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.

modelo

de tela y sintético Número: 863779404

Sexo: mens

Color: BLUE FOX / CERULEAN-COLLEGE NAVY

Hecho en: Vietnam

Marca Nuevo con Original Caja

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 Zapatillas running NIKE para para mujer Lunarepic Low Flyknit de 2 Colegio Navy/ Black Concord 65fb425Updated 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 Zapatillas running NIKE para para mujer Lunarepic Low Flyknit de 2 Colegio Navy/ Black Concord 65fb425

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 Gris blanco , Brooks Womens Mazama Diva Rosa / Antracita / Bluefish , Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / Blanco Zapatillas de running W1260V5 para mujer New Balance Azul plateado Zapatillas de running Mizuno Women s Wave Hitogami 4 Naranja / gris Pantalón largo clásico NIKE Golf para mujer Tech Essentials Black Crimson total Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul Zapatillas de running NIKE Air Huarache para hombre Olive Flax White 302 Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink Short NIKE Tempo para mujer Negro, Racer Pink , Under Armour Women s HOVR Sonic NC Blanco (102) / Elemental , Zapatillas de running adidas Adizero Boston 6 W para mujer Utilidad Negro / Platino / Negro , Zapatillas de running Nike \ Noble Red White Black 600) , Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Polvo negro / asteroide / peltre / blanco Zapatillas de running Under Armour para mujer Toccoa Chambray Blue (400) / Overcast Grey , NIKE Men s Benassi Just Do It Athletic Sandal Negro Zapato de correr ideal para rejilla de mujer Saucony Gris blanco , Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul Zapato deportivo ASICS Gel-Kayano 24 para mujer Porcelana Azul / Humo Azul / Blanco , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Rojo , Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink , Zapatillas de running NIKE Zooms Winflo 4 para mujer Blanco / Negro / Aurora Verde EN Zapatilla Cloudster para mujer Magenta / Berry Zapato de running GEL-Excite 3 para mujer ASICS Rosa fuerte / Blanco / Amarillo intermitente Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Lush / Silver Mink , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono Zapatillas de running NIKE Lunarglide 9 para mujer Gunsmoke / Solar Red / Hot Punch / Vast Grey , Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Negro / Gris Sólido ,

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 NIKE Kids Kawa Slide (GS / PS) Negro , Zapatillas de deporte casual Gazelle de adidas para hombre Colegiado Real / Blanco / Oro Met. Zapato de running ASICS GEL-Venture 5 para mujer Profundidad del Océano / Pink Glow / Aruba Blue Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Caqui (Cargo Khaki / White / River Rock) , Zapatillas de running NIKE para hombre Air Max BW OG Negro / persa Violet-white , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Lobo Gris Puro / Negro-puro Platino-caliente Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco azul Zapatillas de senderismo acuáticas para mujer Astral TR1 Trek Deep Water Navy Zapatillas de running New Balance 990V3 para mujer Gris Alpargata de tela \ Lino azul marino Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Gris pardo , Zapato de entrenamiento WX824 para mujer New Balance Negro , Calzado de espuma fresca New Balance para mujer 1165v1 Gris , Converse Chuck Taylor All Star Core Ox Armada , Bomba de cuña Lansing para mujer Trotters Gamuza marrón oscuro , Clásicos de base Slip-on (Tm) para hombres de Vans (Bum de goma) True White / True White Zapatillas de baloncesto NIKE Air Python PRM - Hombre, de punta redonda, de cuero Hasta / Hasta blanco Zapatillas de ciclismo híbridas ECM Saddle para mujer ECCO Negro Sandalia Zonita Slide de Nine West para mujer Negro , Sandalias planas de piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural Plataforma CLARKS Leisa Grace para mujeres Blanco , Bota de montar de Donnily de Calvin Klein para mujer Shadow Gray 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5004- Gold Sparkle , Tamaris Women s Fee 22425 Pump Chile , Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Amarillo , Jeffrey Campbell Womens Lilith-2 Boot Nuevo Combo floral Kaitlyn Pan Block Heel Microsuede Botas sobre la rodilla ajustadas ejercito verde , Bomba Parigi de LifeStride para mujer Negro brillante ,

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 Zapatillas running NIKE para para mujer Lunarepic Low Flyknit de 2 Colegio Navy/ Black Concord 65fb425 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
1534994

1599677

3622330

3696040

6038004

11570382

12218194

15722427

17429471

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