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.

Cuero, sintético cuero o textil superior dependiendo en color

Espuma entresuela con visible Amortiguación de aire Max

Goma suela con Waffle patrón de durabilidad

Flex surcos permiten natural rango de movimiento

Mostrado: Blanco / polvo / solar Negro rojo

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 running NIKE Air Max B072Y341XQ 90 Air para de mujer Blanco/ negro/ polvo/ rojo solar e0cf44cUpdated 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 running NIKE Air Max B072Y341XQ 90 Air para de mujer Blanco/ negro/ polvo/ rojo solar e0cf44c

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 NIKE Free Run para mujer - SU16-10 - Blanco Dark Grey / Pink Blast / Blanco Zapatillas de running NIKE Air Huarache para hombre Partícula rosa / hueso claro , Zapatillas de running New Balance para mujer Vazee Coast Azul blanco , Nike Men s Air Presto Essential Negro / Verde pino / Gris neutro , adidas Performance Ultraboost X para mujer Gris / Gris / Hi-res Azul Zapato de tenis ASICS Gel-Dedicate 5 para mujer Mónaco Azul / Blanco Zapatillas de running NIKE Air Max 90 para mujer Vela / radiante esmeralda negro , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Blanco / azul índigo / lavanda Zapato de running ASICS Gt-2000 5 para mujer Diva Blue / White / Aqua Splash Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird Zapatillas Skechers Performance para mujer Go Run 400 de running rápido Azul marino / blanco , Nike Men s Air Presto Essential Binary Blue 400 , Glycerin de las mujeres de Brooks 14 Antracita / Azalea / Plata Zapatillas de running New Balance 990V3 para mujer Armada , Zapato deportivo para acolchado 635v2 New Balance para mujer Plata / Amarillo , Adidas Women s EQT Support ADV Originals Zapatillas para correr Correr Blanco / Core Black Amortiguación 635v2 New Balance para mujer Gris Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Nube de lluvia / Artic Fox Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas Wolf Gray / Cool Gray Zapatillas de running ASICS Gel-Contend 4 para mujer Azure Frosted Rose Zapatillas de deporte CL Nylon MH Reebok para mujer Eco azul / azul impresionante / verde azulado / añil / blanco / dólar , Zapatillas para correr NIKE Womens Sock Dart Glacier Blue / White Zapatillas de running Gel-Moya para mujer de ASICS Durazno Pétalo / Blanco Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul humo / azul oscuro / Cantalupo Zapatillas de running GEL-Kayano 25 (2A) para mujer ASICS 1012A019, color carbón / gris medio - 8.5 2A US - , Under Armour Women s HOVR Sonic NC Gris Nublado (101) / Blanco Zapatillas de running New Balance 990V3 para mujer Armada , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro , Glycerin de las mujeres de Brooks 14 Antracita / Azalea / Plata ,

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 Jacquard para mujer Negro / Metalic Plata / Blanco , Zapatillas deportivas 574S Sport New Balance para hombre Blanco Carro de mujer Brooks Chili Pepper / Black Zapatillas de entrenamiento Nike Womens Metcon 3 Persian Violet / Black-cool Gray Zapatillas de tenis ASICS Gel-Game 6 para mujer Gris medio / rosa fuerte / blanco , Chaco Women s Z2 Classic Athletic Sandal Edredón dorado , Zapatillas de running Saucony para mujer Hurricane ISO 2 Azul rosado KEEN Hombres Newport H2 Sandal Midnight Navy / Feather Grey , Sandalia de Hollyleaf de las mujeres de Merrell Gris pardo , Zapatillas de running NIKE Air Max 2017 para mujer Blanco / polvo New Balance Vazee Summit V2 Running Shoe Trail Runner para mujeres Dark Denim / Alpha Pink , Zapatillas de senderismo adidas outdoor Terrex AX2R - Mujer Negro / Negro / Rosa Táctil , Zapatillas de running Mizuno Wave Viper para mujer Negro-fucsia Pruple-plata Inov-8 Womens Trailtalon 250 Neon rosa / negro / verde azulado , Sandalia de vestir de cuero Gona de Nine West para mujer Rosado NIKE Women s Revolution 4 Running Shoe Ambiente gris Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Glaciar Gris / Blanco / Carbono , Zapatilla S Skechers para mujer E Z Flex 3.0 Swift N Sly Negro , SAS Womens Sunny Rojo / Arco Iris , Zapato de malla Nohea de OLUKAI - Hombre Negro Negro , Zapato de entrenamiento WX88V1 para mujer New Balance Azul amarillo Zapatillas de entrenamiento cruzado Realflex Train 4.0 de Reebok para mujer Blanco negro , Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave , Zapato para caminar Skechers Women s You Blanco negro , Vans Unisex Old Skool Zapatillas clásicas de skate marrón , Sandalia de tobogán plano de piscina Iona de Gentle Souls De color amarillo pálido , Rieker Mirjam 11 bomba para mujer Beige , Trotters Lucy Ballet para mujer Hueso Zapatillas de verano mahabis Sala Stone / Borgen Blue , Me Too Womens zena14 Suede Almond Toe Tobillera Botas de moda Gamuza floral negra

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 running NIKE Air Max B072Y341XQ 90 Air para de mujer Blanco/ negro/ polvo/ rojo solar e0cf44c 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