Course:Harris, Fall 08: Diary Week 2

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(short week)


  • Turned back Quadrilaterals Exploration.
    • Groups seemed unclear that U2 could be anything other than a Rectangle.
  • Groups finished up Tessellations (First Look) Exploration, 15-20 minutes.
    • Question: What's a regular hexagon? Or other polygon?
      • All sides congruent.
      • All angles congruent.
      • Are those redundant, i.e., does one imply the other?
        • Example looked at of hexagon with all sides congruent, but not convex.
  • Groups looked at Kali in Symmetric Figures Exploration.
    • Considered what a symmetry of a figure is:
      • a rigid motion of the plane that moves the figure onto itself
    • Kali "rosette group" buttons induced
      • upper row:
        • rotational symmetries
      • lower row:
        • rotational symmetries
        • flip symmetries


  • Turned back Tessellations I Exploration and Polygon Exercises.
    • Exercises (but not Explorations, as a rule) can be redone for additional credit.
  • We spent most of the period going over #3 of Polygon Exercises (parallelograms are the same as quadrilaterals with opposite angles congruent).
    • I noted that there are (or should be, from Quadrilaterals Exploration) two logical directions:
      • (both pairs of opposite angles congruent) ==> (both pairs of opposite sides parallel)
        • proved this last week, using:
          • Parallel lines cut by a third line implies opposite interior angles are congruent.
            • (We didn't say where this theorem comes from.)
          • Angle-sum of a triangle is 180 degrees.
            • (Proved using opp-int-ang theorem, plus construction of a parallel line through a point not on the line--Parallel Postulate.)
      • (both pairs of opposite sides parallel) ==> (both pairs of opposite angles congruent)
        • Use opp-int-ang theorem three times:
          • for parallel horizontal sides cut by one of the diagonal sides;
          • for parallel horizontal sides cut by the other diagonal side; and
          • for parallel diagonal sides cut by one of the horizontal sides.
        • Also use angles making up straight line are supplements (add to 180).
        • Put it all together, and if one angle is A, the neighboring angle is 180 - A, the next one is A, and the next one is 180 - A.
  • Groups started in on Symmetry of Stars/Polygons Exploration, but question was raised about Symmetry Groups:
    • The Symmetry Group for a figure is all the symmetries that apply to it.
      • A symmetry is a rigid motion of the plane. The ones applicable to compact figures are:
        • rotation of the plane about a point (rotation center);
        • reflection of the plane across a line (the reflection axis).
      • If there are just n-fold rotation symmetries, then that group is Cyclic and called Cn.
      • If there are n-fold rotation symmetries and also reflection symmetries, then that group is Dihedral ("two-faced") and called Dn.
  • We'll finish up Symmetry of Stars/Polygons Exploration on Monday (5 minutes?).
  • Rosette Exercises are due Monday.