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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Importado

Prácticamente FITKNIT inconsútil la parte superior es de apoyo, cómodo y transpirable

Ligero, durable y flexible web paramétrica suela proporciona tracción multi-superficie - Golpe de pie medio zona promueve la eficiencia en cada zancada

4mm compensación de gota de talón - Tire de la pestaña del talón función para accesible fácil de encendido y apagado

Ultra ligero, VUELO receptivo Amortiguación GEN entresuela - moldeada contador de talón para un estable y seguro ajuste

Hot melt construcción de collar para mayor comodidad - Plantilla extraíble para un ajuste personalizado

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

Skechers Blanco Women s Go MEB Razor B00PJ89LYY 19120 2 Blanco negro 1892ac0Updated 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.

Skechers Blanco Women s Go MEB Razor B00PJ89LYY 19120 2 Blanco negro 1892ac0

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 Air Max Thea para mujer Blanco / Blanco-blanco Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue Zapato de running ASICS GT-2000 4 para mujer Turquesa / rosa fuerte / azul marino PUMA Womens Fierce Rope Satin EP Melocotón Beige / Puma Blanco / Perla NIKE Men s Flex 2017 RN Hyper Orange / Black , Zapatillas de running Under Armour Speedform Intake 2 para mujer Negro (001) / Acero Zapatillas de running Kineta Relay para mujer Saucony Gris / Heather / Coral , Calzado de running NIKE RN 2017 2017 para mujer Negro / Fuchsia Glow / Hyper Magenta , Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Blanqueado Lila / Rosa Blast-white , Zapatillas de running Saucony Women s Freedom ISO Berry / rosa , adidas Running Womens Pureboost Xpose todo terreno Noble Ink / Legend Ink / Multi , Zapatillas de running Nike Kids Air Max Zero Essential GS River Rock Black Cobblestone 005 Zapatillas de running NIKE Mens Air Zoom Structure 19 Blanco / negro / verde eléctrico , Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Blanco , Zapatillas de running Cortana 4 Saucony para mujer Azul / rosa / negro , Skechers Sport Skech Air Run High Fashion Sneaker Ajuste de malla de carbón / turquesa Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Violeta plata / persa negra / metálica Short NIKE Tempo para mujer Juego Royal / Game Royal / Wolf Gray New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla oceánica / Blanco-azul Glow-hyper Jade , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Cobre rosa / sal marina / chicle New Balance Women s w990v4 Running Shoe Armada , Zapatillas de running Newton Kismet II para mujer Rosa / Blanco , Zapatillas para correr Altra Women s Torin 3.0 Azul / Coral , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Berenjena / Sal marina , Zapatillas de running Pearl NZUMi W Em Road N 2 V3 para mujer Plata / Aqua Mint , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Black / White-hyper Punch Zapatillas de running Kineta Relay para mujer Saucony Fuschia / cerveza ,

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 KEEN Hombres Newport H2 Sandal Realtree / Xtra Green Sandalia Merrell Tetrex Crest Wrap Sport para mujer Negro , Zapatillas de trail running Topo Athletic MT2 - Hombre Azul marino / gris EN Zapatilla Cloudster para mujer Chile / Curacao , Zapatillas de running NIKE Lunartempo 2 para mujer Pink Blast / Clr Jd / Black / Brly Grn , Zapatillas de trail running Ryka Women s Intent XT 2 SR Blanco / plata cromada En la ejecución de calzado de mujer en la nube Blanco negro Zapatillas de running PUMA Women s Ignite Dual WN Quarry / Red Blast , Botín Rubin para mujer Sam Edelman Ante negro Zapatillas deportivas sintéticas Drew Shoe para mujer Flamenco Imprimir , New Balance 611v1 Cush + Cross Trainer para mujeres Negro , BareTraps Sandalia Trudy para mujer Caramelo , Linea Paolo Felicia | Cuero perforado negro Entrenador campeón Gusto Cross Trainer Negro Negro DC Mujer Trase TX SE Tostado / Marrón Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on Gris / rosa , Zapatillas de tenis ASICS Gel-Solution Speed 3 para mujer Plata blanca , Sandalia Otto Slide de mujer Yellow Box Aceituna , Very Fine Dance Shoes 2707 (Grado de competencia) 3 \ Oro , Trotters Lucy Ballet para mujer Hueso , Zapatillas OLUKAI Hi ona - Mujer Bronce / Dark Java Bomba Hastings Sam Edelman para mujeres Patente negra , mejor sentido Unisex Lana clásica Zapatilla para Adultos - Toda la lana natural - Ultra cómodo - Muchos tamaños y Colores Medoc-naranja rojo Trotters Lucy Ballet para mujer Armada Corbata perforada Aquet de mujer ECCO para mujer Oxford Blanco Bota chukka de mujer Timberland Londyn 6 \ Hoja de uva ECCO Footwear Babett Lace Oxford para mujer Negro TOMS Clásicos de mujer Lona azul marino , Zapato Lites Bates para mujer marrón , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro sin blanco ,

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 Skechers Blanco Women s Go MEB Razor B00PJ89LYY 19120 2 Blanco negro 1892ac0 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