Think of the game rock_ paper_ scissors

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					I have 7 triangles, 1 each of:
    acute scalene, acute isosceles, equilateral,
            right scalene, right isosceles,
         obtuse scalene, obtuse isosceles.
If I ask a student to draw any random triangle,
        (1) P(exactly 2 sides congruent) =
       (2) P(at least 2 angles congruent) =
     (3) P(2 different triangles with no sides
                     congruent) =
•   Go over warm up.
•   Exploration 8.1--share answers
•   Review geometry concepts
•   Discuss attributes: Quadrilateral Hierarchy
•   Exploration 8.6.
•   More practice problems.
•   Assign homework.
   How did you group the
• For kids… talk about attributes
  –   Shape: # sides, special quadrilaterals
  –   Convex or non-convex
  –   (1 or 2) Pair of parallel sides
  –   (1 or 2) Pair of congruent sides
  –   (1 or 2) Pair of perpendicular sides
  –   Nothing special about it.
  –   Cannot do any proof or justification if kids can’t
      classify and describe similarities and differences.
How do I use a protractor?
         I forgot!
• Line up the center and line.

                           135˚   90˚
            Can you…
• Sketch a pair of angles whose
  intersection is:
  a. exactly two points?
  b. exactly three points?
  c. exactly four points?
• If it is not possible to sketch one or
  more of these figures, explain why.
        Use Geoboards
• On your geoboard, copy the given segment.
• Then, create a parallel line and a
  perpendicular line if possible. Describe how
  you know your answer is correct.
          Exploration 8.6
• Do part 1 using the pattern blocks--make sure your
  justifications make sense.
• You may not use a protractor for part 1.
• Once your group agrees on the angle measures for
  each polygon, trace each onto your paper, and
  measure the angles with a protractor.
• List 5 or more reasons for your protractor measures
  to be slightly “off”.
             Given m // n.
• T or F:  7 and  4
  are vertical.
                                               7 6
• T or F:  1   4                        3
• T or F:  2   3                      1              n
• T or F: m  7 + m  6 = m  1                   m
• T or F: m  7 = m  6 + m  5
• If m  5 = 35˚, find all the angles you can.
• If m  5 = 35˚, label each angle as acute, right,
• Describe at least one reflex angle.
  More practice problems
• Sketch four lines such that three are
  concurrent with each other and two are
  parallel to each other.
            True or False
• If 2 distinct lines do not intersect, then they are
• If 2 lines are parallel, then a single plane contains
• If 2 lines intersect, then a single plane contains them.
• If a line is perpendicular to a plane, then it is
  perpendicular to all lines in that plane.
• If 3 lines are concurrent, then they are also coplanar.
   Pythagorean Theorem
• Remember the Pythagorean Theorem?
• a2 + b2 = c2 where c is the hypotenuse
  in a right triangle.
• Use your geoboard to make a right
  triangle whose hypotenuse is the
  square root of 5.
• If a2 + b2 = c2 is to be used, we want a
  right triangle whose hypotenuse is
  square root of 5.
• So, a  2 + b2 = 5.

