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.

Sintético y Textil

Importado

Sintético único

Eje mide aproximadamente parte baja del arco

ACTEVA entresuela

ABZORB tacón

Respuesta NB 2.0 Performance insertar

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

Zapato deportivo para acolchado 680v5 B01GE1L0WA New mujer Balance para 15036 mujer Coral cb5a730Updated 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.

Zapato deportivo para acolchado 680v5 B01GE1L0WA New mujer Balance para 15036 mujer Coral cb5a730

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

Zapatillas NIKE Mujer Air Max Sasha Black Lobo gris / blanco , Zapato de running 840v4 New Balance para mujer Gris claro Zapatillas de running Mizuno Wave para mujer con sombra Topacio azul / Coral Fiery / Azul Imperial , Zapatillas de running Saucony Women s Freedom ISO Berry / rosa , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro , Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver Nike Men s Air Presto Essential Blustery / Terra Orange , Zapatillas de running NIKE Lunartempo 2 para mujer Black / Summit White-hot Punch Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Violeta plata / persa negra / metálica NIKE W Air Vapormax Flyknit C para mujer Ah8448-001 Atmósfera Gris / Blanco-blanco-caliente Punch Zapatillas de running EM Road N1 v2 Pearl iZUMi para mujer Grey / Deep Wisteria , Under Armour Threadborne Fortis para mujer Negro (003) / Lapis Blue , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Negro / sal marina / goma de mascar , Zapatillas de running acolchadas 420v4 para mujer New Balance Silver Mink Zapatillas de running NIKE Fs Lite 2 para mujer Imán de luz gris / Hyper Punch / Magnet Gray Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Stealth Gray / Glacier Grey / Pink Chroma Zapatillas de deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Pure Platinum Hyper Violet 001 Zapatillas de running NIKE Roshe Two para mujer Caqui / Negro / Negro / Blanco , Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Blanco / Menta-negro Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Azul marino / rosa / verde azulado Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink , MBT Shoes Mujeres GT 17 Athletic Shoe Cuero / Malla Con cordones Blanco / morado Zapatillas de trail running Inov-8 Inov8 para mujer X-Talon 230 Gris / verde azulado Zapatillas de running NIKE Fs Lite 2 para mujer Gris oscuro / explosión de Jade-rosa / blanco , Zapatillas de running EM Tri N1 v2 para mujer Pearl iZUMi Blue Atoll / Electric Pink , Zapatillas de running W1540V2 New Balance para mujer Blanco azul Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Azul Cloro / Antracita / Gris Fresco Zapatillas de running NIKE Women s Air Zoom Structure 20 Midnight Turquoise / Green Glow / Ghost Green / White

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 OLUKAI Pua Sandal - Mujer Frijol / frijol , Zapatillas de entrenamiento cruzado Heather para mujer New Balance 711v3 Gris oscuro Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa Mizuno Women s Wave Prophecy 5-w Jacinto Violeta / Royal Purple , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Estratos Earth Womens Allegro Ballet Flat Rose Gold Pearlized Soft Leather Under Armour Women s HOVR Sonic NC Brillo (601) / Elemental MAIERNISI JESSI Clásico de mujeres para hombres Estiramiento de dos correas Sandalias de tacón bajo (Las mujeres necesitan 1.5-2 Tamaño abajo) rojo Drew Shoe Women s Paradise II Velco Slip-on Taupe ternero Zapatilla deportiva Sperry para mujer Seacoast Core Gris , Zapatilla Ascot de hombre UGG Dusty Olive / Sahara Prop & eacute; t Zapatillas para caminar TravelFit para mujer Naranja / Rosa Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Plano extra ancho Feet Breeze Womens Zapatos para caminar , Zapatillas de deporte Dr. Comfort Sandy para mujer Gris , Zapatillas deportivas sintéticas Drew Shoe para mujer Flamenco Imprimir , Zapatillas para mujer Del Rey TOMS Desierto de color topo Diadora Unisex Intrepid NYL Blue Estate , Under Armour Threadborne Push para mujer Negro (001) / Acero Lakai Limited Footwear Mens Griffin Negro / Reflective Suede , Lakai Limited Footwear Mens Griffin Textil negro , Zapatillas Casual NIKE WMNS Lunar Force 1 Duckboot para mujer Sidra / Sidra-negro-gris pálido , Birkenstock Mujer Mayari Birko-Flor Sandal Amarillo ámbar Birko-flor , kensie Sandalia Dominic Slide para mujer Floral negro , UGG Sandalia Cindi para Mujer Blanco negro , Zapatillas Dr. Mary Comfort Flauta Lycra Elasticables Diamante Mary Jane Licra Calzas de la dinastía Plata blanca , Bomba de vestido LifeStride Sevyn para mujer Rosa Multi Bota Cobb Hill Joy de Rockport para mujeres Negro , Bomba de vestido de pasión Easy Street 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 Zapato deportivo para acolchado 680v5 B01GE1L0WA New mujer Balance para 15036 mujer Coral cb5a730 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