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.

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mate TPU tres rayas. Moldeado por inyección Entresuela de EVA para amortiguación ligera

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

Camiseta adidas Swift Run Blanco W Run Blanco para mujer Blanco/ Blanco/ Blanco 8883a9aUpdated 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.

Camiseta adidas Swift Run Blanco W Run Blanco para mujer Blanco/ Blanco/ Blanco 8883a9a

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

Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Negro / blanco-cool gris Zapatillas de running NIKE Air Huarache para hombre Partícula rosa / hueso claro HOKA ONE ONE Zapatillas para mujer Arahi Azul Medieval / Lavanda Zapatillas de running Adidas Women s Response 3 con alto hasta el tobillo Negro / Gris oscuro / Gris oscuro Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Choque Verde / Negro , WMNS Zoom Fly Provence púrpura / negro Zapatillas de running NIKE Zooms Winflo 4 para mujer Negro / Blanco / Gris Oscuro Calzado de running Reebok Print Run Prime Ultk para mujer Blanco / Calavera Gris / Melocotón Twist / Acero , PUMA Men s Roma Basic Sneaker Blanco / Peacoat / avena , Zapatillas de running adidas Adizero Boston 6 W para mujer Azul , Under Armour Women s Dash 2 Blanco negro Zapatillas de running Brooks Women s Glycerin 16 Road Ébano / verde / negro Zapatillas para correr ASICS Gel-Venture 6 para mujer Castlerock / Silver / Honeydew Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco negro Nike Hombres Air Zoom Pegasus 33 Concord / Blanco-negro-eléctrico Verde , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul Brooks Womens Adrenaline GTS 18 Púrpura / negro / champán , Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird , ONEMIX Zapatillas para correr Air Cushiong para mujer, Zapatillas deportivas deportivas ligeras Vistoso , Zapatillas de running adidas Women s Cf Racer Tr W, mujer Blanco / blanco / plata mate Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres Negro Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo Calzado deportivo New Balance para mujer Vazee FuelCell V1 Saúco / Plata Zapatillas para correr New Balance Rushv3 para mujer Acero / Negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco / Antracita , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas para correr Saucony Women s Triumph 11 Azul rojo Zapatillas de running acolchadas 420v4 para mujer New Balance Gris azul ,

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 Sandalias KEEN Mujer Whisper-w Higo de verano / Crabapple Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Turq / Black-volt-rio Teal , Adicción a las mujeres de Brooks 13 Evening Blue / Teal Victory / Purple Cactus Flower Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul Sandalia Merrell para hombre All Out Blaze Aero Sport Vapor , Sandalias KEEN Clearwater CNX-W para mujer Higo de verano / Crabapple , Zapatillas de volante Nike Women s Zoom Hyperface 2 Blanco / Menta-lobo gris , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Negro Negro , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón , Sandalia de mujer de nivel medio Aerosoles Tela azul oscuro , Zapatillas de running adidas Ultraboost W para mujer Red Night / Red Night / Black , NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / White Sneachers Sport Premium-Premix Slip-On Sneaker para mujer Naval Blanca , Zapatillas de paseo Slip-on Go Step Lite de Skechers Performance para mujer Teal Hush Puppies Men s Gus Oxford Ante negro Zapatillas adidas Irana 2 Cross-Trainer de adidas para mujer Gray Two / Flash Red / Black , Zapatillas de tenis 696v3 New Balance para mujer Oxblood / Coral vivo , Vans Old Skool Unisex Adults Low-Top Trainers Gris / Violeta Vans Herren Authentic Core Classic Sneakers Negro Negro , Zapatillas de golf ECCO Women s Classic Hybrid II Blanco / caramelo , Sandalias Dior J Pliner Dacia para mujer Piedra Sandalia Raelyn Wedge Slide de Easy Street para mujer Guijarro rojo Zapatos muy finos Dance Sneakers Series SN016 Verde , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1620- Tan Satin , Sunny Equestrian Boot de Tommy Hilfiger para mujer Marino / Chocolate , Bomba de vestido Kalila Total Motion Kalport para mujer Patente Gray Perla Gris Zapatilla de Nalga Gráfica El Naturalista N5271 para mujer Soft / El Viajero Oceano , Zapatillas de running NIKE Men s Train Speed 4 Equipo Maroon / Blanco / Dorado Klogs USA Moro Sneakers Frost Grey Clásicos de base Slip-on (Tm) para hombres de Vans (Tablero de ajedrez) Negro / 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 Camiseta adidas Swift Run Blanco W Run Blanco para mujer Blanco/ Blanco/ Blanco 8883a9a 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

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