• If you do not use
  a geoboard, there
  are lots of answers.
         Van Hiele levels
• Formal study of geometry in high school requires
  that students are familiar and comfortable with many
  different aspects of elementary and middle school
• Visualization, analysis, informal deduction are all
  necessary prior to high school geometry.
• This means students need to categorize, classify,
  compare and contrast, and make predictions about
  figures based upon their attributes.
• Early childhood:
   –   Size: big--little
   –   Thickness: thin--thick
   –   Colors: red-yellow-blue-etc.
   –   Shape: triangle, rectangle, square, circle, etc.
   –   Texture: rough--smooth
   Why do we need this??? READING!!
 Talk about polygons
What is a polygon?
• A simple, closed, plane figure
  composed of at least 3 line segments.
• Why are each of the figures below not
  Convex vs. Non-convex
• Both are hexagons. One is convex.
  One is non-convex.
• Look at diagonals: segments
  connecting non-consecutive vertices.
• Boundary, interior, exterior
        Names of polygons!
•   Triangle
•   Quadrilateral
•   Pentagon
•   Hexagon
•   Heptagon (Septagon)
•   Octagon
•   Nonagon (Ennagon)
•   Decagon
•   11-gon
•   Dodecagon
         Triangle Attributes
•   Sides: equilateral, isosceles, scalene
•   Angles: acute, obtuse, right.
•   Can you draw an acute, scalene triangle?
•   Can you draw an obtuse, isosceles triangle?
•   Can you draw an obtuse equilateral triangle?
One Attribute of Triangles
• The Triangle Angle Sum is 180˚.
• This is a theorem because it can be
• Exploration 8.10--do Part 1 #1 - 3 and
  Part 2.
          Diagonals, and
        interior angle sum
•   Triangle
•   Quadrilateral
•   Pentagon
•   Hexagon
•   Heptagon (Septagon)
•   Octagon
•   Nonagon (Ennagon)
•   Decagon
•   11-gon
•   Dodecagon
 Congruence vs. Similarity
Two figures are congruent if they are exactly
  the same size and shape.
Think: If I can lay one on top of the other, and
  it fits perfectly, then they are congruent.
Question: Are these two
  figures congruent?
Similar: Same shape, but
  maybe different size.
Quadrilateral Hierarchy
• Look at Exploration 8.13. Do 2a, 3a - f.
• Use these categories for 2a:
   –   At least 1 right angle
   –   4 right angles
   –   1 pair parallel sides
   –   2 pair parallel sides
   –   1 pair congruent sides
   –   2 pair congruent sides
   –   Non-convex
          Exploration 8.13
• Let’s do f together:
• In the innermost region, all shapes have 4 equal
• In the middle region, all shapes have 2 pairs of equal
  sides. Note that if a figure has 4 equal sides, then it
  also has 2 pairs of equal sides. But the converse is
  not true.
• In the outermost region, figures have a pair of equal
  sides. In the universe are the figures with no equal
               Warm Up
•   Use your geoboard to make:
•   1. A hexagon with exactly 2 right angles
•   2. A hexagon with exactly 4 right angles.
•   3. A hexagon with exactly 5 right angles.
•   Can you make different hexagons for each
       Warm-up part 2
• 1. Can you make a non-convex
• 2. Can you make a non-simple closed
• 3. Can you make a non-convex
  pentagon with 3 collinear vertices?
       Warm-up Part 3
• Given the diagram at   A
  the right, name at
  least 6 different
                         B       F
  polygons using their
  vertices.                  G

                         C           D   E
•   Go over warm up.
•   Complete discussion of 2-Dimensional Geometry
•   Polyhedra attributes
•   Exploration 8.15 and 8.17
•   Examining the Regular Polyhedra
•   3 Dimensions require 3 views
•   Assign Homework
  Quadrilateral Hierarchy
• Do the worksheet.
Some formulas--know how
       they work.
• Number of degrees in a polygon:
  Take 1 point and draw all the
  diagonals. Triangles are formed. Each
  triangle has 180˚. So, (n - 2)•180˚ is
  the number of degrees in a polygon.
• If the polygon is regular, then each
  angle is (n - 2) • 180/n.
Some formulas--know how
       they work.
• Distance formula: This is related to the
  Pythagorean Theorem.
• If a2 + b2 = c2, then c = a2 + b2 .
• Now, if a is the distance from left to right, and
  b is the distance from top to bottom, then the
  distance formula makes sense.
Some formulas--know how
       they work.
• The distance formula is
                             (x1, y1)• A

                                 (x2, y2) • B
   (x2 - x1)2 + (y2 - y1)2
Some formulas--know how
       they work.
• Midpoint formula: If the midpoint is half
  way between two points, then we are
  finding the average of the left and right,
  and the average of the up and down.
• Midpoint: (x2 + x1) , (y2 + y1)
                2           2
Some formulas--know how
       they work.
• Slope of a line: change in left and right
  compared to the change in up and
• m = (y2 - y1)
      (x2 - x1)
   Discuss answers to
Explorations 8.11 and 8.13
• 8.11
• 1a - c

• 3a: pair 1:
  same area,
  not congruent;
  pair 2: different area, not congruent;
• Pair 3: congruent--entire figure is rotated 180˚.
  More practice problems
• Think of an analog clock.
• A. How many times a day will the minute hand be
  directly on top of the hour hand?
• B. What times could it be when the two hands
  make a 90˚ angle?
• C. What angle do the hands make at 7:00? 3:30?

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