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.

Microfibra

Importada

Get un poco de estilo Caliente eso hará tu mirada caliente, caliente, caliente!

Micro o superior de cuero en un cabestrillo estilo de bomba.

Elasticized correa para la honda para un ajuste flexible.

Pull-on diseño para fácil encendido y apagado.

Transpirable forro de cuero para la comodidad todo el día. Ligeramente plantilla acolchada para la comodidad y apoyo.Cuero suela.Importado.Medidas: Altura del tacón: 2 1 & frasl; 2 in Peso: 8 oz Mediciones del producto fueron tomadas usando tamaño 8, ancho W (RE). que las medidas puede 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

Bomba de vestido caliente de Micro mujer Rose Petals mujer Pewter caliente Micro Point 9f7d807Updated 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.

Bomba de vestido caliente de Micro mujer Rose Petals mujer Pewter caliente Micro Point 9f7d807

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

Botín Hayes Gore de Cole Haan para mujer Ante negro , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro Soft Style de Hush Puppies Women s Jaylene Oxford Gris oscuro Nubuck , ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Pisos tachonado Zapatillas Zapatos Patente negra Sneaker para mujer Earth Serval Armada Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro Aravon Betty-AR Oxford para mujeres Negro Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Cr negro , Dolce Vita Saxon Oxford para mujer Negro , TOMS Kids 10009918 Alpargata-K Pale Blush Metallic Party Dots TOMS Kids 10009918 Alpargata-K Flequillo metálico de granito gris hierro forjado Zapatilla de deporte sin cordones sin mangas lavada Chillax de Keds para mujer Carbón , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Negro / Combo , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Denim Lea / Nubuck , Gravati Womens Calf Leather Wing Tip Oro Zapatos de mujer Cole Haan Zerogrand Wingtip Oxford II para mujer Tawny Port Suede / Blanco Easy Spirit Women s Motion Lace up Oxford Negro Sebago Women s Meriden Kiltie Oxford Cuero tostado / blanco Easy Devitt Oxford Flat para mujer Oro , Ros Hommerson Annie Zapato informal para mujer con cordones de cuero Negro / Combo , Propet Travelactive Slip-On Oxford para mujeres Heather negro / gris , Ros Hommerson Nadia Oxfords para mujer Desnudo, Bronce Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Rosa oscuro / gris suave Aravon Betty-AR Oxford para mujeres marrón , OTBT Womens Lunar Sneaker Paloma gris Easy Spirit Neota Oxford para mujer Cuero negro / negro U-lite para mujeres Medio talón multicolor Cordones perforados Wingtip Leather Flat Oxfords Vintage Zapatos Oxford Brownbiege , Skechers Womens Alba - Cozy Lane Negro Negro , Drew Shoe Women s Pepper Criss Cross cuero, caucho Casual Oxfords Negro ,

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 NIKE Air Max Thea Jacquard para mujer Negro / Negro / Azul Racer / Blanco OLUKAI Pua Sandal - Mujer Frijol negro , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Azul hielo / azul hielo / azul , Zapatillas para correr Saucony Women s Triumph 11 Gris / verde azulado / cidra , Calzado de running New Balance para mujer Vazee 2090 Azul marino / amarillo Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Melocotón espolvoreado , Zapatillas de running adidas Women s Rockadia W Trail Trace Púrpura / Blanco / Core Black , Zapatillas de trail running Ryka Women s Intent XT 2 SR Negro / plata cromo Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro KEEN Women s Uneek 8MM Sandal Anemone / Very Berry Zapatillas de running NIKE Lunar Skyelux para mujer Celadon-azul Zapatillas de deporte con cordones Sneakers de hombre Carson Runner de PUMA Brillante Plasma-blanco Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Super Sock 2 Slip-On Viejo carbón de leña , Teva ALP Premier - Mujeres Negro Sandalia Gora Yoga Sanuk para mujer Olivo oscuro Crocs Flip Crocband unisex Rosa (Paradise Pink / White) , Sandalia Skechers Mujer Reggae T correa Azul marino Multi étnica Imprimir Converse Chuck Taylor All Star Core Ox Rojo / negro / blanco , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Blanco negro , Zapatillas de skate Vans Unisex Old Skool (Double Light Gum) Rosa (goma doble luz) Tiza rosa Qk7) , Clásicos de base Slip-on (Tm) para hombres de Vans PANEL NEGRO NEGRO , Supra mujer Skytop Sneaker Tiza malva / tiza malva NIKE Lunar Command 2 Golf Shoes 2017 Mujeres Blanco negro Zapatillas de ciclismo híbridas ECM Saddle para mujer ECCO Negro , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ Sandalias sintéticas Pleaser, mujer, cielo 301MR Claro, peltre negro Zapatilla de deporte Blair Fashion de Naturalizer para mujer Mármol suave Zapatilla Smoking Boxers Platform para mujeres de Soludos - Arena Beige Zapato de pista Md800v5 New Balance para mujer Llama blanca A.S.98 Botín de mujer Brock para mujer 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 Bomba de vestido caliente de Micro mujer Rose Petals mujer Pewter caliente Micro Point 9f7d807 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