• If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • Stop wasting time looking for files and revisions. Connect your Gmail, DriveDropbox, and Slack accounts and in less than 2 minutes, Dokkio will automatically organize all your file attachments. Learn more and claim your free account.


Calendar of Assignments

Page history last edited by Jonathan Dietz 6 years, 5 months ago

















Assignment Due (to be done before class starts on date listed at left) Readings  Videos 
  • Our course website is a wiki! What is a wiki? Watch the Common Craft video for an explanation. 
  • Sign up to be an "writer" of this wiki webspace by clicking on the link in the email you were sent or by click on the "To join this workspace, request access" link on the right.

Introduction to Euclid as the Father of Geometry

 ( Video)



Active Geometry: The House that Euclid Built




 In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem. It is generally attributed to Thales, who is said to have sacrificed an ox in honor of the discovery, but sometimes it is attributed to Pythagoras.

Khan Academy: Euclid as the Father of Geometry


Lincoln(2012)-Euclid Quote


Euclid and Politics







Carl Sagan Cosmos- Eratosthenes


Theta= 1/50 of 360 degrees


10 stade= 1 mile





Carl Sagan on the death of Hypatia of Alexandria


History Connection: Was Columbus a Great Navigator? 


Catigara is also shown at this location on Martin Waldseemueller’s 1507 world map, which avowedly followed the tradition of Ptolemy.[11] Ptolemy’s information was thereby misinterpreted so that the coast of China, which should have been represented as part of the coast of eastern Asia, was falsely made to represent an eastern shore of the Indian Ocean. As a result, Ptolemy implied more land east of the 180th meridian and an ocean beyond. Marco Polo’s account of his travels in eastern Asia described lands and seaports on an eastern ocean apparently unknown to Ptolemy. Marco Polo’s narrative authorized the extensive additions to the Ptolemaic map shown on the 1492 globe of Martin Behaim. The fact that Ptolemy did not represent an eastern coast of Asia made it admissible for Behaim to extend that continent far to the east. Behaim’s globe placed Marco Polo’s Mangi and Cathay east of Ptolemy’s 180th meridian, and the Great Khan’s capital, Cambaluc (near Beijing), on the 41st parallel of latitude at approximately 233 degrees East. Behaim allowed 60 degrees beyond Ptolemy’s 180 degrees for the mainland of Asia and 30 degrees more to the east coast of Cipangu (Japan). Cipangu and the mainland of Asia were thus placed only 90 and 120 degrees, respectively, west of the Canary Islands.



Christopher Columbus modified this geography further by using 53⅔ Italian nautical miles as the length of a degree instead of the longer degree of Ptolemy, and by adopting Marinus of Tyre’s longitude of 225 degrees for the east coast of the Magnus Sinus. This resulted in a considerable eastward advancement of the longitudes given by Martin Behaim and other contemporaries of Columbus. By some process Columbus reasoned that the longitudes of eastern Asia and Cipangu respectively were about 270 and 300 degrees east, or 90 and 60 degrees west of the Canary Islands. He said that he had sailed 1100 leagues from the Canaries when he found Cuba in 1492. This was approximately where he thought the coast of eastern Asia would be found. On this basis of calculation he identified Hispaniola with Cipangu, which he had expected to find on the outward voyage at a distance of about 700 leagues from the Canaries. His later voyages resulted in further exploration of Cuba and in the discovery of South and Central America. At first South America, the Mundus Novus (New World) was considered to be a great island of continental proportions; but as a result of his fourth voyage, it was apparently considered to be identical with the great Upper India peninsula (India Superior) represented by Behaim—the Cape of Cattigara. This seems to be the best interpretation of the sketch map made by Alessandro Zorzi on the advice of Bartholomew Columbus (Christopher’s brother) around 1506, which bears an inscription saying that according to the ancient geographer Marinus of Tyre and Christopher Columbus the distance from Cape St Vincent on the coast of Portugal to Cattigara on the peninsula of India Superior was 225 degrees, while according to Ptolemy the same distance was 180 degrees.[12]

  Review Week #1     

Introduction to SketchUp 





  •  Design Small house or structure
  • Structure should be solid model
  • Scale Model to < 1" in size 
  • Assemble models as a group 
  • Export as STL file 
  • Save 

