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.

Textil / Sintético

Importado

Suela de goma

Plantilla extraíble

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 de running Adidas Rojo Performance Ultra s Boost M Cardenal Borgoña para hombre Colegial Borgoña/ Cardenal/ Misterio Rojo 5cce745Updated 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 de running Adidas Rojo Performance Ultra s Boost M Cardenal Borgoña para hombre Colegial Borgoña/ Cardenal/ Misterio Rojo 5cce745

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

Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Antracita / Tarta Converse Chuck Taylor All Star Seasonal Color Hola Negro / Blanco / Lava , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco , Zapatillas de running NIKE Roshe Two para mujer Lobo gris / blanco / gris lobo , Zapatillas de running Saucony Women s Triumph ISO 4 Blanco azul , Zapatillas de running Spira Phoenix para mujer con resortes Negro / Neón Amarillo / Blanco , Zapatillas de deporte NTI de PUMA Fierce para mujer NATI Cantera , Under Armour Speedform Slingwrap para mujer Glacier Grey (100) / Acero Zapatillas de running adidas Alphabounce Cr W, mujer Core Black / Core Black / Core Black , Zapatillas para correr Nike Women s Flex Experience Rn 5 Blanco / Blanco / Gris Lobo , Zapatillas MBT Zoom 18 para mujer Zapato deportivo con cordones / Malla con cordones Negro / Malla , NIKE Women s Tanjun Blanco / Plata Metálica Converse Chuck Taylor All Star Seasonal Color Hola Azul Zapatillas NIKE Women s Revolution 3 para correr Gris oscuro / blanco / negro , NIKE W Air Max 97 Ul 17 PRM Mujeres Ao2325-002 Negro / Negro-gran gris Zapatillas de running NIKE Roshe Two para mujer Paramount Blue / Paramount Blue , Zapatillas para correr NIKE Lunarcharge Essential para mujer Negro / Gris Oscuro-negro-voltio Under Armour Women s Charged Rebel Gris Nublado (941) / Blanco Zapatillas para correr A7200WX085 de Apex para mujeres Rosado , Converse Chuck Taylor All Star Seasonal Color Hola Amarillo fresco , Converse Chuck Taylor All Star Seasonal Color Hola Negro / garceta / cuero Cordura negro Zapato de running ASICS GEL-Venture 5 para mujer Gris plateado / turquesa / sacador de lima , adidas Women s Distancestar W Blanco / Hi-res Verde / Real Purple , NIKE Men s Roshe Run Volt / negro-blanco , adidas Originals Superstar de mujer Slipon W Sneaker Blanco / Blanco-blanco roto , Zapatillas de running NIKE Juvenate para mujer Negro / Blanco / Negro / Blanco Carro de mujer Brooks Fucsia NIKE Women s Tanjun Blanco negro Calzado casual tejido NIKE NIKE para hombre Cool Gray / White-black ,

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 Supra Martine Run Skate Shoe para hombre Andora-blanco ALMA NATURAL Sandalia de pescador Beatrice para mujer Chocolate , Sandalias Bali KEEN para mujer Negro / Imán Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth NIKE Women s WMNS Zoom Fly, negro / blanco Negro / Antracita / Lobo Gris / Blanco , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Blanco negro Sandalia de mujer Drew Alana Naranja amarillo , NIKE Mujeres Air Zoom Pegasus 33 Ocean Fog / Black-gamma Blue-volt , Wolky Comfort Jewel 80120 Cuero blanco roto Columbia Montrail Women s Trans Alps F.K.T. Jade Lime, Splash Propet Women s TravelActiv Ss Sandal Púrpura / negro CLARKS Mujer Gracelin Mara Flat Black Smooth Sandalia atlética de la sandalia del deporte de Lila 6 de las mujeres de MBT Negro / Gris carbón Zapatillas de running W775V2 para mujer New Balance Gris / morado , Zapatillas de running NIKE Air Max 90 para mujer Negro / Negro-gran gris , Sandalia Terrell II de Merrell para mujer Negro , Sandalia para mujer Merrell Dwayka Seaway Thong para mujer Cielo azul Calzado de espuma fresca para mujer New Balance 1100V1 Blanco negro Sandalia de gladiador Samina de BareTraps para mujer Mezclilla Zapatillas de skate Etnies para mujer The Scam W Rosado , Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro , Sandalia Juno Slide de Vionic Women s Rest Broncearse Andr y eacute; Oro suave , Sandalias planas Cammie de Jessica Simpson para mujer Desnudo Blush , Vince Camuto Women s Haydan Slide Sandal Karat Gold Zapato de danza de gema Soleil moderno para mujer de PUMA Magenta Púrpura / Magen , J.Renee Womens Bilboa Plata blanca , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2721- Silver Sparklenet & Silver Trim , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Zapatos de mujer - Wichita Negro Botin de pantorrilla de Chinchilla de mujer Acorn Crackleberry

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 de running Adidas Rojo Performance Ultra s Boost M Cardenal Borgoña para hombre Colegial Borgoña/ Cardenal/ Misterio Rojo 5cce745 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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2833637

4912522

5546166

8071127

11171317

13254986

15095051

18861861

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