Download exterior angles - Math GR. 6-8

MA.8.G.2.3
Demonstrate that the sum of
the angles in a triangle is 180degrees and apply this fact to
find unknown measure of
angles, and the sum of angles
in polygons
Block 27
Polygon Capture Game
• In this activity, participants classify
polygons according to more than one
property at a time. In the context of a
game, participants move from a
simple description of shapes to an
analysis of how properties are related.
Note:
Use activity if review of polygons is necessary
If not, skip to slide 8
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Instructional Plan
• The purpose of this game is to motivate
students to examine relationships among
geometric properties of polygons.
• From the perspective of the Van Hiele model
of geometry, the students move from
recognition or description to analysis.
• Middle school students rarely use more than
one property to describe a polygon.
• By having to choose figures according to a
pair of properties, students go beyond simple
recognition to an analysis of the properties
and how they interrelate.
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Pre-requisites for the Game
• Knowledge of the properties of
polygons that include angles, sides,
diagonals
• Use the special quadrilateral
worksheet if a review is necessary
• Familiarity with vocabulary: parallel,
perpendicular, polygon, and
classification of angles
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Materials for the Game
• Game Rules
• Game Cards
• Game Polygons
Each group of two needs one set of
Cards and one copy of the polygons
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Extensions
• The Polygon Capture game cards can
also be used to generate figures. As in
the game, students turn over two
cards. Instead of capturing polygons,
they use a geoboard or dot paper to
make a figure that has the two
properties. Rather than a game, this is
simply an activity to help students
learn to coordinate the features of a
polygon.
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Discussions
• Will students find difficult to coordinate
two properties at a time?
• How could this game by adapted for
different students?
• Is this game best suited for advanced
students?
• Could this game be used as a review
of the lesson?
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Interior Angle Sum of Polygons
• Distribute worksheet Triangulation of
polygons
• Participants, in small groups, work on
the worksheet
• Whole group discussion on patterns
seen in the worksheet
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Exterior Angle Sum of Polygons
Is there an exterior angle sum?
Open a new GeoGebra file
Draw a large polygon
Extend its sides to form a set of exterior angles
Measure all the interior angles
Use the Linear Pair Conjecture to calculate the measure
of each interior angle
• Calculate the sum of the measures of the exterior angles
• Share your results with group members
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Open the GeoGebra file exterior angles, to show the
exterior angle sum conjecture. Notice what happens to
the exterior angles when the vertices get closer to the
point in the center.
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Star Polygons
A star polygon is
formed by
extending pairs of
sides of a convex
polygon that are
connected by a
third side.
Regular Star Pentagon
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Regular Star Pentagon
What is the sum of the angles in the “points of the star?
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Non-regular star pentagon
Hint:
What is the sum of
the angles of the
shaded polygon?
Look at quadrilateral
ABCJ
How many
quadrilaterals can
we have like that?
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What is the sum of the angles in
the points of the star hexagon?
Hint:
Each point
of the star
hexagon is
part of a
pentagon
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Could we generalize?
• Could we have used a star triangle?
• Could we have used a star
quadrilateral?
• What is the pattern?
• Is it possible to find a general formula?
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General formula
• If a star polygon is from from an nsided polygon (n ≥ 5)
• (the sum of the measures of the points
of the star polygon) + (n-2)(sum of the
measures of the angles of the n-gon) =
n(n-3)180°
 n(n  3)180  (n  2)(n  2)180
 (n 2  3n  n 2  4n  4)180
 (n  4)180
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Extension
How does the sum of the
internal angles of a {7, 3} star
compare to a {7, 2} star?
{7, 2} star
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{7, 3} star
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Extension:
• How does the sum of the internal
angles of a {7, 3} star compare to
a {7, 2} star?
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What about the exterior angles?
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Regular polygons and
Tessellations
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Do all regular polygons tessellate?
Which ones do and which ones don’t?
Why?
Can an explanation be given based
on the interior angles?
Open the GeoGebra file polygon
tessellation to very your answers
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