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

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

I.G.S. (Orientación de impacto) Sistema) Tecnología: Filosofía de diseño ASICS que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.

FluidRide Entresuela: FluidRide proporciona lo último combinación de recuperarse y propiedades de amortiguación con peso reducido y excepcional durabilidad.

Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.

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 de Púrpura running ASICS Gel-Cumulus 19 mujer para mujer Gel-Cumulus Azul Púrpura/ Negro/ Flash Coral 95a25fdUpdated 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 de Púrpura running ASICS Gel-Cumulus 19 mujer para mujer Gel-Cumulus Azul Púrpura/ Negro/ Flash Coral 95a25fd

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 New Balance 990V3 para mujer Gris Zapatillas de running New Balance 520v3 para mujer Bronce / Blanqueado Amanecer / Trueno Brooks Womens Ravenna 9 Rosa / Plum / Champagne , Zapatillas de running NIKE Mens Lunarglide 8 Blue Glow , Zapatillas de running NIKE Women s Air Zoom Structure 20 Negro / Blanco-lava Glow-chlorine Blue Zapatillas para caminar Reebok Women s Cloudride LS DMX Blanco negro , Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Verde Ceniza Amortiguación 635v2 New Balance para mujer Negro , New Balance Women s w990v4 Running Shoe Negro / Plata Calzado de running NIKE Kids Roshe One (GS) Rosado Zapatillas de deporte NIKE Womens Air Zoom Pegasus 32 Running 749344 Sneakers Shoes Azul Zapato de running ASICS Gel-Quantum 360 cm para mujer Flash Coral / Negro / Plata , Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Choque Verde / Negro , New Balance Women s w990v4 Running Shoe Gray / Castlerock Zapatillas de running adidas Women s Edge Lux W Colegial azul marino / azul claro / azul claro Zapatillas de running Reebok para mujer Pump Plus Vortex Seaside Grey / Mist / Silver Met , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosa Fuego / Negro / Crema de Melocotón , Zapatillas Reebok para mujer Trilux Run PNT Sneaker Prnt - Negro / Blanco Nike Men s Air Presto Essential Universidad / Rojo / Negro / Negro New Balance Women s w940v3 Running Shoe Plata , Lanzamiento de Brooks Womens 4 Negro / Fucsia Púrpura / Azul deslumbrante , Zapatillas de running NIKE Mens Lunarglide 8 Multicolor , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / desierto Moss-desierto musgo , Zapatillas de trail running Topo Athletic MT2 - Hombre Índigo / Fucsia Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino blanco / negro-sigiloso puro Camiseta adidas Swift Run W para mujer Trace Cargo / Trace Pink / Crystal White , New Balance Women s w990v4 Running Shoe Negro / Plata , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Electro Verde / Negro / Azul Mediano ,

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 Air Max Thea Ultra FK para mujer Blanco negro Zapatillas de running adidas Duramo 8 para mujer Turquesa Zapatillas deportivas 574S Sport New Balance para hombre Cáñamo / incienso Zapatillas de running Questr Cc W de mujer adidas Aero Blue S, Aero Blue S, Colegiata Marina MBT Shoes GT 17 Athletic Athletic Leather / Mesh Lace-up Blanco / morado , Zapatillas de trail New Balance para mujer Summit Arrecife / equinoccio Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Navy Marina , Zapato de running COMUTORA para mujer ASICS Negro / Negro / Blanco Sanda Verra Teva para mujer Surf Negro / Multi Zapatillas Slip-on de mujer Skechers You Inspire gris Mocasines y zapatillas Slip-ONS Monica de TOMS para mujer Chambray azul / Cuero , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy de arena Sandalia de cuña Karmen de mujer Alegria Diamantes para siempre Zapatillas para mujer Reebok Zprint Her MTM para mujer Negro / Gris Ceniza / Blanco , David Tate Wish Sandalia para mujer Plata Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers CHOCOLATE / CASI NEGRO / BLANCO Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Blanco / rojo / azul , Zapatillas de golf adidas W Climacool II para mujer Clear Aqua / Running blanco / plata mate , Sandalia de cuero Nirveli de Nine West para mujer Cuero negro , La Plume Jen de las mujeres Cobra roja , Charles by Charles David Sandalia mujer Kadia Slide Negro , Sandalia de plataforma de playa Coast de Seychelles Blanco Zapato de voleibol Gel-Tactic 2 para mujer de ASICS Rendimiento Negro / Pixel Rosa 50 sombras de blanco 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) 1620- Satén blanco Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Blanco , Regard Clinical Paciente médico Pantuflas / calcetines dobles Zapatilla, Adulto XL, SZ 10 a 11, Azul claro - , Coleean GrandEvOlution Shortwing Oxford para mujeres Pink Glitter , Bota holgazana novata Fergalicious para mujer Negro , CLARKS Mujer Ashland Bubble Slip-On Cuero 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 de Púrpura running ASICS Gel-Cumulus 19 mujer para mujer Gel-Cumulus Azul Púrpura/ Negro/ Flash Coral 95a25fd 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