The purpose of this test experience is to help you understand the material and make connections. Research has shown that the effort you expend in studying for tests and clearly explaining your work solidifies your learning.

At the Exam

  • One 8.5*11 sheet with writing on both sides allowed. You can put anything you like on your sheet.
  • In addition, you may also bring your child's ball with you.
  • One scientific calculator or graphing calculator allowed (but no cell phone nor other calculators bundled in combination with additional technologies)
  • You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be stand alone--no cell phone, internet or other technological connections)
  • Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.

    On the test, your grade will be based on the quality and depth of your responses in the timed environment (the test must be turned in when time is called). Questions will mainly consist of short answer or essay types along with some computations. There is NO need to write in full sentences - bullet points, etc will be fine.

    The test will be a mixture of computational questions as well as critical reasoning and reflection involving the "big picture." You will be expected to answer questions about activities from class or lab and the homework readings and activities.

  • Here is a partial sample test so that you can see and have some practice with some diverse examples of the formatting and style of questions. The actual test will differ and will be about 5 pages long.
  • The main web page and class highlights page contain links to homework readings and class activities, which also serve as a good review.
  • See the ASULearn Review Questions for Test 1 and glossary/wiki
  • Be sure you read through the following overview of topics and that you could respond to questions about these topics on the exam. I want you to understand the material and I am happy to help in office hours or on the ASULearn forum! Computational Questions
  • You could be asked to reflect on Eratosthenes method of calculating the circumference of the earth and/or provide the details of the computation.
    Angle sum, straight feeling paths and parallels
  • You might be presented with one of Escher's works, asked to calculate the sum of the angles in a triangle in the space represented, and specify whether the space is then Euclidean, spherical or hyperbolic. video review of Escher. Embedded in this analysis is the knowledge that the sum of the angles in a triangle is important since it determines the geometry of a space: 180 degrees for flat Euclidean spaces, greater than 180 degrees for spherical spaces, and less than 180 degrees for hyperbolic spaces.
  • Describe an argument of why the angle sum is 180 degrees in Euclidean (folding using parallels, or walking the angle sum) and why it is smaller in hyperbolic and larger in spherical geometry
  • Be able to plug into a spherical, hyperbolic or flat angle sum relationship to solve for the sum or an angle, like A + 90 + 45 > 180 in spherical geometry so A > 45.
  • Be able to analyze straight feeling/symmetric (to a bug) paths and parallels (or a lack of parallels) in an Escher representation, a perspective drawing, or on the earth
    Pythagorean theorem
  • Be able to plug into a spherical, hyperbolic or flat Pythagorean theorem to determine (for instance) problems like the fact that in hyperbolic geometry, the hyperbolic hypotenuse would be longer than a2 + b2 because a2 + b2 < (hyperbolic c)2.
  • Describe a geomeric argument of why the Pythagorean theorem is true in Euclidean geometry (water wheel) and false in spherical geometry (string argument...)
  • Be able to algebraically demonstrate the Pythagorean theorem based on the picture in the Zhou Bi Suan Jing or Chou Pei Suan Ching which had a large square broken up into 4 triangles and a smaller square like on the second slide here
    Wraparound spaces
  • Klein bottle tic-tac-toe: You might be given a partially completed torus or Klein bottle tic-tac-toe game and asked to mark a winning move. You may wish to review by playing a few games. In addition, you should understand the tiling view - where identified moves are marked above, below, to the right, and to the left of the board, as on the lab picture. video review of Klein bottle tic-tac-toe.
    Inverse square law
  • Inverse square law for light: Brightness changes as 1/ the square of the distance for Euclidean, with a < relationship for hyperbolic (ie dimmer at a given distance than in Euclidean) and a > relationship for spherical (ie brighter at a given distance than in flat space). This was used in the analysis of the supernova experiments.

    Other Directed Questions and Short Essay Questions
    Earth and Universe Research
  • Review class discussions and arguments for problems from earth and universe research. For example, you should be able to explain in depth the various ways we explored shortest distance paths on a sphere in class (the car, masking tape, string, Chicago-Rome reading in Heart of Mathematics, symmetry...) and the ways we explored the Pythagorean theorem on the sphere (string, algebra, computer program on a transparency, Futurama Greenwaldian theorem), along with more general questions like describing one of the possible finite shapes without edges for our universe, or providing responses for some of the other questions.
    Cross-sectional views
  • Instead of thinking about what Arthur Square or 2-D Marge would see if an orange passed by them (a sequence of lines/curves that appear, get bigger and disappear), you might be asked what she would see if something more complicated passed through, such as a mug. Also be able to differential between what they would see versus what the full cross sections are.
    Analyzing experiments to determine the geometry of the universe
  • You might be asked to describe in detail some of the methods being used to attempt to discover the shape of the universe, as well as our classroom discussion of those critiques (Gauss' triangle but light bending with gravity and other problems, Rob Kirshner's Supernovae distance/brightness experiments but Supernovae not necessarily exploding at the same brightness, looking for repeated patterns like the quarter-turn space but difficulty recognizing the patterns, WMAP and Planck data but difficulty agreeing on the meaning of the data) as in the homework readings and in class.
    Additional real-life connections
  • You might be asked about real-life applications of some of the material we have been studying, like a dodecahedron, hyperbolic geometry, a hypersphere, higher spatial dimensions or more. A good review for this would be the ASULearn Review Questions for Test 1, but the question wouldn't be in a multiple choice format, instead I would ask a real-life application of the geometric object or concept. You should also know how to describe the spaces themselves.
    Changes in world view
  • You might be asked about some of the changes in world view that came with mathematical discoveries, from the videos and homework readings. A good review for this would be the ASULearn Review Questions for Test 1, but the question wouldn't be in a multiple choice format
    Clicker questions converted to short answer
  • Short answer/essay questions adapted from the clicker questions from class.
    Lab questions
  • Lab questions or reflective questions on the labs: Perspective Drawing and Projective Geometry, 2d Universes Lab or Universe Lab (Note - I will NOT ask you about names of people nor dates).
    Thematic questions
  • You might be asked about thematic issues, like an example of local to global issues and how they related to Eratosthenes, the sum of the angles, the Pythagorean theorem, or people around the world, a reflection from this segment on what is mathematics/what is geometry, what mathematics has to offer and how it is useful, the diverse ways that people succeed in and impact mathematics, or the themes of what a space looks like, how do we know (the theme of truth--When are we convinced? What are the consequences of certain truths? What is the role of chance and probability?) and how do we represent it. For instance, a question could be phrased like: Name and discuss an instance from the geometry segment where probability and chance played a role, or a more directed question, like analyze the role of probability and chance in computing the density and geometry of the universe, how .4% related, how 1 in 3000 chance related, or how 1/180th of a degree related. Similar types of open ended and/or directed questions could ask about the other themes.