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

Sendero específico Sistema ISOFIT con protección adicional que se sienta más cerca al ajuste.

Agregado capa vampiresa de fusión en caliente para reforzado protección

Externa marco de soporte para protección adicional y un encierro ajuste de talón.

3 mm Suela superior EVERUN construcción para amortiguación continua durante el correr

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 11179 para correr Saucony Xodus Iso Saucony 2 para Coral mujer Berry Coral 632556fUpdated 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 11179 para correr Saucony Xodus Iso Saucony 2 para Coral mujer Berry Coral 632556f

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 Fresh Life Arishi V1 para mujer New Balance Gris claro , Zapatillas de running NIKE Air Max 90 para mujer Vela / radiante esmeralda negro , Newton Running Womens Fate II Verde Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blue Vapor / Camp Smoke , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blanco Blanco , Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco Zapatillas de entrenamiento Nike Womens Metcon 3 Negro / Blanco-volt-cloro Azul Zapatillas de running NIKE RN 2017 para mujer gratis Negro / Blanco / Gris Oscuro / Antracita Zapatillas de running NIKE Air Huarache para hombre Gris frío Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Negro / Fantasma Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Core Black / Core Black / Ice Purple , Zapatillas de running New Balance 790v6 para mujer Black / Poison Berry Brooks Womens Hyperion Imperial Purple / Blue Jewel / Vida Nocturna Calzado de running NIKE Kids Roshe One (GS) Azul Calzado de running Pearl iZUMi para mujer Em Road H3 v2 Perla ahumada / vino morado Zapato deportivo para acolchado 635v2 New Balance para mujer Gris Zapatillas de running Reebok Print 2.0 para mujer Nube gris / azul cielo / negro / azul gable / blanco / plata , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de running Altra Women s Intuition 4 Gris / morado Zapato deportivo para acolchado 680v5 New Balance para mujer Coral Zapatillas de running adidas Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco , Zapatillas de running adidas Women s Edge Lux W Gris / gris / blanco cristal Zapatillas de running Flex Ride 630v5 para mujer New Balance Gris oscuro , Zapatillas de running acolchadas 420v4 para mujer New Balance Gris , Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Gris oscuro / Blanco / Negro / Sigilo , Zapatillas de running adidas FLB_Runner Originals para mujer Trace Scarlet / White / Trace Scarlet , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , Zapatillas de running NIKE Air Huarache para hombre Mango brillante / blanco ,

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 New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Saúco mobils by Mephisto - Sandalias de tacón Getha para mujer Black Queen / Magic Leather Zapatillas Drew Shoe para mujer Paige Hueso , Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Brillante Crimson / Prm Rd-atmc Pink-wh Sandalia Chaco Mujer Zvolv X2 W Lluvia Zapato de running FUZETORA para mujer ASICS Negro / Negro / Blanco , Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Gris , Ballet de niña de flores Aerosoles femenino Negro exótico Zapatillas clásicas Vionic para mujer Walker Blanco / azul / gris Sandalias Bernie Mev para mujer, Capri Slide Negro , Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro Chaco Mujer Maya Sandalia Gris , Sandalia Barroll T Correa de Natural Soul para mujer Mezclilla , BareTraps Women s Dasie Platform Sandal Armada Calzado para caminar WW812 para mujer New Balance Negro , Sandalias de mujer Nina, Georgea Slide Negro Zapato de tenis 7 de ASICS Gel-Resolution femenino Plata blanca , Zapatillas deportivas Superga Womens 2750 Cotu Canvas Gris New Balance WX811v2 High Top Cross Trainer para mujeres Gráfico de espacio negro / exterior , Calzado de entrenamiento de cuero Prim WP de mujer Vivobarefoot Negro , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Negro Negro , Sandalia de vestir Lauren de Jack Rogers para mujer Pewter / Pewter , Sam Edelman Women s Bay Slide Sandal Cuero de silla , Ringor Bandit 2.0 Spike- Pitching Azul marino / blanco Pantuflas de lana extrafinas para hombre Foamtreads Armada , Sandalias de plataforma ocasionales flojas del dedo del pie de la tela florentina del coche de las sandalias Chalk Matte Calf / Chalk Lux Snake , Bota de senderismo impermeable X Alp MTN GTX para hombre Salomon Negro / Asfalto / Pulga Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse Newport OXF Oxford de Paul Green para mujeres Grafito , Nine West Toastie Oxfords Womens Flats Cuero Marrón Oscuro

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 11179 para correr Saucony Xodus Iso Saucony 2 para Coral mujer Berry Coral 632556f 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