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.

Cuero y hombre Hecho

Importado

Sintético único

Eje mide aproximadamente parte baja del arco

Transpirable malla con cuero y superposiciones sintéticas.

Memoria plantilla de espuma

Compresión EVA moldeado.

Segundo densidad moldeada EVA

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

Ryka Negro Heather Cross Trainer Ryka para mujeres Negro// Menta 5ff6aedUpdated 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.

Ryka Negro Heather Cross Trainer Ryka para mujeres Negro// Menta 5ff6aed

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

Entrenador cruzado NIKE Flex 8 femenino Negro / Blanco-antracita , Inov-8 Unisex F-Lite 195-U Zapato de entrenamiento cruzado Neon azul / negro / blanco , Zapatillas Cross-Trainer de entrenamiento Newonic Flexonic WX66V2 para mujer Gunmetal / Alpha Zapatillas Adidas Performance Mujer Yvori Cross-Trainer Pop Purple / Super Yellow / White Vapor Skechers Sport Skech Air 2.0 City Love Fashion Sneaker Negro / rosa caliente , PUMA Women s Fierce Rope Pleats Wn Sneaker Paradise Pink-paradise Pink Calzado de trabajo resistente a los resbalones de Fila, Memory Comfort Trainer Blanco / Blanco / Plata , ASICS Gel-Fit Sana 3 Cross Trainer para mujer Invierno Bloom / Plata / Violeta , Entrenador campeón Gusto Cross Trainer Mono azul claro , ASICS para mujer Gel-Fit Yui 2 SE Brote Verde / Flash Coral / Blanco Entrenador cruzado Hydrosphere Ryka para mujeres Puerto / Yucca Mint , Zapatillas de deporte Fit Fitness GEL de ASICS para mujer Gris / Corydalis / Negro , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Nrgy Peach-puma Black PUMA Women s Fierce Rope Pleats Wn Sneaker Paradise Pink-paradise Pink Empuje cargado de Under Armour para mujer Rhino Gray (077) / Acero Zapatillas Adidas Performance Aleki X Cross-Trainer para mujer Bold Blue / Dark Indigo / Black , Under Armour Women s Precision Mid Lux Downtown Green (330) / Marfil , New Balance WX77 Cross Trainer para mujeres Negro / Tie Dye Gray / Gráfico Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Negro Zapato Ryka INFLUENCE2.5 Cross-Trainer para mujer Negro / púrpura , Zapatillas Superga 2750 Cotu para mujer Legión azul , New Balance 711v3 Cush + Cross Trainer para mujeres Blanco Zapatillas Cross-Trainer PUMA para mujer Fierce Swan WN Puma White-puma White Entrenador cruzado Hydrosphere Ryka para mujeres Negro / gris helada Zapatillas Cross-Trainer CrossFit Speed Tr para mujer Reebok Blanco / Negro / Azul Beam / Fire Spark / Pure Silver , Calzado de entrenamiento NIKE Metcon Repper DSX para mujer NIKE Gris oscuro / Lobo Gris-negro-hipervioleta , Empuje cargado de Under Armour para mujer Gala (692) / Glacier Grey Reebok Crossfit Nano 7.0 Track Shoe para mujer Neón / blanco / negro / amarillo solar Zapato de entrenamiento WX824 para mujer New Balance Blanco

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 CLARKS Sandalia Saylie Jade para mujer Arena Saucony Women s Ride 6 Running Shoe Gris / Púrpura / Coral Vizipro Teva Terra-Float Active Lace - Mujeres Azul Wolky Comfort Jewel Beige , Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Azul marino / azul claro , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco , Zapatillas de senderismo V753WW085 para mujer Apex gris CLARKS Womens Un Karely Hail Nubuck azul marino Skechers Women s Define Sneaker Negro Stuart Weitzman Womens Carmina Brillo blanquecino Naturalizer Samantha Pointed Toe Flat para mujer Perla blanca Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Mushroom Nubuck , Zapatillas para caminar Skechers Performance para mujer Go Step Lite Ultrasock Azul marino , Zapatillas de entrenamiento cruzado FM Tazon 6 WN de PUMA para mujer Puma Black-puma Silv Reebok Crossfit Nano 7.0 Track Shoe para mujer Blanco / Negro / Plata Metálico , Drew Shoe Women s Tulip Comfort Ternero blanco , Zapatos de golf AsJei para mujer FootJoy Negro Sandalias planas para piscina Walking Cradles para mujer Black Matte Snake Print / Cork Wrap , CLARKS Leisa Cacti Slide Sandalia para mujer Cuero desnudo Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro Sandalia Dolce Vita Delana Slide para mujer Fuera de cuero blanco Zapatillas de deporte unisex de bajo perfil rojo , Sandalia de plataforma Pleaser Stplash808 / C / B para mujeres Claro y Escarcha TOMS Clásicos de mujer Dp Cobalt Vintage Fl Botas de lluvia impermeables de pantano clásico de mujer con mangas altas de Bogs Black Smooth , Bota de imitación para mujer SoftWalk Gris oscuro Cole Haan Women s 2.Zerogrand Laser Wing Oxford Bronce Metálico Steve Madden Womens Kyra Blanco Multi , CLARKS Mujeres Cheyn Ava Oxford Cuero azul marino , Zapatillas Slip-On Roamer SAS para Mujer 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 Ryka Negro Heather Cross Trainer Ryka para mujeres Negro// Menta 5ff6aed 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
2090769

2983455

4221506

5771255

6120301

7666283

9533956

17414964

17454845

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