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 importado

tiras Soncino alcanza un nuevo nivel de sofisticado sex appeal a su estilo.

Rico parte superior de gamuza con un dedo abierto y correas envolventes en el tobillo.

Ajustable cierre de hebilla en el tobillo.

Suave revestimientos sintéticos.

Ligeramente memoria amortiguada plantilla de espuma y tacón envuelto. Suela hecha por el hombre. Importado. Altura del talón: 2 3 y frasl; 4 en peso: 7 oz Mediciones del producto fueron tomadas usando tamaño 8.5, ancho M (B). 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

J.Renee Womens B0163GIM0Y Soncino Negro Negro Soncino 2 8f9f725Updated 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.

J.Renee Womens B0163GIM0Y Soncino Negro Negro Soncino 2 8f9f725

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

Bernie Mev Women s Comfi Flat Brillo negro J Slides Sandalia Emmie para Mujer Negro , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Coñac Drew Womens Element Nubuck naranja , Sandalia Miel Mooz para mujer Negro , CLARKS Sillian Stork Fisherman Sandal para mujer Negro CLARKS Womens Wynnmere Avah Nobuck gris cálido , Sandalia para tobillo ECCO Women s Touch 25 Camello / Camello , OTBT Nomadic Negro , Sandalia Kinsley de Naturalizer para mujer Papaya Imprimir JF Shoes JF Embellished Sandals Brindan hebillas Extra Comfort para los tobillos Dorado Sandalia Drew Cascade para mujer Denim Blue Nubuck , Bomba de cuña Dreamstride Women s Dreams Classic Navy , Sandalia Janny de BareTraps para mujer Rosa roja Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero negro , Sam Edelman Womens Barry Cuero metalizado pulido de oro fundido oscuro , Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja , Skechers - Sandalia deportiva de 600 brilliancy para mujer Natural , Sandalia Tallis de mujer VANELi Blanco Verano Vip Stuart Weitzman Sandalia Nudistju para mujeres Negro Sandalia CLARKS Wave Grip para mujer Smokey Brown , OLUKAI Pua Sandal - Mujer Deep Red / Bean , Sandalia de cuña Logan Walking Cradles para mujer Cuero Antanado Negro Suave Sandalia Gabby de mujer Cole Haan Ante negro Sandalia Pamela Mujer Mephisto Negro , CLARKS Sultana Rayne Sandal para mujer Cuero caqui Sandalias de cuero de Viv Viv de TOMS Yute metálico natural , Sandalia Minnie Trotters para mujer Nubuck 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 deporte Reebok Print Run Dist para hombre Negro / Aleación Sandalia Uneek O2-w de mujer KEEN Acero Gris / Vapor , Zapatillas de trail running 910 V4 New Balance para mujer Vivid Coral / Clear Sky Zapato de running ASICS GT-2000 3 para mujer Blanco / Rayo / Polvo azul Zapatillas de running adidas Alphabounce Rc W para mujer Gris claro Heather / Light Solid Gray / Medium Solid Gray Zapatillas de baño Skechers Performance para mujer Go Step-Seashore marrón Calzado SCARPA Women s Crux Walking Camel / Poppy Red , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Blanco Chaco Mujer Zcloud Sport Sandal Noche Emplumada Skechers Performance Women s On The Go City 3 Optimize Armada , Naturalizer Samantha Pointed Toe Flat para mujer Caramelo , Zapatillas para caminar WW669V1 New Balance para mujer Gris , Ahnu Women s Yoga Flex Orquídea oscura Skechers Women s Define Sneaker Rosado Zapato para caminar Eden Propet para mujer Marfil ASICS Tiger Womens Gel-Lyte V Skyway / Skyway Vans Unisex Old Skool Zapatillas clásicas de skate Cabernet Mono Bumper / True White Zapatillas de golf adidas W Climacool Knit para mujer Rosa / Coral , Sandalia Mason para mujer Naturalizer Slide Cuero metalizado lila Diapositivas de Sam Edelman Gertrude para mujeres entrecruzadas Avena Kid Gamuza Cuero , Sandalias Jerusalem Sandalias Deborah Slide Miel 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) S92309 Trenza dorada y plateada , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 6030- Tan satinado Zapato Capezio Mary Jane Tap para mujer Negro Zapatillas Pretty You London Crema Zapatilla Mindy Mocasín BEARPAW para mujer Hickory Patchwork Mocasines de conducción Twisted X para mujer, bombardero / naranja marrón , Sandalia de vestir Taimi de Naturalizer para mujer Brillo negro Bota Cwen Gwen de Hush Puppies para mujer Gamuza negra crujiente , Skechers BOBS de Women s Plush-Wag Party Flat Multi ,

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 J.Renee Womens B0163GIM0Y Soncino Negro Negro Soncino 2 8f9f725 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