block 27

MA.8.G.2.3

Demonstrate that the sum of the angles in a triangle is 180-degrees and apply this fact to

find unknown measure of angles, and the sum of

angles in polygons

Block 27

Teacher Quality Grant - AE - FAU 2

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.

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Note:Use activity if review of polygons is necessaryIf not, skip to slide 8


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

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.

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Regular Star Pentagon


Regular Star Pentagon

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What is the sum of the angles in the “points of the star?


Non-regular star pentagon

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Hint:

What is the sum of the angles of the shaded polygon?

Look at quadrilateral ABCJ

How many quadrilaterals can we have like that?


What is the sum of the angles in the points of the star hexagon?

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Hint:

Each point of the star hexagon is part of a pentagon


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 n-

sided 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°

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€

=n(n − 3)180 − (n −2)(n −2)180

= (n2 − 3n − n2 + 4n − 4)180

= (n − 4)180


Extension

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{7, 2} star {7, 3} star

How does the sum of the internal angles of a {7, 3} star compare to a {7, 2} star?


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

• 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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block 27

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