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.

tela

Acolchado y receptivo, 0-drop trail running zapatos

Alto caucho de rendimiento único

Relajado talón de elastano para un más natural sentir

Mejor para trail running

High suela de tracción

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 Runventure para correr Topo Athletic Runventure 2 2 - Mujer Athletic Piedra/ Ciruela 0169210Updated 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 Runventure para correr Topo Athletic Runventure 2 2 - Mujer Athletic Piedra/ Ciruela 0169210

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 running Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño , Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Blanco / Mediano Azul-corredor Rosa Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Dark Raisin / White-purple Earth , Zapatillas de running Kineta Relay para mujer Saucony Negro / Chevron , Zapato de running ASICS Alpine XT para mujer Fucsia Morado / Fucsia Morado / Azul índigo , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Arctic Punch / White , Zapatillas de running Mizuno Wave para mujer Inspire 13 Dark Slate / Raspberry , Nike Hombres Air Zoom Pegasus 33 Wolf Gray / Blanco-cool gris , Zapatillas para correr NIKE Women s Air Vapormax Flyknit Negro / Blanco-negro-corredor Azul Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird , Zapatillas para correr Saucony Women s Cohesion 10 Gris / verde azulado / cidra , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco , Zapatillas de running Mizuno Women s Wave Catalyst 2 Monumento / imán , Amortiguación 635v2 New Balance para mujer Plata , Zapatillas de running NIKE Air Huarache para hombre Partícula rosa / hueso claro Brooks Womens Adrenaline GTS 18 Púrpura / rosa / plata Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Blanco / Gris Oscuro Zapatillas de running Kinvara 8 Saucony para mujer Azul Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , New Balance Women s w990v4 Running Shoe Komen Pink , Zapatillas de running adidas Performance Women s Lite Slip-On Prisma azul / plata / blanco ONEMIX Zapatillas para correr Air Cushiong para mujer, Zapatillas deportivas deportivas ligeras Negro rojo Zapatillas de running adidas Women s Edge Lux 2 Energy Aqua / Trace Blue / Gray One , Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Indigo vintage / Reflexión Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Zapatillas de running Saucony para mujer Hurricane ISO 2 Azul / Plata / Citron ,

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 Newton Running Womens Kismet 4 Primavera Verde / Amarillo Zapatillas de running NIKE Roshe One para mujer Total Crimson / Total Crimson-black Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Negro / Vivid Pink , Zapatillas de deporte Onitsuka Tiger Mexico 66 Amarillo / negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Púrpura Sandalia Archie Fisherman del Dr. Scholl Monumento Nubuck , Skechers Performance On The Go 400 Vivacity Flip Flop Taupe / Blanco Zapatillas de trail running S-Lab Speedcross para hombre Salomon Negro , Zapato para caminar WW877 para mujer New Balance Plata , Zapato de caminar sin tocar Skechers Performance para mujer Go Step Gamuza de carbón , OLUKAI Alohi - Mujeres Fray Brown / Fray Brown Zapatillas deportivas Easy Spirit Punter para mujer Blanco / gris claro Sandalia Tonya de mujer Trotters Equipaje Easy Spirit - Zapatillas Faisal2 para mujer Blanco NIKE Men s rsquo; s Zapatos casuales Darwin Zapatillas deportivas ligeras Confort Running Sail / Sail-Summit White , adidas Athletics 24/7 Tr M Cross Trainer Acero sin Procesar / Acero Crudo / Núcleo Negro , Vans Auténtico Madder Brown / True White , Zapatillas de skate solar NIKE SB Check para hombre NIKE Black Team Red Gum Marrón Claro Zapatillas para correr de tenis Flyknit para mujer, de los hombres de la ciudad, par núm. 66 Zapatillas de deporte, zapatillas de baloncesto Negro , Sandalia Dansko para mujer Sophie Negro Full Grain Sandalia de diapositivas para mujer LifeStride Remix Gris pardo , Zapatillas de volante Mizuno Wave Lightning Z4 para mujer Blanco negro Botas vaqueras de media caña para mujer Carlos Carlos, de Carlo Santana, color cuero Negro , Bota de montar de mujer Arabia de ALDO Negro , Puf lindo de Patricia Green para mujer Blanco Zapatillas Iris Easy Spirit para mujer rojo , Vince Camuto Womens Sessily astuto , Zapatillas para mujer Del Rey TOMS Sub azul Propet Walkers Evolution Oxford lavable de las mujeres Gris / rosa Ballet de Cole Haan Leah Skimmer para mujeres Cuero 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 Zapatillas Runventure para correr Topo Athletic Runventure 2 2 - Mujer Athletic Piedra/ Ciruela 0169210 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
481893

1462333

2252823

2810588

8567186

15693819

16530971

16932333

18009472

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