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.

Sintético

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

Fresco Entresuela de espuma

sin costura aplicación de material

Meta-Lock

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 running New Zapatillas Balance Trueno Boracay V3 para mujer mujer Plata Metálica/ Trueno af9f649Updated 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 running New Zapatillas Balance Trueno Boracay V3 para mujer mujer Plata Metálica/ Trueno af9f649

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 para correr Brooks Cascadia 12 para mujer Azul marino / azul / menta Zapatillas de running NIKE Lunar Skyelux para mujer Blanco / puro Platinum-off blanco , Zapatillas de running cross cross para mujer ASICS Cross Faire 2 para mujer Polvo azul / blanco / azul marino , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer BLANCO NEGRO Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Negro / Onyx / Plata , nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Hematita Metálica / Negro-gris oscuro Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Zapatillas de entrenamiento Nike Womens Metcon 3 Glacier Blue / Mica Blue Brooks Women s PureFlow 6 Antracita / Plata / Flor de Cactus Púrpura NIKE Women s WMNS Air Max 90, Negro / Negro-Cool Gris-Negro Negro / Negro / Gris Fresco Zapatillas de running adidas Ultraboost para mujer - SS18 Blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Plata Metálica Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / Metálico Dorado-blanco Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Blanco / Racer Pink / Melon Tint / Wolf Gray , 361 Zapatillas de running para mujer 361-Enjector Sparkle / Phlox Zapatillas de running adidas Alphabounce Em W para mujer Energy Blue / Energy Aqua / Mystery Petrol , Nike Men s Air Presto Essential Dusty Gray / University Red / Hyper Cobalt , NIKE Men s Roshe Two Running Shoe Iguana / Black-sail-volt Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird , Zapatillas de running Saucony para mujer Echelon 6 Gris rosado , Short NIKE Tempo para mujer Negro / rosa , Zapatillas de running NIKE Air Max Jewell para mujer Medium Olive Dark Stucco 204 Nike Men s Air Presto Essential Negro / Negro-blanco , adidas Performance Ultraboost X para mujer Gris / Gris / Hi-res Azul , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Blanco Zapatillas para correr New Balance Rushv3 para mujer Acero / Negro , HOKA ONE ONE Mujeres Clifton 2 Rosado , Zapatillas de running NIKE Air Max Flair para hombre Light Bone / White - Cool Gray ,

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 KEEN Hombres Newport H2 Sandal Azul marino / gris medio Earth Origins Sophie Sedona Brown Zapatillas de running NIKE Air Max 2017 para mujer Negro / Blanco / Antracita Sandalias Under Armour UA Ignite VII SL para mujer Negro / Neo Pulse Salomon Women s Speedcross 4 CS W Trail Runner Sombra , Corredor Trail ECCO Terracruz II de Mujeres Negro Negro , Zapatillas de trail running Inov-8 Women s Race Ultra 270 Negro / Azul / Berry , Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos Negro Sandalias de mujer Ros Hommerson Mellow Armada Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Dragon Fruit / Sea Salt , Zapatillas de deporte Skechers Performance Go Walk 4 Exceed con cordones para mujer gris Mercado de Tierra Santa Adultos / niños unisex piel genuina Sandalias Bíblicas / Flip fracasos (Jesús - Yashua) Pastor Estilo de campo II Camel Trademark Zapatillas para caminar Skechers Performance para mujer Go Train Turquesa (Turq) , Easy Spirit Women s Traveltime Mule Negro Gamuza / Blanco , adidas Crazytrain Pro W Cross Trainer para mujer Noble Ink / Cyber Gold / Super Purple Reebok Zapatillas para caminar de estilo libre para mujer Plata blanca , Zapatillas de entrenamiento Nike Air Zoom Strong 2 Metallic para mujer US) Negro Supra Vaider LC Sneaker Vintage Khaki Charcoal White Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Gris pardo Así que Danca CH50 Character Shoe con 1.5 Heel Broncearse , Trotters Lucy Ballet para mujer Hueso Ariat Unisex Scout Chap Half Chaps Negro , Sandalia Birkenstock de cuero Florida para mujer Cuero aceitado negro , FRYE Melissa Pull on Fashion Boot para mujer Fatiga Sandalias de la diapositiva del dedo del pie abierto de la moda de las mujeres de XYD Deslizamiento de cuero en las zapatillas de la zapatilla del vestido de la puerta Negro , Zapatilla Sebina Pour La Victoire para mujer Jazmín Zapatillas de deporte lavables Walker Propet para mujer Bronce , Pikolinos Womens Sitges W7J-4665CL Piedra Mocasines casuales Alpargata Deconstructed de TOMS para mujer Oxford Tan Suede ,

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 running New Zapatillas Balance Trueno Boracay V3 para mujer mujer Plata Metálica/ Trueno af9f649 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