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.

Caucho

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

NIKE Men s 19571 Flex 2017 RN Negro Gris oscuro 2017 B017O26ZQY Gris oscuro 8fd6a7cUpdated 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.

NIKE Men s 19571 Flex 2017 RN Negro Gris oscuro 2017 B017O26ZQY Gris oscuro 8fd6a7c

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 NIKE Mens Air Zoom Structure 19 Negro / Blanco-Gris Oscuro-Gris Fresco Supra Martine Run Skate Shoe para hombre Caqui Vintage / Caqui Vintage , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Trabajo azul / gris Supra Martine Run Skate Shoe para hombre Gris blanco Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul Mediano / Negro-aluminio-electro Verde Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Trueno / Blanco Calzado de running NIKE Kids Roshe One (GS) Coastal Blue / White-blue Spark Under Armour Men s Speedform Intake Gris Nublado (941) / Glacier Grey , Zapatillas de running NIKE Mens Air Zoom Structure 19 Laguna Gris Oscuro / Azul / Volt / Negro , Zapato de running Saucony Zealot ISO 2 para mujer Coral / púrpura / algodón Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo , Zapatillas de running adidas Alphabounce Em W para mujer Mystery Petrol / Gray Two / Petrol Night Zapatillas de running NIKE RN Flyknit 2018 Nylon para mujer Pulso carmesí / Vela / Híper carmesí / Negro , Zapatillas de running Saucony Women s Freedom ISO Negro , Zapatillas de running NIKE Air Max Thea para mujer Fucshia Flash / Fuchsia Glow En la ejecución de calzado de mujer en la nube menta NIKE Men s Roshe Two Running Shoe Wolf Gray / Dark Gray / Volt Nike Men s Air Presto Essential Velvet Brown / Light Bone , Zapatillas de running NIKE para mujer Air Huarache Run Ultra Elemental Rose / Wolf Gray / White , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Gris oscuro / Barely Volt / Wolf Gray Skechers GOrun 5 Zapatos de mujer Negro Negro Calzado de running NIKE RN 2017 2017 para mujer Solar Red / Black / University Red Supra Martine Run Skate Shoe para hombre Rosa / Rosa-blanco adidas Women s Edge Lux 2 Sol / Plata / Naranja Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de running Saucony Women s Breakthru 2 Rosa / azul marino , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Espuma de mar / cielo cubierto , Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa ,

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 Zapato de running ASICS GT-1000 6 para mujer Insignia Azul / Plateado / Rojo Rouge , Chaco Women s ZX2 Classic Athletic Sandal Crest Citrus , Zapatillas de baloncesto Jordan Kids Jumpman Team II Wolf Gray Infrared 23 Black , Zapato Casual de Cuero Cortez Clásico NIKE para Mujer Pure Platinum / Pure Platinum Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Gris oscuro / Blanco / Negro / Sigilo Zapatillas de running Under Armour para mujer Toccoa Negro (002) / Gris Nublado Zapatillas de running adidas Duramo Lite W para mujer Azul / blanco / colegial azul marino Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo , Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Negro / Negro / Blanco OLUKAI Nohea Nubuck - Mujeres Dark Java / Tan , Sandalia Ombre Original Teva Mujer Original Deep Teal Finn Comfort Women s Ikebukuro Oxford Notte / atlántico Cris / patente , Bomba de vestir Berlin-Su de mujer Pelle Moda Arena / puesta de sol , Sandalia de mujer UGG Holly Ocre suave Revere Miami sandalia de confort para la mujer w. Rojo / Croc Zapatillas clásicas Vionic para mujer Walker Blanco azul , Zapatillas para caminar Skechers Go Flex Train-14825 para mujer Blanco negro Entrenador cruzado NIKE Flex 8 femenino Negro / Blanco-antracita , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Gris oscuro , Zapatillas de senderismo Zen Pro para hombre SCARPA Carbón de leña / tónico Apis Mt. Negro , Zapatillas de deporte Fenty x Trainer Mid de PUMA para mujer Puma Black / Puma White , Zapato de skate casual Heathrow DC para hombre Azul marino / caqui Zapatillas de tenis NIKE Women s Court Lite Blanco / Mate Plata-medium gris Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Negro / Tweed , Sandalias de plataforma deslizante con plataforma Forella de Gentle Souls para mujer Flor negra Bota de sarga Rockel Total Motion para mujer Rockport Cuero negro Blundstone Hombres 1464 Nubuck Boots Gris Nubuck , Botín de arranque corto patriótico United Old Blue Gringo para mujer - Bl2976-1 Azul , Propet Walkers Evolution Oxford lavable de las mujeres Gris claro / azul claro ,

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 NIKE Men s 19571 Flex 2017 RN Negro Gris oscuro 2017 B017O26ZQY Gris oscuro 8fd6a7c 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
4308850

6036900

6961299

11042388

14573577

14659211

15280560

17419400

17587092

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