Discovering the Sum of the Interior Angles of a Polygon with Geogebra Class: High School Geometry Grades: 9th-11th Unit: Polygon Properties

Introduction:

This lesson has students working with a free online program, Geogebra, to discover the formula for the interior angle sum of a polygon. This lesson will help the teacher become a facilitator of learning rather than the typical lecture-based class which allows students to take control and show initiative in their own learning. It is important for teachers to stress the fact that through the day’s activities, if prompted with the number of sides of a polygon, students should be able to determine/calculate the Interior Angle Sum. That is to say, at the end of the activity students should discover the formula that allows us to calculate the interior angle sum of a polygon. Though this lesson is not directly related to the real world problem, this lesson makes a connection between Geometry and Algebra. Though the students have some options in terms of how to find the function for finding the interior angle sum of a polygon, most strategies are Algebra related.

Setting up the Technology: To prepare for the lesson, the teacher must ensure that all software has been uploaded to student computers. Geogebra is a free online program that can either be downloaded to a computer or simply run off the internet. All pictures/diagrams were created using Geogebra, a link is attached. http://www.geogebra.org/cms/

Standards: G.1.B Use inductive reasoning to make conjectures, to test the plausibility of a geometric statement, and to help find a counterexample G.3.G Know, prove, and apply theorems about properties of quadrilaterals and other polygon

Lesson:

As an introduction to the lesson, the teacher should discuss what an interior angle of a polygon is and what it looks like. To help get students engaged in the lesson, the teacher should focus the students by emphasizing the purpose of the activity, to find the relationship between the number of sides and the sum of the polygon’s interior angles. Of course, this should be described in student friendly terms, emphasizing that our input should be “n”, the number of sides, and our function’s output should be the interior angle measure of the polygon.

Since students should already know the interior angle sum for a triangle and quadrilateral, the teacher can review this quickly, and use the quadrilateral as an example of how to use the program. The teacher should introduce the computer software, Geogebra. If students have not used this software before in a previous lesson, the teacher should explain the basics.

The teacher should lead the class through the following steps:

Polygon: Creating a quadrilateral using the polygon feature

Angle: There are two different ways to find the measurement of each angle. First is clicking the angle tab, then selecting on the polygon. After doing this, all the interior angle measurements will be given. Second, each angle can be measured individually, by selecting three vertices or two sides. It is the teacher’s decision which method students should use.

Angle Sum: The students should add up all the angle measurements and write down the measurement in their table.

Once students understand how to use this program, the teacher should send the students off to work on their table, which should guide them to finding the formula for interior angle sum. Students will likely find a pattern but will struggle with making the pattern into a formula using only n as a variable. It is the teacher’s decision to determine how many hints or guiding questions to use (if any) or when to share them with the class. I find it is best for the students to struggle with finding the relation to n before giving away any hints that will limit student thinking.

Possible Guiding Questions:

What is the biggest number that is divisible by all of your “interior angle sums”? Can you factor that number out? What number is left? How does this number relate to the number of sides of the polygon? (How does this number relate to n?)

Can we represent the data algebraically? Graphically? Find the function!!

Number of sides of the polygon

Interior Angle Sum

3

4

5

6

7

10

15

n

Assessment: Students can be assessed in a variety of ways. Students will show the teacher that they know how to use Geogebra by including at least four diagrams of polygons (each with different number of sides) with the interior angle measurements included. Students must complete their table and make a conjecture about the interior angle sum for an n-gon.

Reflection: I think this lesson increases the rigor in a mathematics classroom because it changes how mathematics is traditionally taught. Since various theorems, postulates or properties are traditionally just given to students in a lesson, this lesson is substantially more difficult because students are not just given the formula of how to find the interior angle sum of polygons. This is relevant because we are forcing students to think for themselves rather than being force fed various properties of polygons. Students will be more interested in discovering how something works rather than listening to a teacher trying to explain it. This lesson is also relevant because it incorporates technology, so students will be working with real moving mathematics. This gives students of different learning styles to shine. Since students are forced to come up with the function on their own, it is up to the student to determine strategies to get to the end result. It is important for teachers to be prepared to lead the class with some guiding questions, but students also need to be given the opportunity to think about strategies to solve the problem. Though this lesson is not directly related to solving a real world problem, this lesson does make the connection between Geometry and Algebra. Depending on the students’ strategy to solving the problem, finding the relationship between the number of sides of a polygon and the interior angle sum can be very algebraically focused.

