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.

Cinta de poliéster

Hecho en los EE. UU. o importados

Caucho único

Rendimiento sandalia deportiva con sistema de cordones bungee y tracción trasera lazo

Soporte arco contorneado ; sandalias de agua son lavable en la lavadora. Usa una pequeña cantidad de detergente, lavado en ciclo suave y secar al aire. Consejo: este estilo está ejecutando un 1/2 talla pequeña. sugerir ordenar un 1/2 de tamaño más grande que tu usual tamaño

metatarsiano soporte de cresta

Cuidado - Lavable en la lavadora, usa una pequeña cantidad de detergente, lavado en ciclo suave y secar al aire

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

KEEN Nova Hombres Newport Newport H2 Sandal Raven// Bossa Nova da8140fUpdated 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.

KEEN Nova Hombres Newport Newport H2 Sandal Raven// Bossa Nova da8140f

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

Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Malla negra , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Negro , Chaco Mujer Diana Sport Sandalia Pulse Eclipse , KEEN sandalia de Venecia para mujer Mineral azul / ámbar , Sandalia Yucatan ECCO Mujer Abedul Sandalia de susurro de mujer Keen Negro / Gris neutro , Chaco Women s ZX2 Classic Athletic Sandal Negro Sandalia de mujer Drew Alana Pewter-bronce , Sandalias KEEN Venice II H2-W para mujer Negro / gris acero Sandalias KEEN Womenport Newport Hydro-W Negro / azul nórdico Wolky Comfort Jewel 70000 canales negros , Sandalia de mujer Bellini Nickel Negro OluKai Ohana Sweet Lime / Black Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Malla negra adidas Sandalia atlética Anyanda Flex Y W para mujer Medium Heather gris / Easy Coral / White , Sandalia Drew Angela para mujer Dusty Multi Metallic , Orthofeet Verona Comfort Ortopédica Diabética Fascitis Plantar Mujer Sandalia Pescador , OluKai Kulapa Kai Carmesí / negro Sandalias Bali KEEN para mujer Ágata gris / aceituna oscura , Sandalia Helen de Propet para mujer Gunsmoke , Sandalia de mujer KEEN Rose Aluminio / gris neutro Sandalias adidas de mujer Adilette Cf + Armad Athletic Slide Maroon / Ice Purple Colegiate Navy , Sandalia KEEN Women s Uneek Slide-w Aqua Sea / Turquesa Pastel , Sandalia deportiva Zvolv X de Chaco para mujer Arco caliente , KEEN Hombres Newport H2 Sandal Negro Teva Terra-Float Active Lace - Mujeres Paloma Salvaje , Sandalia Drew Element para mujer Nubuck azul marino , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Bosque ártico , Sandalia de mujer Drew Alana Pewter-bronce ,

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 para correr Nike Women s Free Rn Flyknit 2017 Multi Color 999 Zapatillas de running Adidas Performance Ultra s Boost M para hombre Negro / Negro / Gris Oscuro , Sandalia Omnium de mujer Teva Pizarra , Sandalia KEEN Women s Venice H2 Gruta Celeste / Azul Chaco Mujer Diana Sport Sandalia Avo florentino Sandalia Drew Angela para mujer Cuero liso negro Zapatos descalzos Signa de mujer Vibram FiveFingers y paquete de toesock premium Amarillo negro Zapato de agua Fathom Aq Fitness para mujer Speedo Azul marino , Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa , Zapatillas de deporte cruzadas WX822V3 de New Balance para mujer Negro / Moire Graphic , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris negro , MBT Zapatillas para caminar Sirima 6s MJ para mujer Negro , Mizuno Women s Wave Prophecy 5-w Gunmetal / Fuchsia Purple , Sandalia casual destello de las mujeres de ECCO Duna Tecnología Vionic Men s Orthaheel White / Navy Walker Azul / verde azulado , Zapatillas de deporte Romy de Easy Spirit para mujer Blanco / rosa oscuro Sandalia Drew Cascade para mujer Sand Nubuck Easy Spirit Women s Traveltime Mule Gamuza morado oscuro / púrpura oscuro Zapato para caminar Sky Bolt para mujer Ryka Negro / Gris / Menta Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Negro Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Negro / Negro Negro Antracita , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma True White , Sandalia de tacón dulce de mujer Freebird Coñac , Sandalias Ava Sandalias Sandalias de Jerusalén rojo , Calzado de senderismo Nine West Women s Palyla de cuero Blanquecino Bomba de vestido de pasión Easy Street para mujer Nueva Marina Sperry Top-Sider Women s Angelfish 2-Oat Oat Slip-On Loafer Lino / avena Sandalias romanas para mujeres LIBERTY BLACK Vegas Faggio - Boot Round - Lb712320b marrón , Blundstone Womens BL1383 Brodo Vintage , Drew Shoe Women s Hope cuero terapéutico moda Oxfords 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 KEEN Nova Hombres Newport Newport H2 Sandal Raven// Bossa Nova da8140f 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