5d1

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

Ripa para hombre / para mujer Himalayan Shearling Slipper Sandal Granate , Bota Deia para mujer de TOMS Desert Taupe Suede / Mud Hut , Toque Ups Sandalia de vestir Eileen Mujeres Patente desnuda NAOT Sandalia plana Nala para mujer Beige Nubuck / hilos de oro de cuero , Taipán de mujer de David Tate Verona Cordero negro Sandalias de cuña con espalda descubierta Stuart Weitzman para mujer Multi Noir Sandalia de cristal con puntera de cuero ajustada FitFlop para mujer Blanco Urbano , Sandalias Bernie Mev para mujer, Capri Slide Negro , Yo también mujer Nakira Rose Gold Matte Metallic , Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Oro , Zapato de malla Nohea de OLUKAI - Hombre Blanco Northside Women s Trinidad Sandal Marrón / rosa Sandalia Gabby de mujer Cole Haan Ante negro , Sandalias mujer La Plume Jen champán Bomba Nikole para mujer Via Spiga Cuero del desierto Sandalias con plataforma SAS de cuero con punta abierta para mujer Negro L`Amour Des Pieds Womens Amadour Sandal Negro rojo Sandalia plana Nanci de Naturalizer para mujer Plata , Sandalia de patchouli OTBT para mujer Zinc , NAOT Womens Tucson Khaki / Gold Combo , Sandalia Tallis de mujer VANELi Blanco Verano Vip Sandalia del vestido de Fe de Naturalizer Azul, nubuck tranquilo , CLARKS Lexi Myrtle Sandal para mujer Cuero de arena , Earth Orígenes Nellie Rojo brillante Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Sandalias de tacón Gretchen de mujer de Jack Rogers Blanco , Sandalia de cuña Bolivia SoftWalk para mujer Broncearse Sandalia de tacón de gamuza Salomé de mujer Anne Klein Verde medio J.Renee Womens Soncino Blanco / Kidskin

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 Zapatillas de running adidas Adaptive Adapt para mujer Blanco / Blanco / Gris Brooks Womens Ariel 14 Perla ahumada / Malvavisco / Violeta Zapatillas de running Nike Kids Air Max Zero Essential GS Blanco / lobo blanco gris Amortiguación 635v2 New Balance para mujer Gris , Top corto de mujer Saucony Bullet Dalia Sandalia KEEN Women s Venice H2 Negro / Gris neutro Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Botas de trabajo Lugz Women s Shifter 6 \ Golden Wheat / Bark / Cream / Gum Thermabuck Zapato deportivo artesanal de mujer superfeet Addy Bálsamo / Blanco , Zapatillas de deporte Converse Chuck Taylor All Star Coral Ox para mujer Negro / Blanco / Negro CLARKS Womens Trigenic Flex Combinación grisácea , Saucony Women s Ride 10 zapatillas para correr Gris , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Toalla de vestir Dulce para mujer Touch Ups Champagne Shimmer Easy Spirit Women s Traveltime Mule rojo OLUKAI Alohi - Mujeres Barro rojo / barro rojo Vans Auténtico Abstracto / negro , Sandalia de mujer Frisco Slide de Franco Sarto Mezclilla Sandalia de diadema Florita Carlos de Carlos Santana Negro , VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe Negro Negro , TOMS - Slip-On de lienzo clásico para hombre Blue Marl , Bomba Lloret para mujer J.Renee True Red , Bella Vita Womens Tex-Italia Grey Suede , Botas de montar de mujer Tommy Hilfiger Ilia2 Black Multi Bomba de vestido exquisita Aerosoles mujeres Cuero negro , Mocasín de gamuza Kilty de Minnetonka para mujer Gris pardo U-lite para mujeres Medio talón multicolor Cordones perforados Wingtip Leather Flat Oxfords Vintage Zapatos Oxford Brownbiege , ED Ellen DeGeneres Mujeres Madaleina Camello / Blue Multi Oiled Calf Suede / Selvage Webbing Botas Carmina Hardy para mujer Mezcal marrón oscuro , Splendid Midi Perf Vionic para mujer Slip-On menta ,

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 Caterine de de cuero de charol Caterine para Patent mujer Chalk Patent 38ae5d1 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
1361573

1760851

6610986

10921242

11076925

15351839

15971501

16320807

17391229

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