Course:Harris, Fall 08: Diary Week 2
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(short week)
Wed:
- 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.
- Question: What's a regular hexagon? Or other polygon?
- 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
- upper row:
- Considered what a symmetry of a figure is:
Fri:
- 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.)
- Parallel lines cut by a third line implies opposite interior angles are congruent.
- proved this last week, using:
- (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.
- Use opp-int-ang theorem three times:
- (both pairs of opposite angles congruent) ==> (both pairs of opposite sides parallel)
- I noted that there are (or should be, from Quadrilaterals Exploration) two logical directions:
- 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.
- A symmetry is a rigid motion of the plane. The ones applicable to compact figures are:
- The Symmetry Group for a figure is all the symmetries that apply to it.
- We'll finish up Symmetry of Stars/Polygons Exploration on Monday (5 minutes?).
- Rosette Exercises are due Monday.
