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.

Flyknit

Goma suela

Nike Air Max Thea Ultra Flyknit corriendo Zapatos tomarán tu del gimnasio a las calles en llamativo estilo

Elegante y sofisticado este Thea está hecho más y vestido impresionar con rapidez Flyknit, mientras Cables Flywire entregar segunda piel apoyo

felpa y cómodo, vas a obtener una nube sentir gracias a la combinación ganadora de espuma Phylon y una unidad Max Air, mientras que el núcleo entresuela reduce peso

El goma mínima suela ofrece peso pluma tracción para su Día ocupado

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 gris de running NIKE Air Max Thea B06X6JJ1DX Ultra FK Thea para mujer Pasas oscuras/ Pasas oscuras-blanco-pálido gris 6ce8427Updated 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 gris de running NIKE Air Max Thea B06X6JJ1DX Ultra FK Thea para mujer Pasas oscuras/ Pasas oscuras-blanco-pálido gris 6ce8427

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

Calzado de running NIKE Kids Roshe One (GS) Racer Pink / Blanco-negro-blanco , Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / antracita , Zapatillas de running PUMA Women s Speed 500 Ignite Tobillo-Alta rojo , Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa Zapatillas de running Nike Kaishi para hombre Negro Zapatillas de running TOPO FLI-Lyte 2 para mujer Coral / Aqua NIKE Women s Tanjun Partícula rosa / blanco , Nike Men s Air Presto Essential Blustery / Terra Orange Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris oscuro , Calzado deportivo New Balance para mujer Vazee Rush v2 Negro / Plata , Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / gris antracita-cool Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas Glacier Blue / Vast Grey / Pure Platinum Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo , Calzado de running Ryka Ultimate Form para mujer Negro / Plata , Zapatillas de running GT-3000 5 Performance para mujer ASICS - T755N.9611 Gris / Gris / Canteloupe , Zapatillas de running adidas Alphabounce Em W para mujer Easy Coral / Black / Easy Orange Zapatillas para correr Mizuno Women s Wave Paradox 3 Gris-rosa Zapatillas de running definitivas Ryka para mujer Gris negro , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas de running Reebok Print 2.0 para mujer Nube gris / azul cielo / negro / azul gable / blanco / plata Zapatillas de running W1540V2 New Balance para mujer Blanco azul , Zapatillas de running New Balance 990v4 para mujer Dark Mulberry , Skechers Performance Women s Go Run 400 Obstruct Rosa caliente Zapatillas para correr ASICS Gel-Venture 6 para mujer Negro / Island Blue / Pink Zapatillas para correr Altra Women s Torin 3.0 Azul / Coral Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco-negro-corredor Azul Zapatillas de running Saucony para mujer Omni 16 Teal / Citron , Zapatillas para correr Nike Women s Free Rn Distance 2 Hyper Violet / White Dark Iris Zapatillas de running NIKE Zooms Winflo 4 para mujer Antracita / gris oscuro / negro ,

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 ASICS Gel-Contend 4 para mujer Blanco / Aruba / Plata Converse Chuck Taylor All Star Seasonal Color Hola Icy Pink , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Dark Grey , Zapatillas de deporte Onitsuka Tiger Mexico 66 Abedul / blanco NIKE Women s Wmns Air Max Jewell SE, Negro / Negro-Gum Med Marrón-Marfil Negro / Negro-goma Med Brown-marfil , adidas Performance Ultraboost X para mujer Gris Tres / Gris Tres / Gris Perla , Zapatillas de running NIKE Air Max Flair para hombre Blanco / puro platino , Top corto de mujer Saucony Bullet Lakeside Print Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Oliva negro Sandalia Terrell Merrell para mujer Tierra oscura , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Gray 789 , Zapato acuático de fitness Speedo Hydroforce Xt para mujer Blanco negro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Azul marino / gris , Zapatillas de running Altra Women s Intuition 4 Ocean / Teal , Hush Puppies Walker clásico de las mujeres Blanco YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Amarillo Chancleta Yoga Sling 2 Sanuk para mujer Citrus Lanai Blanket , Columbia Womens Heavenly Chimera Omni-Heat Outdry Nori / Rusty Ahnu Women s Yoga Flex Pimiento rojo , Zapatillas de skate Vans Unisex Authentic Platform Púrpura (Petunia True White Qa1) 7270 , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Verde , Vans Unisex Old Skool Zapatillas clásicas de skate Negro Negro , Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Monocromo negro SCHUTZ Sandalia Yaslin Slide para mujer Negro , Zapatillas Iris Easy Spirit para mujer Oro , Bota de mujer Patricia Nash Sabrina Negro , Bota de mujer L19 de Arcopedico rojo , Sorel - Bota de mujer Tivoli Iii Premium Non Shell Negro / Tetera , Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra 07 Sandalia Birkenstock de cuero Florida para mujer Sfb Blue Birko-flor ,

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 gris de running NIKE Air Max Thea B06X6JJ1DX Ultra FK Thea para mujer Pasas oscuras/ Pasas oscuras-blanco-pálido gris 6ce8427 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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1416499

3259204

7059623

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10140940

10635767

13592996

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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