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.

Parte superior sintética

Importada

Phylon suela de espuma

de una sola pieza cuero sintético correa tiene jersey forro para un suave sentir

inyectado Entresuela de Phylon se dobla como una suela para peso ligero amortiguación

Flex surcos ayudan al deslizar el movimiento con tu pie

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

NIKE Men s Benassi NIKE Rojo Just Do It Athletic 15384 Sandal Negro/ Rojo de desafío 46a41f9Updated 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.

NIKE Men s Benassi NIKE Rojo Just Do It Athletic 15384 Sandal Negro/ Rojo de desafío 46a41f9

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 Under Armour Speedform Slingshot para mujer - SS16 Negro / Rojo , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Blanco, lobo gris brillante carmesí , Zapatillas running para mujer Saucony Liteform para mujer borgoña Zapatillas de running NIKE Lunar Skyelux para mujer Blanco / Platino puro / Blanco apagado , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Carbono / Rojo , Zapatillas para correr Mizuno Women s Wave Rider 21 Patriot Blue / White , Zapatillas de running New Balance Boracay V3 para mujer Blanco negro , Mujer adidas Cloudfoam Qt Racer Negro / Cobre Metálico / Blanco Zapatillas para correr New Balance Rushv3 para mujer Poisonberry / Black , Zapatillas de running NIKE para mujer Lunarepic Flyknit Shield Ocean Fog Blue 400 Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Abismo / llovizna , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Dark Grey , Zapatillas de running NIKE Free Run para mujer - SU16-10 - Blanco Negro / Lava Glow / Off White , Supra Martine Run Skate Shoe para hombre Corcho / hueso Zapatillas de running adidas Alphabounce Rc W para mujer Núcleo Negro / Plata Metálico / Gris Cinco , Zapato de running ASICS GT-1000 6 para mujer Negro / Negro / Plata Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro , NIKE Women s Revolution 4 Running Shoe Negro / Pulso carmesí / Antracita Brooks Womens Adrenaline GTS 18 Púrpura / negro / champán Zapatillas de running NIKE para mujer Epic React Flyknit Rosado Zapatillas de running New Balance 990V3 para mujer Gris Zapatillas de running Under Armour Speedform Intake 2 para mujer Azul , PUMA Men s Roma Basic Sneaker Cuero blanco / negro , HOKA ONE ONE Zapatillas para mujer Arahi Sunny Lime / Blue Topaz , Zapato deportivo para acolchado 680v5 New Balance para mujer Silver Mink , Calzado de running Jukenate PRM NIKE para mujer Negro / lobo negro gris , Brooks Women s PureGrit 5 Azul pavo real / Rosa virtual / Azul patriota

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 Azul marino / gris medio , Calzado de trabajo con base de gel para mujer ASICS - , Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Onyx / Carbono , Sandalia atlética Salomon Women s Crossamphibian Swift W Negro NIKE Kids gratis Rn (grande) Ember Glow / Metallic Silver OluKai Kulapa Kai Burbujeante / negro Wolky Comfort Jewel Beige 1 , Sandalia de mujer Drew Alana Verde / Fucsia , Zapatillas de running Saucony Women s Triumph ISO 2 Gris / azul / limo , Saucony Women s Guide 10 Zapatillas para correr Gray Teal , Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul PUMA Mujer Basket Corazón NS WN Sneaker Puma White-puma White , Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Blanco negro Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral Calzado para caminar WW812 para mujer New Balance Negro , Sandalia Huarache Easy Street Allure para mujer Natural Drew Shoe Women s Tulip Comfort Brown Croc , nativo AP Mercury Liteknit Jiffy Black / Shell White / Natural Rubber Sandalia de mujer Lucienne Suede Slide de Nine West Gamuza rosa Sandalia para mujer SCHUTZ Margarit Slide Negro Zapatillas de deporte CVT-Hemp de mujer Vibram Caqui Zapatilla Stella Nd52 de El Naturalista para mujer Negro , 206 Cojín de mocasín de piel de oveja Pearson de mujer Ante azul marino , Bomba de cuña Farrow para mujeres LifeStride Piedra TOMS - Slip-On de lienzo clásico para hombre Cayenne Canvas Tile Corral Boots Womens E1403 Turquesa Bota del oeste de Denver de Dan Post para hombres Bay Apache , Bota de ante Sadie de Jack Rogers para mujer Roble Prop & eacute; t Mabel Oxford para mujeres Vino , CLARKS Women s CloudSteppers Sillian Paz Slip-on Loafer Nubuck sintético de arena

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 NIKE Men s Benassi NIKE Rojo Just Do It Athletic 15384 Sandal Negro/ Rojo de desafío 46a41f9 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