Transcript Think of the game rock, paper, scissors

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
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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
polygons?
• For kids… talk about attributes
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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.
0˚
180˚
45˚
135˚
90˚
135˚
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
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
3
5
• T or F:  1   4
4
2
1
• T or F:  2   3
m
• T or F: m  7 + m  6 = m  1
• 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.
n
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
2
2
• So, a + b = 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
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:
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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?
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!
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Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon (Septagon)
Octagon
Nonagon (Ennagon)
Decagon
11-gon
Dodecagon
Triangle Attributes
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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
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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
Quadrilaterals
• Look at Exploration 8.13. Do 2a, 3a - f.
• Use these categories for 2a:
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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
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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
quadrilateral?
• 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
the right, name at
least 6 different
polygons using their
vertices.
A
F
B
G
C
D
E
Agenda
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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
down.
• 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?
2:06?