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.

Poliéster

Importado

Goma único

Eje mide aproximadamente 0,88 del arco

NOTA: 'W' mencionado en la caja del producto y el producto denota 'Mujeres, sin ancho

Clásico sandalia atlética presentando gancho y lazo correas de correas con talón acolchado correa

Contorneada Plantilla de EVA y entresuela

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 original Teva Women s original Original Sandalia Monterey Tan Tan a1a5bddUpdated 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 original Teva Women s original Original Sandalia Monterey Tan Tan a1a5bdd

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

Sandalia KEEN Women s Uneek Slide-w Aqua Sea / Turquesa Pastel , Sanda Verra Teva para mujer Surf Taupe Multi OluKai Ohana Basalto / gris Zapato de agua H2 Go Skechers Performance para mujer gris Sandalias KEEN Mujer Whisper-w Ágata gris / azul ópalo , Sandalias KEEN Venice II H2-W para mujer Negro / gris acero Wolky Comfort Jewel 70000 canales negros Sandalia de mujer Drew Alana Pewter-bronce , Sandalia atlética de Rka Women s Glance Azul / verde azulado , Chaco Mujer Diana Sport Sandalia Avo florentino , Teva Terra-Float Active Lace - Mujeres Paloma Salvaje Sandalia NIKE Kawa Slide para mujer Cloro Azul / Blanco / Obsidiana , Sandalia Merrell para mujer Terran Ari Post Sport Acuífero Sandalia Terrell Convertible II Merrell para mujer Negro CLARKS Sandalia Saylie Jade para mujer Cuero marrón caído Sandalia Drew Angela para mujer Dusty Multi Metallic Leather , Sandalias Newport H2 de KEEN para mujer Cuervo / Capri OluKai Kulapa Kai Negro plata Sandalia Savannah de Ryka para mujer Gris blanco , Sandalia deportiva Chaco Zvolv X2 para mujer Dash Black , KEEN sandalia de Venecia para mujer Tierra oscura / Mar Caribe Chaco Women s Z1 Classic Athletic Sandal Verdure Eclipse Sanda Verra Teva para mujer Surf Negro / Multi , Sandalias Therafit Summer Women s Slingback Sport - para fascitis plantar / dolor en el pie Negro , Sandalia atlética de Rka Women s Glance Negro / rosa , KEEN - Sandalia de agua EVOFIT ONE para mujer para aventuras al aire libre, 6 M US, Triple Black - , Skechers Performance On The Go 400 Vivacity Flip Flop Negro Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Pimenton , Sandalias Bali KEEN para mujer Nórdico azul / azul ópalo ,

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 Chaco Zcloud 2 Sport Sandal para hombre Salute Forest Zapatillas de running Hupana 2 para mujer de HOKA ONE ONE KEEN - Sandalia de agua EVOFIT ONE para mujer para aventuras al aire libre, 6 M EE. UU., Jaspeado negro / imán - , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Negro Zapatillas de paseo Slip-on Go Step Lite de Skechers Performance para mujer Heather negro / gris Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo , Trotters Mujer Simmy Mary Jane Flat Ante negro SAS Womens Sunny Azul profundo Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra , Saucony Women s Guide 8 Running Shoe Blanco / Crepúsculo / Rosa Sandalia de cuña Bolivia SoftWalk para mujer Broncearse Skechers Hombres Merodeador Mershon Azul marino / negro , Zapatillas de entrenamiento cruzado Froma FM de PUMA Tazon 6 para hombre Puma Black Zapato Dantera, para hombre, L Low Waxed Court Negro / Blanco / Negro , Zapatillas de skate DC Evan Hi Tx Se W para mujer Mezclilla , Sandalia Rockdale Lanea Woven Slide para mujer Tejido azul marino , Birkenstock Arizona Sandalia de ante con suela suave Taupe Suede , Sandalia para mujer Unterman de ALDO Blanco Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Azul marino / gris , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Brillante Crimson / Ice Blue / Sail , Mocasín Havlinger ASV MC para mujer Gris Sandalia Birkenstock de cuero Florida para mujer Sfb Blue Birko-flor , Bota del oeste de Denver de Dan Post para hombres Chocolate Zapatilla de deporte Cotleyu 2750 Superga para mujer Khaki / Gold Ta-Randi Tahari para mujeres Negro Ziera Alley Women s Lace Up Negro / Stingray , Sandalia de gamuza Poppy de TOMS para mujer Sombra de gamuza / lino Trotters Liz Loafer para mujer Armada Mocasín slip-on Jill de Steve Madden para mujer Red 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 Sandalia original Teva Women s original Original Sandalia Monterey Tan Tan a1a5bdd 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