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

Goma único

Modelo Número: 831070405

Sexo: para mujer

Color: OCEAN FOG / NEGRO MANGO BRILLANTE

Hecho En: China

Marca Nuevo con Original Caja

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

Zapatillas Free Flyknit para correr Nike Women s Free Rn Flyknit para 2017 Gris 62fa0ebUpdated 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.

Zapatillas Free Flyknit para correr Nike Women s Free Rn Flyknit para 2017 Gris 62fa0eb

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 de running adidas Ultraboost W para mujer Blanco / blanco / blanco cristal , NIKE Men s Flex 2017 RN Negro / Metálico Hematite / Tough Red , Zapatillas de running W775V2 para mujer New Balance Azul blanco Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Obra azul / obsidiana oscura , Zapatillas de running Reebok Print 2.0 para mujer Negro / azul polar / peltre Zapatillas para correr Mizuno Women s Wave Rider 17 Blanco Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta Zapatillas para correr Mizuno Women s Wave Paradox 4 Patriot Blue / White Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel , Zapatillas de running adidas Originals Tubular Defiant PK Originals para mujer Sun Glow / calzado blanco , Brooks Womens Adrenaline GTS 18 Blanco / morado / gris Zapatillas de running Gobiv2 para mujer New Balance Poisonberry / Black , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello NIKE Women s Revolution 4 Running Shoe Blue Recall / Pure Platinum / Purple Slate Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Carbon / Black / Aruba Blue Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal Zapatillas para correr Nike Downshifter 6 para mujer Negro / antracita / blanco MBT Shoes Mujeres GT 17 Athletic Shoe Cuero / Malla Con cordones Gris / morado Calzado de running NIKE Flyknit Lunar 2 para mujer Rosado , Brooks Women s Dyad 9 Ombre Blue / Festival Fuchsia / Mood Indigo Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Solar Rojo / Negro / Mango Under Armour Women s Thrill 3 Lapis Blue / Mdn / Icelandic Rose , New Balance W880gb7 para mujeres Azul claro , Saucony Women s Guide 8 Running Shoe Blanco / Crepúsculo / Rosa Zapatillas de running NIKE Air Huarache para hombre Hyper Turquesa / Blanco , Zapatillas de running Reebok Print 2.0 para mujer Negro / azul polar / peltre , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Gris oscuro / Azul Racer / 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 para mujer Lunarepic Low Flyknit 2 Azul profundo profundo / azul mediano Zapatillas de running adidas ZX Flux para mujer Menta / Blanco / Verde , Zapatillas de running Kinvara 6 Saucony para mujer Rojo / naranja / menta , OluKai Kulapa Kai Blanco negro Mizuno Women s Wave Hayate 3 Trail Runner Dark Shadow / Tangerine Tango Helly Hansen 11308 Zapatillas Santiago L.20 para mujer PINGÜINO / NIMBUS NUBE Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On Negro , Brooks Womens Adrenaline GTS 18 Blanco / negro / gris , Zapatillas de running ASICS Gel-Kinsei 6 para mujer Rosa fuerte / Blanco / Amarillo intermitente Sandalia Imogene Gladiador de Naturalizer Camelot Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Azul (estilete) Walking Cradles Women s Bounce Flat Tela elástica negra Zapatillas de deporte Fila Philadelphia s Cage Delirium Dark Shadow, Pink Glo, Royal Blue , Zapatillas de tenis acolchadas 786v2 de New Balance para mujeres Blanco azul , New Balance Wx577v4 Cross Trainer para mujeres Negro / Alpha Pink Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Blanco Zapatillas de tenis NIKE Women s Air Zoom Prestige Hardcourt - Blanco / Hot Lava / Pure Platinum / Blue Nebula , Zapatillas de tenis adidas Ligra 5 W para mujer Blanco / Plata Metálica / Plata Metálica Zapatillas de skate Vans Unisex Authentic Platform Cadmio Verde 7295 , Sandalia Mason para mujer Naturalizer Slide Negro ALMA NATURAL Sandalia Adalia Slide para mujer Negro , Sandalias de la diapositiva del dedo del pie abierto de la moda de las mujeres de XYD Deslizamiento de cuero en las zapatillas de la zapatilla del vestido de la puerta Plata Zapatos de ciclismo de interior de Louis Garneau Actifly para mujeres, una colaboración con Reebok TOMS Clásicos de mujer Gris Zapato CLARKS Wallabee para mujer Suede verde oscuro Blundstone Super 550 Series Boot Rustic Brown Plantilla de fútbol adidas Performance Women s F30, con suela firme, tierra W Solar Red / Running Blanco / Negro , Zapatillas de deporte Easy Spirit AP1 Sport Blanco Merrell Women s Jungle Moc Slip On Medianoche , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Hojas de encaje 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 Zapatillas Free Flyknit para correr Nike Women s Free Rn Flyknit para 2017 Gris 62fa0eb 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
48878

130223

419912

1992037

2510124

6393785

10375801

12520451

15588393

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