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

Ligero, conducido por estilo de vida superior

Corte bajo diseño

Nuevo esculpido y en capas entresuela

Goma en áreas de alto desgaste abajo

Nuevo esculpido y en capas suela

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 para caminar B077XBR55H Sublite Speedpak Pigment MTM para mujer/ Reebok Pigment Purple/ Lilac Metallic/ Crisp Blue 2343e90Updated 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 para caminar B077XBR55H Sublite Speedpak Pigment MTM para mujer/ Reebok Pigment Purple/ Lilac Metallic/ Crisp Blue 2343e90

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 de deporte Skechers Performance Go Walk 4 Exceed con cordones para mujer Negro , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy Skechers Women s Walk Walk Joy-Invite Sneaker Negro / rosa Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Negro Metálico , Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Negro / Microdot , Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Black Suede / White Mesh Zapato de agua Easy Spirit para mujer First Walker Azul-menta , Skechers Performance - Go Walk 4 Majestic Walking Shoe para mujer gris , Calzado SCARPA para mujer Hydrogen GTX Gris medio / Agua estancada , P W Minor Leisure Mujer Terapia Casual Extra Profundo Calzado Cuero Cordones Negro , CLARKS Janey Mae Oxford para mujeres Cuero negro Finn Comfort Women s Ikebukuro Oxford Plata , Zapatillas para caminar Skechers Go Step Lite-Wander para mujer Negro , Sandalias Harper Huarache de SoftWalk para mujer Negro / peltre , Calzado de espuma fresca New Balance para mujer 1165v1 Gris blanco , Zapatilla deportiva Skechers You-14958 para mujer gris , Earth Origins Tova Azul cobalto , Dr. Comfort Annie-X Mujer diabética terapéutica Doble Profundidad Zapato Lycra Velcro Beige Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 Blanco / Puro Platino / Plata Metálica , Skechers para el trabajo de la mujer Albie Walking Shoe Negro Zapatillas de paseo Skechers Go Walk Lite-15435 para mujer Mezclilla , Zapato atlético para mujeres con neuropatía de Ped-Lite - Adele Negro Bernie Mev Women s Gummies Gem Flat Bronce Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Negro , Zapatillas para caminar New Balance 85v1 para mujer Tan / joya , Zapatillas para caminar RYKA Women s Seabreeze antideslizantes Gris oscuro Zapato de caminar sin tocar Skechers Performance para mujer Go Step Gamuza de carbón , Easy Spirit Mujer Ezcool Mule Azul marino Combo de gamuza Skechers Women s Walk Walk Lite-15375 Sneaker Negro / gris

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 Chancleta Eezay adidas para mujer Blanco / negro / negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Birch / India Ink / Latte Earth Origins Sophie Women s Sandal Platino , Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón Merrell Women s Mqm Flex Vapor , Zapatillas de trail running Vivobarefoot para mujer Trail Freak todoterreno Dark Purple / Navy , Zapatillas de running Nike Air Huarache Run Ultra para hombre Negro / Negro / Negro , New Balance WT510RS3 Trail Running Shoes para mujer Silver Mink Zapatillas Skechers Go Walk Lite-15360 para mujer Negro Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Blanco Sandalia CLARKS Wave Grip para mujer Cuero blanco 2 , Zapatillas de running Reebok Women s Club C 85 Blanco / gris claro , Zapatillas de deporte cruzadas WX40V1 de New Balance para mujer Negro / aleta , Zapato para caminar WW877 para mujer New Balance Plata CLARKS Janey Mae Oxford para mujeres Ante azul marino , Converse Chuck Taylor All Star Core Ox Negro , Skechers Performance Women s Go Walk 2 - Zapato de marcha atrás para caminar Azul marino / rosa , Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Peacoat / Blanco , BareTraps Women s Rebecca Slide Sandal Castaño , Zapato de skate DC Mujer Trase V SE Blanco / Blanco / Rojo Atlético Sandalia Dolce Vita Jene Slide para mujer Satén medianoche Sandalia Otto Slide de mujer Yellow Box Sonrojo , Sansha Hi-Step Dance Sneaker Negro Sunny Equestrian Boot de Tommy Hilfiger para mujer Marino / Chocolate Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 9691EB Cómodo-Muy Fino 2.5 \ de 5] Satén negro CLARKS Cheyn Work Tobillera para mujer Cuero Marrón Oscuro , Under Armour Women s Lax Finisher MC Blanco (101) / Negro Mephisto Sabatina Tuxedo Oxford para mujer Patente de cuero del vino Bernie Mev Women s Brighten Fabric Casual Estaño Trotters Liz Loafer para mujer Armada

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 para caminar B077XBR55H Sublite Speedpak Pigment MTM para mujer/ Reebok Pigment Purple/ Lilac Metallic/ Crisp Blue 2343e90 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
1595691

7423563

8212145

8903190

10048859

11928700

13379236

16417861

18372726

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