In this guided investigation of geometry, students will explore polygons to see the interrelationship between figures and deduce a formula for the patterns that they find. This lesson is part of a unit that is targeted to teach the exploration of geometric principles at the middle school level. The unit will integrate the skills of critical thinking and mathematical communication.
In the lesson "Determining the Sum of the Interior Angles in a Polygon", the students will construct a four-sided polygon. They will explore this polygon on Geometer's Sketchpad by measuring the four angles and calculating the sum of these angles. Then, they will construct segments from one point to the other points in the figure. This will divide the polygon into two triangles. Using their knowledge from the lesson "Determining the Sum of the Interior Angles of a Polygon", the students will calculate the sum of the internal degrees in the polygons by multiplying the number of triangles by 180 degrees. They should compare this calculation to the previous one. After exploring a variety of polygons, they should make a conjecture that the number of triangles formed from a polygon is two less than the number of sides in a polygon. From this pattern, they will be able to generalize the formula for the sum of an n-sided polygon: The sum of the internal angles of a polygon is the number of sides minus two times 180 degrees or (n - 2) 180 where n is the number of sides.
For those students that work at a quicker pace, there will be a challenge question that asks them to apply the formula they deduced. In their formula, given the number of sides in a polygon, the sum of the internal angles can be found. The Explore More question, asks them to find the number of sides in a polygon by working backwards from the sum of the angles. It will reinforce the knowledge that has been developed but will not disadvantage a student that does not reach it.
After completing this exercise, the students will have begun to move from the first van Hiele level to the second. This is accomplished by having the students develop an understanding of the interrelationship between figures. Also, they will use deductive reasoning at an informal level by determining the formula for a general polygon from the patterns that form. The development of these critical thinking skills is an important part of the curriculum.

Title: Determining the Sum of the Interior Degrees of Polygons.

Time Allotment: Two fifty-minute class period

Prerequisite Skills: These mini-lessons are a part of a unit for analyzing polygons. Before starting, students should have a basic understanding of how to use Geometer's Sketchpad.

Objective: The learner will demonstrate an ability to determine the sum of the internal degrees of any convex polygon by exploring a variety of polygons and deducing a formula for an n-sided polygon from any patterns that occur.

Activities: In these exercises, students will explore a variety of polygons on Geometer's Sketchpad. Then they will make a conjecture to determine the number of degrees in an n-sided polygon.

Materials: Each group of students will need a copy of the guided investigations and a computer with The Geometer's Sketchpad software.

Teaching Strategies:
*Involvement of students in active learning
*Promotion of group work (collaborative and cooperative learning)
*Enhancement of written and oral communication
*Discovery approach
*Integration of mathematics with other disciplines
*Teaching to encompass a variety of learning styles
*Relating new knowledge to old
*Promoting higher level thinking
*Use of technology for computation and exploration
*Designing lesson to move from one Van Heile level to the next.

Assessment: Comprehension will be determined by their written explanation and algebraic representation of their findings.
Determining the Sum of the Interior Angles of a Triangle.

2. Set Preferences to Display Points and Straight Objects and make sure that the angle units are Degrees.

3. Construct triangle ABC.
*Choose the point tool.
*Draw three points.
*Select the three points with the arrow tool.
*Choose Construct Segment

4. Measure all three internal angles of your triangle.
*Select all three points, in any order.
*Choose Angle from the Measure menu. The angle measured will display.
*Select the three points in a different order until you have measured the three possible angles.

5. Calculate the sum of the degrees in the three angles.
*Select the angle measurements and choose Calculate from the Measure menu.
*Click the measurement for the first angle, the plus, the second angle, the plus, the third angle, and then the O.K. The sum of the angle measurement will display.

Investigate

1. What is the sum of the angle measurements? _______________

2. Try moving the parts of the triangle around. Explain what you've found. Is it always true? ______________________________________________________________

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Determining the Sum of the Interior Angles of a Polygon.

1. Choose a New Sketch from the File menu.

2. Set Preferences to Display Points and Straight Objects and make sure that angle units are in Degrees.

3. Construct a four-sided polygon ABCD.
*Choose the point tool.
*Draw four points.
*Select the four points with the arrow tool.
*Choose Construct Segment.

4. Measure the internal angles of the polygon.
*Select three consecutive points.
*Choose Angle from the Measure menu. The angle measurement will display.
*Measure the other three angles (there are four angles in this polygon.)

5. Calculate the sum of the internal angles.
*Select the angle measurements and choose Calculate from the Measure menu.
*Click the measurement for the first angle, the plus, the second angle, the plus, the third angle, the plus, the fourth angle, and then the O.K. The sum of the angle measurements will display.

Investigate

1. What is the sum of the internal angles in a four-sided polygon? ________

2. Transform the polygon (move, flip, turn, or resize proportionally) to observe the effects the transformations have on the measures of the angles.
Does the sum always stay the same? ________________________________
3. Construct a segment from point A to all other points on the polygon. Use the knowledge that the sum of the internal angles of a triangle equals 180 degrees to prove that the sum of the four-sided polygon equals 360 degrees.

4. Construct several more polygons with different numbers of sides. Connect point A with every other point.

Complete the following table:

5. Using complete sentences, explain any patterns you've found. ___________

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6. Predict the sum of the interior angles for a 10-sided polygon, a 25-sided polygon, and a n-sided polygon.

7. How can you find the number of degrees in each angle of a regular polygon with 3, 4, ... , n sides? ____________________________________

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Explore More

Since the sum of the interior angles of a polygon can be determined by knowing the number of sides in the polygon, determine the number of sides in a polygon that has 3240 degrees? ______________

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