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.

Gamuza

Importada

Goma único

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Ofertas una lengua acolchada y cuello.

Fácil cierres con cordones para un ajustable sin embargo, ajuste seguro.

felpa y transpirable revestimientos de malla. Acolchado plantilla con agregado gel en la entresuela y retropié para comodidad todo el día. Duradero y ligero suela de goma con tracción.Importado.Medidas: Peso: 12 oz Mediciones del producto fueron tomadas usando tamaño 10 hombres, Mujer 11.5, ancho Medio. tenga en cuenta que las mediciones puede variar por tamaño

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

Onitsuka Tiger Asics Onitsuka/ Unisex Gris Tiger Ally Gris topo/ gris topo e9452e9Updated 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.

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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 Women s Tanjun Antracita Wht Black Platinum Zapatillas de deporte Onitsuka Tiger Mexico 66 Abedul / blanco , NIKE Men s Lunar Fingertrap Cross Trainer Cool Gray / Black-speed Red Zapatillas para correr NIKE Lunarcharge Essential para mujer True Blue / Hyper Verde Sandalia deportiva NIKE Kawa Slide para hombre Lobo gris / antracita / gris lobo , Under Armour Threadborne Slingflex para mujer Flint / Rhino Gray / Lavender Ice Zapatillas de running New Balance Fresh Foam Lazr V1 Blanco negro Zapatillas de mujer CVR-Hemp Descalzo Vibram FiveFingers Black 39 & Premium Toesock Bundle - , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Energía / Energía / Negro Zapato multideporte Columbia Conspiracy IV Outdry para mujer Gris Zapatillas NIKE Women s Revolution 3 para correr Negro / Fresh Mint / White Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre rojo , NIKE W Air Max 97 Ul 17 PRM Mujeres Ao2325-002 Negro / Negro-gran gris , Zapatillas deportivas 574S Sport New Balance para hombre Granate , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Turtledove encerado , Zapatillas de entrenamiento de vuelo Jordan Nike para hombre Cool Gray / Cool Gray / Wolf Gray Zapatillas de running Solana Zoot para mujer Black / Punch / Solar Flare , Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / acero , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Asfalto Carro de mujer Brooks Chili Pepper / Black MBT GT femenino 18 Negro , Zapatillas deportivas 574S Sport New Balance para hombre Blanco Converse Chuck Taylor All Star Seasonal Color Hola Hyper Orange Zapatillas de running Adidas Performance Ultra s Boost M para hombre Medium Heather gris / Medium Gray Heather / Dark Heather gris , NIKE Women s WMNS Zoom Fly, negro / blanco Blanco negro Zapatillas de running NIKE Juvenate para mujer Lobo gris / blanco / gris lobo / blanco Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco , Zapatillas de running NIKE Unisex Flyknit Racer Universidad Red Black 608 Zapatillas de running NIKE Unisex Flyknit Racer Green Strike / Black / Blue Lagoon ,

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 Zapatillas de running AMPLICA para mujer ASICS Blanco / Plata / Porcelana azul Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Plata / rosa Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron Zapatillas de deporte ligeras y amortiguadas de YiDiar para mujer Zapatillas de deporte de gimnasia Zapatillas de deporte atléticas exteriores Zapatillas de deporte Negro / rosa Zapatillas de running New Balance para mujer 590v3 Negro / rosa Chaco Women s Zx2 Classic Athletic Sandal Wink Blue , Zapatillas para mujer Cloud 2568 Caliber Slide Plata , Zapato de running COMUTORA para mujer ASICS Negro / Negro / Blanco Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Negro / oscuro / gris / antracita Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones rojo Walking Cradles Women s Bounce Flat Negro 1 , Bomba de plataforma FLY London Yadi718fly para mujer Cupido / Mousse negro , Zapatillas Superga 2750 Cotu para mujer Blanco negro , Clásicos de base Slip-on (Tm) para hombres de Vans Azul marino (lienzo) , Zapatillas de tenis adidas Asmc Barricade 2017 para mujer Legblu / Super Purple / White Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Tierra roja / Blanco-negro Zapatillas de golf Skechers Women s Go Golf Birdie Azul marino / Mint Tropic Aerosoles A2 by Cool Breeze Slide Sandalia para mujer Bronceado oscuro Lacoste Women & rsquo; s Fraisier Slides Negro / Pnk sintético Sandalia para mujer FRYE Ally Criss Cross Slide Negro , Zapato de skate casual Heathrow DC para hombre Blanco negro , Zapatillas de beisbol Synchro MX para mujer Mizuno Blanco negro , Bota de invierno UGG Classic Mini II para mujer Girasol , Zapatilla de UGG Women s Lane castaña , FRYE Bota de tobillo corto con botones Malorie para mujer Negro Zapato clásico Haflinger para mujer Natural , Zapato de uniforme de poliéster Thorogood Mujeres Poromeric Gloss Black Lace Oxford 531-6103 - Mephisto Poppy Tuxedo Oxford para mujer Seda blanca / mano de avellana , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Birch Retro Floral Trotters Liz Loafer para mujer Negro

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 Onitsuka Tiger Asics Onitsuka/ Unisex Gris Tiger Ally Gris topo/ gris topo e9452e9 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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3003031

3039290

4286284

6731825

11148917

11533773

12183906

17103457

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