# 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,
find:
(1) P(exactly 2 sides congruent) =
(2) P(at least 2 angles congruent) =
(3) P(2 different triangles with no sides
congruent) =
Agenda
•   Go over warm up.
•   Review geometry concepts
•   Exploration 8.6.
•   More practice problems.
•   Assign homework.
How did you group the
polygons?
• 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
–   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.

45˚
135˚   90˚
0˚
135˚
180˚
45˚
180˚
0˚
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
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
4
5
2
• 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,
obtuse.
• 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
parallel.
• If 2 lines are parallel, then a single plane contains
them.
• 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.
Solution…
• If a2 + b2 = c2 is to be used, we want a
right triangle whose hypotenuse is
square root of 5.
5
• So, a  2 + b2 = 5.

• If you do not use
a geoboard, there
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
geometry.
• 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.
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!!
What is a polygon?
Polygon
• A simple, closed, plane figure
composed of at least 3 line segments.
• Why are each of the figures below not
polygons?
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
•   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
proven.
• Exploration 8.10--do Part 1 #1 - 3 and
Part 2.
Diagonals, and
interior angle sum
•   Triangle
•   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.
• 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
sides.
• 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
sides.
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
case?
Warm-up part 2
• 1. Can you make a non-convex
• 2. Can you make a non-simple closed
curve?
• 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
Agenda
•   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
• 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
down.
• m = (y2 - y1)
(x2 - x1)
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?
2:06?

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 views: 4 posted: 7/4/2012 language: pages: 37