Introduction to 3-D Printing 


  • Download and install BFB Axon 2 Software (PC only, sorry)
  • Open STL file, scale, set build parameters, "build", save build file as BFB file per Axon2 Manual
  • Setup Rapman 3.2 printer per Rapman manual
  • Insert build file into printer, hit 'print'






Design a small house and small furniture item, and fabricate on 3-D printer 



Introduction to Art of Problem Solving-Geometry


Geogebra- Geometry Software


Optional--Khan Academy Intro to Geometry (take notes)



Richard Feynman on youtube on beauty


Opening Scene of "Infinity"

Feynman Diagrams 


Video Example: Two Column Proof


Latex for Geometry Proof


Reading: Chapter 1, Sections 2.1-2.5


Do notes and  exercises


Art of Problem Solving Introduction to Geometry Begins


  •  Week 1: Angles
    • Points, lines, rays, segments, planes
    • Measuring angles
    • Parallel lines
    • Angles in a Triangle 

Sections 2.6-2.7, 3.1-3.5

  •  Week 2: Congruent Triangles
    • Exterior angles
    • Congruent Triangles-SSS, SAS, Funky Triangles  




Sections 2.6-2.7, 3.1-3.5

Khan Academy: Triangles 

Enrichment: Visit from Henry Skupniewicz, MIT Department of Architecture





 Arranged by Jeanette Nip- Science PTO



Enrichment: Mathemusician Vi Hart


Computational Origami









  •  Week 3: Area
    • Isoceles and Equilateral triangles 
    • Perimeter and Area 
Section 3.6-3.7, Chapter 4  
  •  Week 4: Similar Triangles
    • What is Similarity?
    • Types of Similarity-AA, SAS, SSS
    • Construction techniques  


Chapter 5  
10/22  Chapter 5, Sections 6.1-6.2   
10/29  Sections 6.3-6.6   












Origami with Catherine Zhao 











































Catherine's Slide Show of Work





Math for Halloween:



  •  Week 7: Special Parts of a Triangle
    • Bisectors
    • Perpendicular Bisectors
    • Angle Bisectors
    • Medians 



Compass Constructions 

Geometry of Gothic Architecture

Sections 7.1-7.4   
  •  Week 8: Special Parts of a Triangle
    • Altitudes
    • Constructing Bisectors 


Leonhard Euler 



Video: Between the Folds





Food for Thought: A Mathematical Thanksgiving Feast




Mathematical Science Fiction:



Mathematical Songs:


Sections 7.5-7.7   


  • Week 9: Quadrilaterals 
    • Quadrilateral Basics
    • Trapezoids
    • Parallelograms 


  • The median is parallel to the bases and its length is the average of the lengths of the bases.
  •  The area of a trapezoid equals the product of the length of the median and the distance between the two parallel sides (which we call the height).
  •  If the angles of a trapezoid along the same base are equal, then the trapezoid is an isosceles trapezoid. In an isosceles trapezoid, the legs are equal in length, as are the diagonals
  •   two general strategies that are useful in trapezoid problems: (1) use what we know about parallel lines and similarity, and (2) drop altitudes to form right triangles.
  • A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel (hence the name).
  • In a parallelogram, opposite sides are equal in length and opposite angles are equal in measure. Also, the diagonals of a parallelogram bisect each other.
  • The area of a parallelogram equals the product of a side length and the height from that side to the opposite side.
  • A rectangle is a quadrilateral with all equal angles. Rectangles are parallelograms, inheriting all of those properties. They additionally have equal diagonals.
  • A rhombus is a quadrilateral with all equal sides. Rhombi are also parallelograms, and their diagonals are perpendicular.
  • The area of a rhombus, or in fact any quadrilateral with perpendicular diagonals, is half the product of the diagonals.



