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

FlyteFoam Tecnología de suela intermedia

Fluido Suela interior Ride: Fluid Ride proporciona la combinación definitiva de rebote y amortiguación propiedades con peso reducido y excepcional durabilidad.

trasero pie y antepié GEL acolchado Sistemas: Atenúa choque durante el impacto y fases de convergencia, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.

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 de running ASICS B01HIJCAC6 Gel-Nimbus 19 para 19 mujer mujer Insignia Blue/ Glacier Sea/ Crystal Blue a0fe2c7Updated 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 de running ASICS B01HIJCAC6 Gel-Nimbus 19 para 19 mujer mujer Insignia Blue/ Glacier Sea/ Crystal Blue a0fe2c7

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

NIKE W Air Max 97 Se Mujeres Aq4137-101 - Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Max naranja / negro / blanco Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Gris / sal marina , Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres gris Zapatillas para correr mujer Gaviota HOKA ONE ONE Zapatillas de running New Balance 990V3 para mujer Blanco azul , Zapato de running ASICS Gel-Quantum 360 cm para mujer Melocotón negro / Onyx , Short NIKE Tempo para mujer Negro 010 Saucony Women s Ride 10 Running Shoe Azul marino / mezclilla , Zapatillas de running adidas Mana Bounce de adidas Performance para mujer Bold Pink / Prism Blue / Shock Pink , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Negro , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Utiblk, Ftwwht, Mgsogr , Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura , Zapatillas de running NIKE Lunarglide 9 para mujer Negro / blanco / gris oscuro / gris lobo , Rieker womens Slipper oldgold / fango-silver Oro viejo / Fango-plata ASICS 1012A019 Zapatillas para correr GEL-Kayano 25 (2A) para mujer, color carbón / gris medio - 10 2A ES - Zapatillas de running Mizuno Wave Viper para mujer Negro-fucsia Pruple-plata , Zapatillas de running TOPO FLI-Lyte 2 para mujer Azul rosado Brooks Womens Mazama Flor de Cactus Púrpura / Peacoat / Hot Coral , ASICS Kanmei Shoe Casual para mujer Negro / Negro / Blanco Zapatillas de running Under Armour Charged Bandit 3 para mujer Grosella negra (923) / arenas rosadas Zapatillas de running adidas Ultraboost W para mujer Blanco / blanco / blanco cristal , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco volt gris oscuro , Zapatillas de running adidas Cloudfoam QT Racer para mujer Triple blanco Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul amarillo

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 Chaco Mujer Z Eddy X1 Sport Sandalia Cipher Gold Converse Chuck Taylor All Star Seasonal Color Hola Monocromo negro , Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Core Green / Chalk White / Easy Green , Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / Navy , Zapatillas de running NIKE LD Runner LW para mujer Blanco / Soar - Electro Green Sandalia Drew Angela para mujer Bone Smooth Leather Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory Zapatillas para correr Mizuno Wave Inspire 12 para mujer Blanco negro Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Encaje blanco Sandalia Huarache Easy Street Allure para mujer Natural Zapatillas de running NIKE Air Max 1 Ultra 2.0 SI para mujer Mushroom / Deadly Pink , Sandalia de cuero original premier original Teva W de mujer Negro Zapato para caminar WW411v2 New Balance para mujer Gris , Zapatos de aproximación Arc teryx Acrux SL - Mujer Nebula de Orión Vans Auténtico (Tarde en la noche) Negro / Hamburguesas Zapatilla de pies Cush N de Prop & eacute; t para mujeres Negro Zapatillas para mujer Del Rey TOMS Natural.metallic.linen Vans Unisex Old Skool Zapatillas clásicas de skate C & l Chambray Blue Sandalia plana de satén Ivarene de Nine West para mujer Negro , Zapato de voleibol Boost W Volley Response 2 de adidas Performance para mujer Poder rojo / plata mate / negro , Botines metálicos A-Game para mujer Boombah - 6 opciones de color - Varios tamaños Blanco , Kmeioo Slip On Mocasines, mujeres Bordado mula Zapatos con felpa Terciopelo de piel de cordero Sin respaldo Toe Mule Slides Zapatilla floral negra Zapatillas Ixia de Easy Spirit para mujer Negro , Bomba de vestido animado de LifeStride para mujer Lux Navy Botas tobilleras Adaline Bluff de Merrell para mujer Negro Hush Puppies Womens Cesky Perf Oxford Nubuck gris helada , Calvin Klein Womens Narla Deep Navy Velvet Snake Print Cole Haan Women s 2.Zerogrand Laser Wing Oxford Nubuck de piedra de hierro , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Infinity Blue Abstract Leaf Zapatillas de deporte KR Strikeforce Ladies Starr Performance - Derecha ancha Tamaño 6.5 ,

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 de running ASICS B01HIJCAC6 Gel-Nimbus 19 para 19 mujer mujer Insignia Blue/ Glacier Sea/ Crystal Blue a0fe2c7 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