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.

Textil

Importado

Sintético suela

Plataforma mide aproximadamente 0.75 pulgadas

Memoria plantilla de espuma

Sport Suela lite

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

Sandalia deportiva Nolita Nolita para mujer Anne Klein Anne Klein Tela negra 58a830dUpdated 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.

Sandalia deportiva Nolita Nolita para mujer Anne Klein Anne Klein Tela negra 58a830d

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

Wolky Comfort Jewel Beige Chaco Women s Wrapsody X Sandalia Key Teal , Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón Sandalias KEEN Venice II H2-W para mujer Paloma / Turquesa Pastel , OluKai Kulapa Kai Coral / Negro OluKai Kulapa Kai Pale Jade , Sandalia Terra-Float Livia de Teva para mujer Negro Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador Mordenmiss - Sandalias Huarache de cuero, para mujer, de verano, zapatos planos de tacón Estilo 2-amarillo , Chaco Women s ZX2 Classic Athletic Sandal Motif Eclipse , Sanda Verra Teva para mujer Atlántico Norte , Skechers Performance On The Go 400 Vivacity Flip Flop Blanco / azul claro Nike Womens s Benassi JDI Print Slide Negro / Fireberry- Blanco Chaco Women s ZX2 Classic Athletic Sandal Credo Golden Sandalias KEEN Mujer Whisper-w Ágata gris / azul ópalo Sandalia de mujer KEEN Rose Aluminio / gris neutro , Chaco Women s Z2 Classic Athletic Sandal Florista Chaco Mujer ZX3 Classic Sport Sandalia Shiver Pine Wolky Comfort Jewel Fantasía de serpiente blanca / negra posters , Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador , Sandalia de mujer KEEN Rose Boysenberry / Red Violet , NAOT Womens Shore - Fuente de Negro , Zapatillas de vela HH 5.5 M Helly Hansen para mujer Off White / Light Aqua , Sandalia Yucatan ECCO Mujer Pastel multicolor Chaco Women s Zx2 Classic Athletic Sandal Wink Blue Wolky Women s Rio Sandalia 40-912 Multi Chaco Women s ZX2 Classic Athletic Sandal Palm aguacate KEEN Hombres Newport H2 Sandal Imán / Capuchina ,

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 NIKE Mujeres Air Zoom Pegasus 33 Blue Glow / Racer Blue / Bluecap / Blanco , Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico Zapatillas de running Under Armour UA Micro G Speed Swift para mujer Azul , Zapatos OLUKAI Pehuea - Mujer Océanos / océanos , Sandalia de mujer de nivel medio Aerosoles Tela negra , Tecnología Vionic Men s Orthaheel White / Navy Walker Naval Blanca Zapatillas para caminar Reebok Women s Cloudride LS DMX Polvo de asteroide / nube gris / blanco / negro Calzado de espuma fresca New Balance para mujer 1165v1 Gris blanco , Sandalias FitFlop Womens Cova con cordón cerrado Rosado oscuro , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris Splendid Sandalia de mujer Farley champán , La sandalia de Miz Mooz para mujer Cielo , NIKE Men s Benassi Just Do It Athletic Sandal Binario azul / menta fresca Toalla de vestir Dulce para mujer Touch Ups Champagne Shimmer , Sam Edelman Women s Gigi Marfil moderno , Sneachers Sport Premium-Premix Slip-On Sneaker para mujer Naval Blanca , Ryka Women s Destiny Cross Trainer Gris / Rosa Clásicos de base Slip-on (Tm) para hombres de Vans (Checker Emboss) Fallen Rock / Blanc De Blanc Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco Sandalias Lloret-5 de mujer Toni Pons Gris pardo Zapatillas deportivas sintéticas Drew Shoe para mujer Flamenco Imprimir , Sandalia Thorne de Badgley Mischka Jewel para mujer Sonrojo Sandalias planas de piscina Walking Cradles para mujer Metálico , Sandalia para mujer PUMA Popcats Slide Blanco negro , Zapatilla Wade para mujer de Badgley Mischka Terciopelo morado Zapatilla de mujer verde de Patricia Green Lino Bota de chukka CLARKS para mujer Desert Boot Gamuza gris oscuro Blundstone Super 550 Series Boot Nuez , MONTAÑA BLANCA Piso de mujeres de Summit Belton Cuero negro Betunia Loafer de mujer Aerosoles Oro ,

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 Sandalia deportiva Nolita Nolita para mujer Anne Klein Anne Klein Tela negra 58a830d 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

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