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.

Hecho en EE. UU. O Importado

Diseño está en los detalles. Completa tu look en estos simplemente Me alineé también y comercio; pisos.

Cayó parte superior de cuero.

Fácil diseño sin cordones.

Redondo silueta del dedo del pie con cadena vamp detalle.

Hecho por el hombre forro. Ligeramente plantilla acolchada.Flexible suela de goma.Importado.Medidas: Peso: 6 oz Mediciones del producto fueron tomadas usando tamaño 7.5, ancho M. Tenga en cuenta que las medidas puede variar por tamaño

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

Me Too OLYMPIA1 Too para Zapato plano de cuero B06ZZCF9XW para mujer Equipaje c99c838Updated 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.

Me Too OLYMPIA1 Too para Zapato plano de cuero B06ZZCF9XW para mujer Equipaje c99c838

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

Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Chocolate , Skechers Women s Go Flex Max Loafer Armada Zapatillas de senderismo Pro Pro 3D Pro de mujer Ryka Gris / rosa , Calzado de mujer de fila Cress Walking Blanco / ciruela de azúcar , Zapatos para caminar de cuero Stretch Walker de Pedors Womens Negro , Drew Shoe La alegría de las mujeres Blanco Prop & eacute; t Zapatillas para caminar TravelFit para mujer Verde Sandalia Juno Slide de Vionic Women s Rest Cereza Skechers Women s Walk Walk Lite-15375 Sneaker Negro / gris , Prop & eacute; t Olivia Oxford para mujeres Bronco Brown , Alegria Meri Pewter Dazzeler para mujer de ancho Pewter Dazzle Earth Orígenes Sizzle para mujeres Coral , Zapato atlético para mujeres con neuropatía de Ped-Lite - Adele Blanco Linea Paolo Felicia | Suede perforado rosa claro , Drew Shoe Women s Paradise II Velco Slip-on Ternero negro , Drew Shoe Women s Tulip Comfort Estiramiento negro / negro , Prop & eacute; t Olivia Oxford para mujeres Bronco Brown Skechers Women s Walk Walk Joy-Invite Sneaker Azul marino / Multi Propet Walker de la estabilidad de las mujeres Blanco deporte Trotters Mujer Simmy Mary Jane Flat Moho , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Negro , Zapatillas para caminar Travellite Propet para mujer Negro Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco Zapato para caminar Skechers Women s You borgoña , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Piso de mujeres de Coral Pies extra anchos Diabético ortopédico Zapatos atléticos Sneakers para caminar gris Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Mushroom Nubuck Calzado de espuma fresca New Balance para mujer 1165v1 Gris blanco Zapatillas para caminar Skechers Women s Go 4-Brisk Blanco negro Zapatilla de pies Cush N de Prop & eacute; t para mujeres Arena ,

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 de running NIKE Air Max Flair para hombre Negro / Neo Turq , Zapatillas de running adidas Adizero Boston 6 W para mujer Noble Indigo / Blanco / Acero sin Procesar Zapatillas de running Newton Kismet para mujer Azul , Zapatillas MBT Zoom 18 para mujer Zapato deportivo con cordones / Malla con cordones Negro / Malla Chaco Women s Z2 Classic Athletic Sandal Swell Melocotón OluKai Ohana Dark Java / Dark Java , Sandalias sintéticas Teva para mujer Terra Fi Lite Atlántico Norte Zapatillas de running Z-Pump Fusion para mujer Reebok Gris plano / grava / cerezo de neón / blanco Zapatillas de running NIKE Zoom para mujer Negro Negro , Sandalia de mujer KEEN Rose Aluminio / gris neutro , Zapatillas de running NIKE Lunarglide 9 para mujer Pure Platinum / Chrome Zapatillas de running Mizuno Wave para mujer con sombra Fucsia Púrpura / Plateado / Azulejo Azul MBT Mujeres Baridi Mary Jane Azul , Calzado de senderismo Go Joy 15603 de Skechers para mujer Negro / rosa caliente , Earth Origins Tova Azul cobalto Zapatillas para caminar Brooks Women s Addiction Walker Negro , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Gris oscuro , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Vestido Blues Zapato deportivo artesanal de mujer superfeet Addy Negro APL: Athletic Propulsion Labs Zapatillas de running mujer Techloom Phantom Marble Green , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Armada Sandalia New Balance para mujer City Slide Plata Sandalia Vince Westcoast para mujer Vince Goma crema Sandalia Bali Yellow Slide para mujer gris , Sandalia de mujer Lucienne Suede Slide de Nine West Dark Natural Suede Bota Ecuestre Xandy LifeStride para mujer Negro , Zapatillas de ciclismo Pearl XZUMi W X-ALP Drift IV para mujer Rojo Rouge , Predicciones Comfort Plus Women s Karmen Pump Patente desnuda Earth Orígenes Raven Womens Bootie Negro , Zapatillas Mujer Teva W DE LA Viña Dos 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 Me Too OLYMPIA1 Too para Zapato plano de cuero B06ZZCF9XW para mujer Equipaje c99c838 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