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.

Lona

pálida rubor metálico lunares de fiesta superior

Clásico Silueta de Alpargata

Vegana

Moldeada plantilla extraíble

Elástico \\ u201cv \\ u201d para fácil encendido / apagado

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

TOMS Alpargata-K Kids 19849 10009918 Alpargata-K Pale Blush Metallic Party Pale Dots 478624aUpdated 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.

TOMS Alpargata-K Kids 19849 10009918 Alpargata-K Pale Blush Metallic Party Pale Dots 478624a

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

Timberland Womens Lakeville Oxford Marrón medio gamuza , Aravon Revsmart Oxford para mujeres Gris pardo , Propet Walkers Evolution Oxford lavable de las mujeres Mora azul Skechers Women s Bikers Winner Sneaker Lycra negro / gamuza , Zapatillas de deporte sin mangas para mujer de bajo perfil de Mónaco Jiffy Black / Shell White / Bone White G.H. Azul , Zapatillas de deporte de mujer Alegria Essence White Shimmer Franco Imagine Imagine Oxford de Franco Sarto Blush Taupe , Tiempo libre para mujeres de SAS Moca Total de sobre de movimiento total de Rockport para mujer Serpiente negra / múltiple , Funtasma de Pleaser Men s Jazz-02 Platform Oxford Verde Cole Haan ZeroGrand Laser Wingtip perforado zapatos Oxford gris / rosa - Zapatillas de deporte Alfie-CV de LOEFFLER RANDALL para mujer Eclipse / tela cosida de marfil , PIC / PAY Jonas - Cordones de las mujeres Oxford - Clásico Ropa de hombre de cuero plana Zapatos estilo holgazán (Nueva caída) Gamuza burdeos , Mocasines Slip-on Pam de Adrianna Papell para mujer Natural , Odema Cuero Oxfords Mujer Perforado con cordones Wingtip Talón bajo Talla Brogue Zapatos de vestir Oxfords Marron gris , Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Cuero de peltre Softspots - Mujeres - Angie Negro , Prop & eacute; t Olivia Oxford para mujeres Gris pardo , Klogs USA Moro Sneakers Frost Grey Nueve Siete Genuino Cuero de las mujeres Square Toe Chunky Heel Dressy Handmade Moda linda Bombas Zapatos Amarillo claro Zapatillas de deporte adidas Women s Courtset Fashion Claro Onix / Plata / Blanco , Klogs USA Moro Sneakers Castillo Fg , Isa de Franco Sarto Negro Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón , TOMS Kids 10009918 Alpargata-K Marina de guerra Ditzy Daisy / Bow Mephisto Sabatina Tuxedo Oxford para mujer Patente negra , Bruno Bianci Womens Vegan Oxford Brogue Plástico transparente Ver a través de Clear Zapatos perfectos para Colorido feliz Calcetines , Cole Haan Women s 2. Zerogrand Stitchlite Ox Oxford Tejido Blanco Óptico / Sol ,

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 OluKai Kulapa Kai Dark Java / Dark Wood Sandalias Swiftwater Mesh de Crocs para mujer Jean azul Zapatillas deportivas 574S Sport New Balance para hombre Azul marino / Vintage Indigo , Zapatillas de deporte Rockport XCS Urban Gear para caminar de Rockport Cuero marrón Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / gris antracita-cool , Sandalia original Teva Women s Original Pirámide azul , Zapato acuático NRS para hombre GUNMETAL , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Rosa caliente , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul Zapatillas de running NIKE Air Huarache para hombre negro , JSport by Jambu Women s Wendy Flat Gris claro , Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño MONTAÑA BLANCA Miso Mule cuero de mujer Teniente Brown , Zapato de entrenamiento cruzado HIIT XT de Altra para mujer Negro / púrpura , NIKE Condición de zoom para mujer 2 Cross Trainer Lobo Gris / Plata Metálica - Blanco , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Apenas Verde / Negro , Zapatillas DC Danni TX para mujeres Blanco , Zapato Dantera, para hombre, L Low Waxed Court Negro / Blanco / Negro , Sandalia Aerosoles Midday Women Slide Tela negra Loeffler Randall Gertie-Ks Sandalia deslizante para mujer Melón , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2702- Silver Stardust & Silver Leathe , Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ Seradark Tan Satin Muk Luks Gloria Grey Slipper para mujer Gris , Zapatilla de arranque Whitecap de mujer Woolrich Hojaldre de crema Zapatillas Nike Zoom para hombre Superfly Elite, para pista y campo Negro / Blanco-volt-gris oscuro , CORRAL tobillera con flecos turquesa de mujer punta redonda - A3196 Coñac Motociclistas mujeres de SKECHERS - Decisión dividida Cuero negro / malla , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Ocean Fog / University Red / Squadron Blue , Zapatilla Moccasin Moxasin de piel de oveja Acorn para mujer Violeta , ZXD Men Chic Low Heel Slip On Mocasines Planos de punta redonda Casual Zapatos de barco Estampado de leopardo ,

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 TOMS Alpargata-K Kids 19849 10009918 Alpargata-K Pale Blush Metallic Party Pale Dots 478624a 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