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 / Vegano

Sintético único

Eje mide aproximadamente 2.5 desde arco

Rediseñado Parallel Wave intercalado entre U4ic y La entresuela U4iX proporciona lo último en suavidad.

Nuevo Tablero EVA Strobel para más bajo comodidad

rediseñado superior para un lujoso y felpa como un guante en forma.

Transpirable Malla de aire con 3D reflectividad

Plano, quedarse atados cordones

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 para correr Mizuno Women Wave s Wave 6 Enigma 6 correr Púrpura/ blanco e321bb5Updated 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 para correr Mizuno Women Wave s Wave 6 Enigma 6 correr Púrpura/ blanco e321bb5

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 adidas Alphabounce Beyond para mujer Blanco / Gris / Gris , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Azul , Saucony Women s Cohesion 8 Running Shoe Zapato para correr ASICS Conviction X 2 para mujer Negro / Carbón / Flash Coral , Zapatillas de running adidas Cloudfoam Xpression W para mujer Blanco / Blanco / Haze Coral Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Gris oscuro / Azul Racer / Negro Zapatillas de running New Balance para mujer 940v3 Azul negro , Zapatillas de running NIKE Air Huarache Run para mujer Dark Stucco / Pale Gray Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird , Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Aruba azul / azul de Aruba / azulejo turco Nike Hombres Air Zoom Pegasus 33 Gris oscuro / negro - blanco , NIKE W Air Huarache City Low Women Ah6804-100 Blanco / Blanco-blanco , Brooks Womens Ghost 10 GTX Negro / Azul pavo real / Malvavisco , Nike Men s Air Presto Essential Oro negro / negro metálico , Zapatillas de running Mizuno Wave para mujer Wave Horizon Blanco negro , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Blanco / Racer Pink / Melon Tint / Wolf Gray , Zapato de running ASICS GT-1000 6 para mujer Negro / Cantel / Carbono , Zapatillas de running Nike Air Max Fury para mujer Blanco , Zapatillas de running Nike Air Huarache Run Ultra Br para mujer Platino Blanco / Blanco-puro Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Core Black / Core Black / Ice Purple , NIKE Air Max 90 Womens Style: 325213 Mujeres 325213-048 Lobo gris / blanco-gris oscuro Calzado de running NIKE Kids Roshe One (GS) Vino de Oporto / Pink Prime , Zapatillas de running Saucony para mujer Hurricane ISO 2 Negro / Carbón / Coral , Zapatilla para mujer Saucony tipo A8 Blanco | , Zapatillas de montaña de montaña Akyra La Sportiva para mujer Slate / Jadegreen , Zapatillas de running NIKE Lunar Skyelux para mujer Negro / Blanco / Antracita , Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / Begonia rosa

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 Short NIKE Tempo para mujer Negro Nike Men s Air Presto Essential Gris suave / Clorofila , Converse Chuck Taylor All Star Seasonal Color Hola Carbón , Converse Chuck Taylor All Star Seasonal Color Hola Monocromo negro Sandalia NIKE Kawa Slide para mujer Cool Grey / Hyper Turquoise / Cool Grey Wolky Comfort Jewel Estampado de flores blancas Sandalia del vestido de Flaunt de las mujeres de LifeStride Plata Zapatillas para caminar ProFit TravelFit Slide para mujer Oro / negro , CLARKS Womens Trigenic Flex Nubuck blanco , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Azul Indigo / Plata / Rosa Brillo , Zapato de running Saucony Zealot ISO 3 para mujer Gris azul , Drew Womens Element Nubuck negro , Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow Easy Street 30-8320 Sandalia airosa para mujer Estiramiento natural Charles by Charles David Mujeres inquietantes Mary Jane Nude Microsuede OLUKAI Pua Sandal - Mujer Frijol / frijol HEAD Women`s Revolt Pro Zapatos tenis blanco y azul cian- (274115WHCB-S15) Revere Miami sandalia de confort para la mujer w. Rojo / Croc Vans Herren Authentic Core Classic Sneakers Negro Negro Supra Skytop Skate Shoe Negro / Gris oscuro / Blanco , Twisted X Ladies Nude Print Driving Mocasines Bombardero / Desnudo , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Vino tinto , Western Chief Women Impermeable de cuatro ojos con cordones Duck up Marron oscuro , Timberland PRO Riveter Alloy Toe SD + Zapato industrial y de construcción Black Swank Full Grain Leather , Zapatillas de running NIKE Men s Train Speed 4 Gimnasio Rojo / Gimnasio Rojo-prisa Azul-blanco , Tiempo libre para mujeres de SAS Moca , Lauren por Ralph Lauren Womens Barrett-SH Marrón / marrón Burn Calf / Kid Suede , Bota de media caña Georgia GB00156 Marron oscuro Vionic Womens Elevated Becca Wedge con cordones Bronceado suave Zapatillas Slip-On Roamer SAS para Mujer Sabio ,

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 para correr Mizuno Women Wave s Wave 6 Enigma 6 correr Púrpura/ blanco e321bb5 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
2080967

6542051

9835543

11370148

11378020

16731255

16868237

18472698

18807444

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