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Einstein Said…

Einstein Said…. “…imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.”

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Einstein Said…

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  1. Einstein Said… • “…imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” Scott, Michael (2011-05-24). The Warlock (The Secrets of the Immortal Nicholas Flamel) (p. 320). Random House, Inc.. Kindle Edition.

  2. How big is a trillion? How long would it take me to snap my fingers a million times, at a rate of one snap per second? • A million seconds is about 11.6 days. How about for a billion? • A billion seconds is about 32 years! …and a trillion? • A trillion seconds is about 31,709 years!! *Discussion primarily drawn from The Week.

  3. MATH/IDST 120 Symmetry, Shape and Space “Mathematics is a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols; and a tool.“ North Central Regional Educational Laboratory http://www.ncrel.org/sdrs/areas/issues/content/cntareas/math/ma3ques1.htm

  4. Course Description Highlights • This is a writing-intensive class. • The course format emphasizes guided exploration and critical thinking. • Students are required to demonstrate understanding through journal writing, presentations, and both primary and secondary resource research.

  5. Topics Covered • Euclidian Geometry • Mathematics and Social Satire (Flatland) • Ruler-and-compass Constructions • The Golden Ratio • Mathematics and Art (Tessellations) • And one other topic, time permitting, that varies from semester to semester.

  6. Texts • Symmetry, Shape, and Space, Kinsey and Moore, New York: Wiley Publishing, 2002 with the related software, Geometer’s Sketch Pad (custom version these days.) • Flatland - A Romance in Many Dimensions, Abbott, Edwin A. New Jersey: Princeton University Press, 1991 (or any other un-annotated copy).

  7. Satisfies • 100-level mathematics course • 100-level humanities course • Required in • LS - Arts Concentration • Composites Technology

  8. Required Tools • A three-ringed binder, a reliable compass, protractor, scissors, straight-edge and #2 pencils including colored ones, index cards for semester research project, are required.

  9. Backing up… • But I’m getting ahead of myself.

  10. Becoming… • Late in the 90’s the Southern Maine Technical College began offering a Liberal Studies degrees. • 2003 - Change of mission and name - Southern Maine Community College • The call went out for more LS courses. • In the meantime….

  11. Flatland, Edwin Abbott • Characters are plane figures • Carefully crafted scale • Satire • Women’s plight • Caste systems • “Irregulars” • 4th physical dimension • Before Einstein • Progressions

  12. And…. The Paul/Elder Model www.criticalthinking.org

  13. Critical Thinking • Critical Thinking (reasoning) is a mental process of analyzing and evaluating information in order to improve thinking (reading, writing, listening.)

  14. A course was born!....

  15. Activities • Let’s spend the rest of our time doing some of the activities.

  16. Memorize or Understand? • A Demonstration • Students want to memorize a process for doing mathematical things. • I think this is a mistake. • With real understanding comes ease.

  17. Experiment • Have a sheet of paper and something to write with (no writing yet though!) • I’m going to display a series of letters for 15 seconds. Try to memorize as many of them in order as you can. • I will then give you a chance to write down what you remember. • Ready?

  18. Memorize! ZOT TFF SSE NTE TTF FSS

  19. Write! • Write down what you remember. • Would it help to UNDERSTAND the pattern? • What if I told you the first three letters represented the words zero, one and two?

  20. Understand! ZOT TFF SSE NTE TTF FSS

  21. Euclidean Geometry • Perimeter • Area • Volume

  22. Perimeter • Distance Around • Perimeter =

  23. Composite Figures • Find the perimeter of the figure below. cm Ans: 33.7 cm

  24. Area • Area = bh • We end up discovering that almost everything can be found using base times height in some form.

  25. How much space? • How much of the space within the rectangle is taken up by the shaded region? Problem devised by Paul Lockhart in A Mathematician’s Lament.

  26. A Challenge • What if the shaded region was obtuse? • Good journal fodder!

  27. Formula – Trapezoid (two opp sides parallel) • one trapezoid • two trapezoids, one rotated • When we join the two figures, what do we get?

  28. Devising a Formula • Bases are labeled. Each trapezoid is ½ of the parallelogram. • Base = b1 + b2 • Height = h • Area = h(b1 + b2) • Areatrap = ½◦h(b1 + b2)

  29. A Challenge • Could this work? • You tell me!  (Good fodder for journaling if you want a challenge)

  30. Pythagorean Theorem

  31. Do you have your scissors? • Let’s see if we can prove to ourselves that the Theorem is correct.

  32. Another Proof

  33. Simplify

  34. This next slide caused some… • …consternation among the conference attendees. I do understand what was being said and perhaps my picture is mis-labeled. I will look at that when there’s a little time. In the meantime, I’ve included all the slides for this exercises. I hope you will agree with me that the proof does actually work.

  35. Student Proof (Darcy Stillman) • Let’s manipulate another figure prove to ourselves that the Theorem is correct. • Handout! http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.html

  36. Student Proof • On a blank 3” by 5” index card, mark a diagonal (segment AC) as shown in the figure. • Mark a perpendicular line from segment AC that passes through point B. • Label the three resulting triangles as noted below on both sides with angle letters and triangle numbers. • Cut the pieces apart.

  37. Equal Sides Hint: State all names of sides in alphabetical order to avoid confusion! • What two pairs of sides are congruent?

  38. Fill in Blanks Hint: State all names of sides in alphabetical order to avoid confusion! • Triangle #3 is always in the numerator.

  39. Substitute Congruent Sides

  40. Cross Multiply

  41. Add these equations! Now factor out (AC) from the left side.

  42. Student Proof

  43. Conclusion?

  44. Student Proof (Darcy Stillman) • Let’s manipulate another figure prove to ourselves that the Theorem is correct. • Handout! http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.html

  45. Student Proof • On a blank 3” by 5” index card, mark a diagonal (segment AC) as shown in the figure. • Mark a perpendicular line from segment AC that passes through point B. • Label the three resulting triangles as noted below on both sides with angle letters and triangle numbers. • Cut the pieces apart.

  46. Equal Sides Hint: State all names of sides in alphabetical order to avoid confusion! • What two pairs of sides are congruent?

  47. Fill in Blanks Hint: State all names of sides in alphabetical order to avoid confusion! • Triangle #3 is always in the numerator.

  48. Substitute Congruent Sides

  49. Cross Multiply

  50. Add these equations! Now factor out (AC) from the left side.

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