Field Trip to MIT 




Meeting with Marty and Erik Demaine



Humanoid Robot PR2: 


Video of PR2 Baking a Cake



SEG Origami Robot:( see http://people.csail.mit.edu/cagdas/SEG/SEG__MITs_Origami_Inspired_Submission.html )



LegoNxT Finite State Machine Presentation- Lindsay Sanneman and Dylan Hadfield, MIT  PowerPoint


LegoNxT Program, Road Clearing

Sections 8.1-8.3   


Geometry of Soap Bubbles



Note 120 degree angles


Geometry of Soap Bubbles


Exploratorium: When Bubbles Meet Bubbles


The Shape of Bubbles



Regardless of their relative sizes, the bubbles will meet the common wall at an angle of 120 degrees. This is easy to see in the bubble picture to the right. All three bubbles meet at the center at an angle of 120 degrees. Although the mathematics to prove this are beyond the scope of this article, the 120 degree rule always holds, even with complex bubble collections like a foam. 







Bubble Colors


"...If the crests of two or more waves are in step (yellow and magenta waves top), or almost in step, they can combine into a larger or more intense effect (red wave top.) This is called "constructive interference." If the crest of one wave meets the valley of another (yellow and magenta waves bottom), they cancel each other out (red wave bottom.) When two light waves cancel each other, the result is darkness and this is called "destructive interference."   

  • Week 10: Quadrilaterals 
    •  Rhombi
    • Rectangles
    • Squares 
    • If and Only If 



Sections 8.4-8.8   



Music and Mathematics:



Vi Hart at Gel 2011 from Gel Conference on Vimeo.


Vi Hart Videos on Mathematics and Music:













  • the shortest distance from a point outside a circle to the circle is along the segment connecting the point to the center of the circle.  
  •  the sum of the interior angles of a polygon with n sides is 180(n-2) degrees, and
    • the sum of the exterior angles of a polygon is 360 degrees.
    • A regular polygon has all its sides equal and all its angles equal.
    • the measure of each interior angle of a regular polygon is 180(n-2)/n and the measure of each exterior angle is 360/n.
    • a polygon with n sides has n(n-3)/2 diagonals.
    • the area of a regular polygon equals half the product of the perimeter of the polygon and the distance from the center of the polygon to one of the sides of the polygon. (This distance is called the apothem.) 
Chapter 9   


Vi Hart: Snowflakes, Starflakes, and Swirlflakes


Vi Hart: Gauss Christmath Special

Chapter 10   

Vi Hart: Dialogue for 2



Gothic Geometry- Rose Window




  • Week 13: Circles and Funky Areas 
    • Arc Measure, Length, and Circumference
    • Area
    • Funky Areas  
Chapter 11   
  • Week 14: Circles and Angles 
    • Inscribed Angles
    • Angles Inside and Outside Circles  


RIP Aaron Swartz 

War for the Web - Excerpts from Aaron Swartz interview, July 10, 2012 from War for the Web on Vimeo.


Vi Hart- Spherical Snowflakes


Snowflake Gallery


Sections 12.1-12.2   




Alternate Segment Theorem 

Sections 12.3-12.5   


Week 16: Power of a Point  




See also: http://www.regentsprep.org/Regents/math/geometry/GP14/PowerTheorem.htm


Google Science Fair


Pizza Physics

Chapter 13 

Khan Academy: Vision


Vi Hart's Guide to Comments


MIT Admissions Director at Maker Faire

How to Get Into MIT








Chapter 14 

Paper Polyhedra 


Live-in Polyhedra


Vi Hart-Fruit Polyhedra


Soda Straw Tensegrity Structures


Pumpkin Polyhedra

  • Week 18: 3-D Curved Surfaces 
    • Cylinders
    • Cones
    • Spheres 
Chapter 15   
  • Week 19: Transformations  
    • Translations
    • Rotations
    • Reflections
    • Dilations
    • Transformations 
Chapter 16   
  • Week 20: Coordinate Geometry I 


DesCartes and the Fly 



Week 20 Proof 10:



Sections 17.1-17.3, 17.5   
  • Week 21: Coordinate Geometry II 
    • Proofs
    • Advanced analytic Geometry Problems  


DesCartes and the Fly


Sections 17.4, 17.6   
  • Week 22: Introduction to Trigonometry 
    • Trigonometry and Right Triangles
    • Not just for right triangles
    • Law of Sines and Cosines  






Rocket Flight Equations( NASA):  http://exploration.grc.nasa.gov/education/rocket/rktpow.html



Initial Mass of Rocket: 22 grams

Weight of Fuel: 6.24 gm

Average Thrust= 6 Newtons

Burn Time: 0.8 seconds

Apogee reached: Approx 3.5 seconds

Estimated Height: 70 meters


 Pi Day 2013


Pi Day.org


Pi Day 2013 at the Exploratorium


Vi Hart: Pi is Still Wrong 

Chapter 18   
  • Week  23: Problem-Solving Strategies in Geometry  
    • The Extra Line
    • Assigning Variables
    • Proofs  



Chapter 19   
  • Week 24: More Problem-Solving Strategies in Geometry 




Honors Geometry Review Problems



Celebrating the Completion of the AOPS geometry course!


