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.

Césped de la dinastía PITCHING

Superior Material: Cuero

Suela: Turf

Opciones de tamaño: Mujeres 5.5-13 (Corre un tamaño medio pequeño)

Ancho: Estándar

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

Ringor Dynasty Turf de Shoe- Lanzamiento Dynasty 15034 Carbón de leña/ negro 2eaa658Updated 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.

Ringor Dynasty Turf de Shoe- Lanzamiento Dynasty 15034 Carbón de leña/ negro 2eaa658

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

Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Azul marino / rojo Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Blanco Ringor Bandit 2.0 Spike- Pitching Rojo blanco , Zapatillas de beisbol Synchro MX para mujer Mizuno Blanco negro , Zapatillas de beisbol de softbol de lanzamiento rapido FUSEV1 metal New Balance para mujeres Blanco Calzas de la dinastía Púrpura / blanco , Calzas de la dinastía Azul marino / blanco Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Blanco 3N2 Mofo Turf Trainer Armada Under Armour Women s Glyde RM Softball Cleat Blanco negro Ringor Bandit 2.0 Spike- Pitching Azul marino / blanco Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Real / Blanco , Zapatillas de beisbol de softbol de lanzamiento rapido FUSEV1 metal New Balance para mujeres Blanco Calzas de la dinastía Carbón de leña / negro , Mizuno (MIZD9) 9-Spike Advanced Finch Franchise 7 Fastpitch Cleat Softball Shoe para mujer Blanco / rosa , Ringor Dynasty Turf Shoe- Lanzamiento Blanco negro 3N2 Mofo Turf Trainer Blanco , Mizuno Women 9 Spike Swift 4 Fast Pitch Metal Softball Cleat Real / Blanco , Botines moldeados del dedo del pie de las mujeres de Boombah - 8 opciones del color - tamaños múltiples Azul marino / blanco Zapatillas de beisbol PowerAlley 4 W TPU de adidas Performance para mujer Real-blanco Boombah Women s A-Game - Juego de calas intermedias moldeadas - 8 opciones de color - Varios tamaños Negro / Rojo Mizuno Women s 9 Spike ADV Finch Elite 2 Juego de sordina de softbol moldeado rápido Blanco negro , Ringor Dynasty Turf Shoe- Lanzamiento Carbón de leña / negro , Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Azul marino / blanco Calzas de la dinastía Púrpura / blanco , Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Blanco Blanco , Pinza de softbol con interruptor Finch Elite de mujer Mizuno Real / Blanco , Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Azul marino / naranja New Balance Sp4040g1 para mujer Gris ,

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 Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Zapatillas de running NIKE para hombre con versatilidad de tren gratis Negro / Blanco / Total Crimson / Hyper Pink Zapatillas de running ASICS Gel-Nimbus 20 para mujer Negro / Negro / Carbono Zapatillas de running Reebok Print 2.0 para mujer Negro / azul polar / peltre Sandalia de Gladiador de Piedra Lunar de Ahnu para Mujer Agave Green , adidas Sandalia atlética Anyanda Flex Y W para mujer Easy Mint / Silver Metallic / White Reebok Women s Ridgerider 2.0 Trail Runner Negro / Rosa Solar / Plata / Estaño , Salomon Women s X-Mission 3 W-w Acuario , Reebok Women s Ridgerider 2.0 Trail Runner Piedra de hierro / Mar Gris / Carbón / Calavera Gris / Negro / Plata / Peltre Zapatillas NIKE Zoom s Out Low 2 para mujer negro , Calzado de running NIKE Zoom All Out Low para hombre NIKE Obisidian / Paramount Blue-black , Entrenador de Mujeres Cassidy Cuero negro , Comfortiva Mujer, Faye Sandalia de tacón medio Antracita Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Baya Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Negro Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Negro Metálico Sandalia Blue Slide de Musse & Cloud para mujer Negro , Sandalia FLY London Women s Whin176fly Slide Rose Cupido , GUESS Zapatillas bajas con adornos en las mujeres Cuero Multi blanco Zapato de pista Md800v5 New Balance para mujer Llama blanca , Miz Mooz Cato para mujeres Whisky , Bomba Parigi de LifeStride para mujer Gris pardo Western Chief Women Impermeable de cuatro ojos con cordones Duck up Armada Bota de moda de lona negra Winsted Dr. Martens Negro Bota del oeste de Denver de Dan Post para hombres Chocolate Cobb Hill Rockport - Aubrey Dress Pump para mujer Broncearse , Bates Lites Black High Gloss Oxford Negro Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Raya de peltre Botas a la rodilla con cordones Wild Diva Timberly-65 Black Pu ,

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 Ringor Dynasty Turf de Shoe- Lanzamiento Dynasty 15034 Carbón de leña/ negro 2eaa658 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