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

El hermosa flor diseño de esto sandalia única en su especie lo hace un verdadero cabeceo

Cuero superior con dos Velcro® ajustable correas para una costumbre ajuste

Cuero completo revestimientos y un natural, dedo oblicuo

Grueso Poliuretano ULTRON® plantilla con gamuza cubierta superior

extraíble Con cierre de Velcro®, plantilla de doble profundidad acomoda la prescripción ortesis

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 mujer de mujer Sandalia Drew Alana Drew Pewter-bronce d5be6aaUpdated 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 mujer de mujer Sandalia Drew Alana Drew Pewter-bronce d5be6aa

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

MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Verde Sandalia deportiva Tirra para mujer Teva Negro / Blanco Multi Chaco Mujer Zcloud X Athletic Sandalia Acción azul , Encaje de quimera de Columbia para mujer Arrecife / Nivel del mar , Sandalias Newport H2 de KEEN para mujer zen Sandalia de susurro de mujer Keen Licor de café / amarillo Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad , Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Laguna azul , Chaco Mujer Z Eddy X2 Sport Sandalia Columnas Eclipse , Earth Origins Sophie Women s Sandal Platino , Revs Seva Sandals, Sandalias de reflexología Para hombres y mujeres. Absorción de impactos, Cojín Comfort & Soporte del arco marrón Chaco Zcloud 2 Sport Sandal para hombre Varsity Sun , Skechers Women s-Go-600-15315 Sport Sandal gris Sandalia KEEN Clearwater CNX para mujer Gárgola / azul nórdico , Sandalia de mujer Drew Alana Pewter-bronce , KEEN Hombres Newport H2 Sandal Tinta / óxido de la India Sandalias KEEN Mujer Whisper-w Higo de verano / Crabapple , Earth Origins Sophie Women s Sandal Sedona Brown KEEN Hombres Newport H2 Sandal Negro / Gris piedra , Sandalias de tirantes Sunstone para mujer Merrell Vertiver Chaco Women s Z2 Classic Athletic Sandal Swell Melocotón Sandalia de deslizamiento ultra Comfort NIKE para mujer Racer Pink / Sunset Glow Sandalia deportiva Marc Jacobs Women s Somewhere Crema Multi , Chaco Z / Canyon 2 Negro acolchado Sandalia atlética Salomon Women s Crossamphibian Swift W Mallard azul Skechers Performance Women s The Go 400 Radiance Flip Flop Negro , Sandalia deportiva Zvolv X de Chaco para mujer Kaleido Eclipse , Chaco Women s Zx2 Classic Athletic Sandal Wink Blue , Earth Origins Sophie Women s Sandal Sedona Brown

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 Teva Terra Sandal para hombre Rocio Olive Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco / Blanco / Negro Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul , KEEN Hombres Newport H2 Sandal Bossa Nova / Gárgola , Sandalia Terrell Post II Femenina de Merrell Pizarra , Zapatillas de running Adidas Women s Energy Cloud WTC W Negro / naranja / blanco , Zapatillas de running NIKE Lunar Skyelux para mujer Gris oscuro / blanco / antracita / explosión rosa Zapatillas de running NIKE para mujer Air Max Motion LW Aurora Verde / Jade Iced / Blanco New Balance Women s w940v3 Running Shoe Plata Finn Comfort Women s Ikebukuro Oxford Marino Royal Nubuck , Bota Deia para mujer de TOMS Negro Negro Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón , OLUKAI Pua Sandal - Mujer Frijol / frijol , Zapatillas de running adidas Women s Edge Lux W Negro / Blanco / Dgh Gris Sólido , Sandalia Gora Yoga Sanuk para mujer Marron oscuro Chaco Zcloud 2 Sport Sandal para hombre Rocket Eclipse CLARKS Sillian Stork Fisherman Sandal para mujer Nubuck sintético azul / gris Clásicos de base Slip-on (Tm) para hombres de Vans Tumble Black , Vans Unisex Old Skool Zapatillas clásicas de skate Leopardo metálico / negro , Sandalia de mujer Hana Slide de Via Spiga Cuero naranja caliente , Sandalias Jerusalem Sandalia Joanna para mujer Negro Bota Ecuestre Xandywc LifeStride para mujer Black Wc Regimiento de mujeres de Lugz Hi Fleece Water Resistant Crema de chocolate Bota de cordones con estampado victoriano reinventado 1460 de Dr. Martens para mujer Cuero liso negro Bates Lites cuero negro Oxford Negro , Bota de mujer Rieker Felicitas 84 Wine Eagle / mogano Ambor / multi Polar , CLARKS Women s CloudSteppers Sillian Jetay Flat Black Snake Print Propet Women s Tour Walker Strap Sneaker Negro , Zapato femenino Keli de Alegria Quarry Crackle ,

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 mujer de mujer Sandalia Drew Alana Drew Pewter-bronce d5be6aa 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
1645528

4664404

5824475

8361978

9013511

10133055

14426099

18539172

18949986

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