Chapter 19   



Textbook on Groups-Part I








 Art of Problem Solving Outline of Group Theory










Professor Javed Aslam introduces students to binary arithmetic for computing.


Math Aside: Binary Arithmetic


Discrete Math Aside May 2nd-


Notes on Transistors and Logic Gates


Boolean Algebra


DeMorgan's Theorem


May 17th- Algorithms




Introduction to Arduino and Physical Computing 



Why are we doing this?









  • Arduino Software:  http://www.arduino.cc/  - Go to "Download"; download Arduino IDE Folder; Follow Instructions under "Getting Started"
  • Fritzing ( Circuit Design Software):  http://fritzing.org/ - Go to "Download"






 Math Aside:



Notes- Number Representations 





1. Basic Circuit Theory






Examples of Arduino Circuits 

2. Schematics 


3. Introduction to Arduino : Blink

 http://www.youtube.com/watch?v=xKwox3dd-dE&feature=share&list=PL8CD32146ED5CD04E  (2:30)


Also:Intro to Arduino http://youtu.be/pMV2isNm8JU



  •  Challenge 2: Make row of blinking LEDs-see answer

4. Sensors 1: Analog Read Serial- Reading an analog voltage, displaying it on the Serial Monitor 




Discrete Math Aside May 2nd-


Notes on Transistors and Logic Gates

Boolean Algebra

DeMorgan's Theorem



Introduction to Breadboarding:  http://www.youtube.com/watch?v=CssOBb-qaX0&feature=share&list=PL8CD32146ED5CD04E 



Challenge 4: Read a Button : If/Else


Reading: Arduino in a Nutshell pp. 9-10








Arduino-Bot: Video


Challenge: Build a robotic vehicle that can follow a wall at a fixed distance.


                    Add lights and buzzers that indicate if it is turning left, right, or going straight.


                    Add a servo motor that enables robot to scan left and right, and find its way

                    through a maze- see Explorer Bot


Example Code

Controlling Motors with an H-Bridge:























Creating Functions (useful for organizing code) 


Adjusting Output Speed with Pulse-width Modulation(PWM)


  analogWrite(pin,  value 0-255);




Challenge 5: Control Tone with  Light Sensor


Challenge 6: Using Ultrasound



Challenge 7: Using Motors







  • Arduino Software: http://www.arduino.cc/ - Go to "Download"; download Arduino IDE Folder; Follow Instructions under "Getting Started"
  • Fritzing ( Circuit Design Software): http://fritzing.org/ - Go to "Download"



Geometry Review:







Bonus Project :Sea Perch:



MIT Sea Grant's new SeaPerch program introduces pre-college students to the

 wonders of underwater robotics. Part of the Office of Naval Research's initiative,

"Recruiting the Next Generation of Naval Architects," this program teaches students

 how to build an underwater robot (called a SeaPerch), how to build a propulsion system,

how to develop a controller, and how to investigate weight and buoyancy. This endeavor

is one of many exciting new projects funded by the Office of Naval Research as part of its

National Naval Responsibility Initiative. The initiative focuses on bringing academia,

government and industry to work together to ensure that the talent needed to design

 the Navy's next generation of ships and submarines will be there when needed.


MIT Sea Perch http://seaperch.mit.edu/


Videos: Building the Sea Perch http://www.phillyseaperch.org/construction-videos.html











Salman Khan, of Khan Academy, interviewed by Rafael Reif, president of MIT




Civil Engineering students at MIT display model bridges utilizing Catenary Arches , which are self-supporting


















Surveyor using triangulation

to map locations

Comments (0)

You don't have permission to comment on this page.