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

Importado

adidas zapatos

Boost es nuestro más receptivo amortiguación alguna vez: más energía das, más usted obtiene; suela de goma de la web flexiona bajo los pies para un energizado paseo

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 adidas de running adidas Ultraboost W para 16734 mujer Blanco Zapatillas/ Blanco-2/ Blanco 24d0ab8Updated 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 adidas de running adidas Ultraboost W para 16734 mujer Blanco Zapatillas/ Blanco-2/ Blanco 24d0ab8

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 Kineta Relay para mujer Saucony Negro adidas Running Womens Supernova Sequence 9 Gris sólido / plata mate / blanco , Zapatillas de running Saucony Women s Peregrine 7 Ice + Running Azul Zapatillas de running NIKE Air Max 90 LX para mujer Partícula Rosa / Partícula Rosa Camiseta adidas Swift Run W para mujer Blanco / Blanco / Blanco , Zapatillas de running NIKE Roshe Two Flyknit V2 para mujer Blanco / Lobo Gris-blanco-blanco Zapatillas de running New Balance Urgev2 para mujer Silver Mink / Alpha Pink Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow NIKE Women s Air Max 95 , Brooks Womens Adrenaline GTS 18 Negro Negro Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento Zapatillas de running New Balance Boracay V3 para mujer Dark Denim / Sunrise Glow Zapatillas para correr Nike Zoom All Out Flyknit para mujer Bluecap / White-deep Royal Blue , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Sunset Pulse / Arctic Punch-white Zapatillas para correr Nike Downshifter 6 para mujer Negro / Hyper Punch / Antracita , New Balance Women s w940v3 Running Shoe Plata Zapatillas de running Saucony para mujer Stretch N Go Breeze Negro PUMA Men s Roma Basic Sneaker Negro Negro , Nike Hombres Air Zoom Pegasus 33 Lobo gris / negro / gris oscuro / azul NIKE Mujeres Air Max 95 LX , Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres gris Zapatillas de running Saucony para mujer Omni 16 Gris azul Zapatillas de running adidas Women s Edge Lux W Negro / Blanco / Dgh Gris Sólido , Zapatillas de running NIKE RN 2017 para mujer gratis Vela / Plata Metálica / Gris Pálido / Blanco Cumbre Zapatillas de running Saucony para mujer Stretch N Go Breeze borgoña Zapatillas de running Nike Women s Zoom Vomero 9 Mgnt Gray / Hypr Pink-hyper Cobalt-rf Camiseta adidas Swift Run W para mujer Trace Cargo / Trace Pink / Crystal White , Zapatillas de running acolchadas 420v4 para mujer New Balance Gris azul Zapatillas de running NIKE Air Huarache para hombre Blanco / Negro / Gum Amarillo / 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 Lanzamiento de Brooks Womens 5 Negro / verde azulado / blanco Birkenstock Arizona Cuero de gamuza con 2 correas Sandalias, Taupe, Unisex, 39 N UE Ancho estrecho - Mujeres de EE. UU. 8-8.5 - , Zapatillas NIKE Kids Revolution 3 (GS) para correr Wolf Grey / Vivid Pink / Blanco / Ghost Green , Zapatillas para correr Pacev2 para mujer New Balance Sea Spray / Black , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo , JSport by Jambu Women s Clare Flat Rosa / Multi Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Wolf Gray / Cool Gray / Black Zapato de caminar WW928HW2 para mujer New Balance Blanco , Supra Martine Run Skate Shoe para hombre Vintage Khaki-blanco , Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Coral vivo / negro / blanco Munsell , Walking Cradles Womens Henna Azul marino Nubuck Estampado , Prop & eacute; t Olivia Oxford para mujeres Hueso Botín para mujer Pleaser Asp1018 / Bpu / M Black Faux Leather / Black Matte , Zapatillas de entrenamiento cruzado Realflex Train 4.0 de Reebok para mujer Negro / carbón / blanco Zapatillas para caminar Ryka Women s Devo Plus 2 Blanco azul , Vionic Women s Flex Sierra con cordones Dark Purple / Pink , Salomon Womens Speedcross 3 GTX Scuba Blue / Cosmic Purple / Black Sandalia Victoria Slide de mujer SCHUTZ Perla Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero negro , Sandalia Dolce Vita Jene Slide para mujer Tela blanca a rayas , Zapato de bailarina Swing Tic-Tac-Toes para mujer Negro BlueBell Blue Bell Zapatos hechos a mano Salón de baile de mujeres Concurso de Boda Salsa Zapatos de baile estilo: MAIA 3.5 \ - Plata - Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Negro / Rojo Bansi Platform Pump Jessica Simpson para mujer Oro , Zapatilla Elodie de UGG Australia para mujer Crema , Toni Pons Womens Lugano Vintage azul Trotters Lucy Ballet para mujer Hueso , Rockport Mujeres Ayva Oxford Negro , Mocasines Slip-on Pam de Adrianna Papell para mujer Black Herring Bone Leather Sandalias con cordones Joanie de SOREL para mujer Camel Brown ,

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 adidas de running adidas Ultraboost W para 16734 mujer Blanco Zapatillas/ Blanco-2/ Blanco 24d0ab8 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