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.

100% Cuero

Importado

Goma suela

Cuero upper

Hook y cierre de lazo

Float-lite suela

Etileno plantilla de acetato de vinilo

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 Teva Teva Mujer 19034 Terra-Float Livia Lux Lux Negro 643373cUpdated 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 Teva Teva Mujer 19034 Terra-Float Livia Lux Lux Negro 643373c

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

Chaco Women s ZX2 Classic Athletic Sandal Mosaico del desierto , HOKA ONE ONE Recuperación ORA para mujeres Blanco negro , Sandalia NIKE Kawa Slide para mujer Negro / Vivid Pink Sandalia de mujer Drew Alana Verde / rosa , Sandalia KEEN Clearwater CNX para mujer Celestial / Vapor , Sandalia Terrell II de Merrell para mujer Dark Slate , Chaco Mujer ZX1 Classic Sport Sandalia Aguacate estático , Sandalia deportiva Tirra para mujer Teva Galaxy Blue Multi Sandalia atlética de la correa de Barraca de las mujeres de Colombia Zing / Iceberg , Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink , Sandalia Savannah de Ryka para mujer Moonrock / Snowline Ecru Fabulicious Women s Caress 401Dm sandalias de diapositivas Clear / Silver Multi Rhinestone Sandalias Bali KEEN para mujer Red Violet / Boysenberry Sandalia Yucatan ECCO Mujer Paloma medieval / salvaje , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Bosque ártico , Sandalia Terrell Ivy Wrap Sport Merrell para mujer Sodalita Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia gris Sandalia de deporte Merrell, para mujer Terran Ari Backstrap Paloma , Sandalia de vestir Tillman de SoftWalk para mujer Lavado de oro Sanda Verra Teva para mujer Negro. , Teva Terra-Float Active Lace - Mujeres Chispas de chocolate Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva Sandalia deportiva Tirra para mujer Teva Atlántico Norte , Sandalia original Teva Women s Original Pirámide azul , Sandalia de deslizamiento ultra Comfort NIKE para mujer Negro / Blanco / Negro Sandalia NIKE Benassi JDI para mujer Habanero Rojo / Habanero Rojo , Sandalia Uneek O2-w de mujer KEEN Anemone / Bright Rose , Skechers Performance On The Go 400 Tropical Flip Flop Grey Tropical , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Negro

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 Savannah de Ryka para mujer Gris oscuro , Sandalia Yucatan ECCO Mujer Paloma medieval / salvaje , Earth Origins Sophie Negro , CLARKS Sandalia plana Brizo Ravena para mujer Gris , Zapatillas de tenis ASICS Gel-Game 6 para mujer Plata blanca , Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Plateado / Flash Coral , OluKai Miki Li - Mujeres Pale Gray / Carboncillo Zapatillas de running NIKE Air Huarache para hombre Hyper Turquesa / Blanco Drew Shoe Women s Paradise II Velco Slip-on Blanco Drew Shoe La alegría de las mujeres Hueso de gamuza / malla Sandalias FitFlop Bumble de mujer con diapositivas en cristal Negro Gorra adidas Iniki Runner gris para mujer , Tecnología Vionic Men s Orthaheel White / Navy Walker Negro Negro , Sandalia para mujer Merrell Dwayka Seaway Thong para mujer Merrell Oak , Nueve West Womens Xaeden Gamuza rosa Zapatillas para caminar Go Flex 2-14992 de Skechers Performance para mujer Gris pardo , Zapatillas de deporte Arizona Trotters para mujer Negro Zapato Dantera, para hombre, L Low Waxed Court Oro blanco / ámbar Sandalia de diadema Florita Carlos de Carlos Santana Plata Sandalia de la diapositiva de Sunvue de las mujeres alrededor de la ciudad de Merrell Textura negra , Birkenstock Arizona Sandalia de cuero con plantilla suave Marrón oscuro Birko Flor , Sandalia Dolce Vita Jene Slide para mujer Denim azul claro Sandalia Bobbi Slide de Dolce Vita para mujer Lino de arena Naturalizer Michelle s Dress Dress Pump Negro brillante Trotters para mujer Dea Slingback Pump Negro Negro , Pearl iZUMi W ELITE RD III Plata blanca , Botas de cuero Fly London Womens Yogi Rojo Córdoba , Arranque de moda Raine de China Laundry para mujer Red Stretch Patent Zapatillas de deporte Onitsuka Tiger Men s Mexico 66 Fashion Moonrock / Vaporous Grey Sneaker de moda de rayas de playa lavada campeón de mujer de Keds rojo ,

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 Teva Teva Mujer 19034 Terra-Float Livia Lux Lux Negro 643373c 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