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.

100% Textil

Importado

Caucho único

-Water-listo cinta de poliéster la parte superior es extremadamente durable y seca rápidamente después de recibir mojado

Fácil gancho y lazo cierre viene y apagado rápidamente y consigue el ajuste simplemente bien

-Injected Suela de espuma EVA para peso ligero, confort acolchado

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 Teva universal W Sanborn de Teva Blue para mujer 10759 Nautical Blue 83e5f0fUpdated 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 Teva universal W Sanborn de Teva Blue para mujer 10759 Nautical Blue 83e5f0f

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

Sandalias fabulosas para mujer Caress 401LS Claro Negro / Claro Sandalia para mujer NIKS Benassi Solarsoft Slide Arctic Pink / Black , Sanda Verra Teva para mujer Orquídea púrpura , KEEN Hombres Newport H2 Sandal Azul marino / gris medio Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Helecho , ALMA NATURAL Sandalia de pescador Beatrice para mujer Rosa , Sandalias Columbia Sorrento Flip Athletic para mujer Negro, grafito , Sandalia Jacardia Merrell para mujer Armada , Teva Terra Sandal para hombre Cross Terra Black Revs Seva Sandals, Sandalias de reflexología Para hombres y mujeres. Absorción de impactos, Cojín Comfort & Soporte del arco marrón , Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón , Sandalia de mujer KEEN Rose Aluminio / gris neutro , CLARKS Womens Wave Grip Sabio , Sandalia de espalda de mujer Merrell Women s Town marrón Sandalias Bali KEEN para mujer Red Violet / Boysenberry , Sandalia de mujer KEEN Rose Púrpura (Sabio púrpura oscuro / púrpura) Sandalia Drew Element para mujer Nubuck marrón CLARKS Sandalia Saylie Jade para mujer Arena , Sandalia Uneek O2-w de mujer KEEN Anemone / Bright Rose Tacones altos Mujeres Chanclas Sandalias de verano Plataforma Cuñas Zapatillas Chica39; s Fashion Beach Shoes rojo Sandalia NIKE Kawa Slide para mujer Blanco negro , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Pimenton Sandalia Uneek O2-w de mujer KEEN Estate Blue / Harvest Gold Revs Seva Sandals, Sandalias de reflexología Para hombres y mujeres. Absorción de impactos, Cojín Comfort & Soporte del arco Negro , Northside Womens Savannah Sport Closed Toe Sandal Gris / Aqua , Sandalia Drew Angela para mujer Cuero liso negro , Sandalia KEEN Women s Exo-W Uneek Negro / gris acero Wolky Comfort Jewel 12750 Cal y Nubuckleather , BZees Sandalia con cuña de beso para mujer 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 Earth Origins Sophie Platino , Zapato de running GEL-Excite 3 para mujer ASICS Rosa fuerte / Blanco / Amarillo intermitente , Sandalia deportiva Nolita para mujer Anne Klein Tela negra , Brooks Women s Dyad 9 Ombre Blue / Festival Fuchsia / Mood Indigo Pink Palms mujeres zapatos de verano café remaches de cuña de mujer decoración alta tacones peep toe correa de tobillo cómoda sandalias de fiesta Skechers Performance Mujeres H2 GOGA Laguna Flip Flop Negro / gris , Zapatillas de running NIKE Air Zoom Vomero 12 para mujer Platino blanco / negro puro La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Fjord / Malibu Blue Under Armour Ua Thrill para Mujeres 2 Gris , Zapatillas para caminar Go Walk 4 Super Sock 4 de Skechers Performance para mujer Negro Calzado para caminar CUSH + WA365v1 New Balance para mujer Gris blanco , Sperry Top-Sider Women s Sunkiss Pearl Sandalia Marfil / Rosa Skechers Performance para mujer Go Step Lite Deluxe Carbón Naturalizer Samantha Pointed Toe Flat para mujer Satén negro Zapatillas Cross-Trainer PUMA para mujer Fierce Swan WN Puma White-puma White , Sandalia de 2 piezas Cobb Hill Women s Hollywood Cuero caqui Jessica Simpson Sandalia de tacón de mujer Mitta Pulir FitFlop Women s The Skinny con tachuelas Sandalia Aterciopelado , Zapatillas tenis NIKE Air Zoom Ultra para mujer Gran gris / negro / blanco , Zapatillas de tenis 696v3 New Balance para mujer Pigmento / coral vivo Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer gris , Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Azul marino / blanco , Sandalia de diapositivas de ocio para mujer de Seychelles Broncearse , Calvin Klein Dalana Slide Sandalia para mujer Plata / Oro rosa / Estaño Birkenstock Arizona Sandalia de ante con suela suave Cuero Habana Aceitado , Birkenstock Mules para mujeres Azul , Zapatillas de ciclismo de Triatlón Air Carbon T4 para mujer Sidi Fucsia Blanco , Twisted X Ladies Tough Basket Driving Mocs Bomber / Neon Pink Leather , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Juanetes caminando Whitney Womens Sneakers Athletic Zapatos Negro Zapatillas de deporte Easy Spirit AP1 Sport Vino ,

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 Teva universal W Sanborn de Teva Blue para mujer 10759 Nautical Blue 83e5f0f 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
11642628

11726354

12454652

12982671

14109296

15671607

17759256

18017163

18317841

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