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

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. 2/7/10Teacher Quality Grant - AE - FAU2 Note: Use activity if review of polygons is necessary If 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. 2/7/10Teacher Quality Grant - AE - FAU3

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 2/7/10Teacher Quality Grant - AE - FAU4

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 2/7/10Teacher Quality Grant - AE - FAU5

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. 2/7/10Teacher Quality Grant - AE - FAU6

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? 2/7/10Teacher Quality Grant - AE - FAU7

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 2/7/10Teacher Quality Grant - AE - FAU8

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. 2/7/10Teacher Quality Grant - AE - FAU9

Star Polygons A star polygon is formed by extending pairs of sides of a convex polygon that are connected by a third side. 2/7/10Teacher Quality Grant - AE - FAU10 Regular Star Pentagon

2/7/10Teacher Quality Grant - AE - FAU11 What is the sum of the angles in the “points of the star?

Non-regular star pentagon 2/7/10Teacher Quality Grant - AE - FAU12 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? 2/7/10Teacher Quality Grant - AE - FAU13 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? 2/7/10Teacher Quality Grant - AE - FAU14

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° 2/7/10Teacher Quality Grant - AE - FAU15

Extension 2/7/10Teacher Quality Grant - AE - FAU16 {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? 2/7/10Teacher Quality Grant - AE - FAU17

What about the exterior angles? 2/7/10Teacher Quality Grant - AE - FAU18

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 2/7/10Teacher Quality Grant - AE - FAU19