Discovering the Sum of the Interior Angles of a Polygon with GeogebraClass: High School Geometry

Grades: 9th-11th

Unit: Polygon Properties

Introduction:This lesson has students working with a free online program, Geogebra, to discover the formula for the interior angle sum of a polygon. This lesson will help the teacher become a facilitator of learning rather than the typical lecture-based class which allows students to take control and show initiative in their own learning. It is important for teachers to stress the fact that through the day’s activities, if prompted with the number of sides of a polygon, students should be able to determine/calculate the Interior Angle Sum. That is to say, at the end of the activity students should discover the formula that allows us to calculate the interior angle sum of a polygon. Though this lesson is not directly related to the real world problem, this lesson makes a connection between Geometry and Algebra. Though the students have some options in terms of how to find the function for finding the interior angle sum of a polygon, most strategies are Algebra related.

Setting up the Technology:To prepare for the lesson, the teacher must ensure that all software has been uploaded to student computers. Geogebra is a free online program that can either be downloaded to a computer or simply run off the internet. All pictures/diagrams were created using Geogebra, a link is attached.

http://www.geogebra.org/cms/

Standards:G.1.BUse inductive reasoning to make conjectures, to test the plausibility of a geometric statement, and to help find a counterexampleG.3.GKnow, prove, and apply theorems about properties of quadrilaterals and other polygonLesson:As an introduction to the lesson, the teacher should discuss what an interior angle of a polygon is and what it looks like. To help get students engaged in the lesson, the teacher should focus the students by emphasizing the purpose of the activity, to find the relationship between the number of sides and the sum of the polygon’s interior angles. Of course, this should be described in student friendly terms, emphasizing that our input should be “n”, the number of sides, and our function’s output should be the interior angle measure of the polygon.

Since students should already know the interior angle sum for a triangle and quadrilateral, the teacher can review this quickly, and use the quadrilateral as an example of how to use the program. The teacher should introduce the computer software, Geogebra. If students have not used this software before in a previous lesson, the teacher should explain the basics.

The teacher should lead the class through the following steps:

Once students understand how to use this program, the teacher should send the students off to work on their table, which should guide them to finding the formula for interior angle sum. Students will likely find a pattern but will struggle with making the pattern into a formula using only n as a variable. It is the teacher’s decision to determine how many hints or guiding questions to use (if any) or when to share them with the class. I find it is best for the students to struggle with finding the relation to n before giving away any hints that will limit student thinking.

Possible Guiding Questions:

Assessment:Students can be assessed in a variety of ways. Students will show the teacher that they know how to use Geogebra by including at least four diagrams of polygons (each with different number of sides) with the interior angle measurements included. Students must complete their table and make a conjecture about the interior angle sum for an n-gon.

Reflection:I think this lesson increases the rigor in a mathematics classroom because it changes how mathematics is traditionally taught. Since various theorems, postulates or properties are traditionally just given to students in a lesson, this lesson is substantially more difficult because students are not just given the formula of how to find the interior angle sum of polygons.

This is relevant because we are forcing students to think for themselves rather than being force fed various properties of polygons. Students will be more interested in discovering how something works rather than listening to a teacher trying to explain it. This lesson is also relevant because it incorporates technology, so students will be working with real moving mathematics. This gives students of different learning styles to shine.

Since students are forced to come up with the function on their own, it is up to the student to determine strategies to get to the end result. It is important for teachers to be prepared to lead the class with some guiding questions, but students also need to be given the opportunity to think about strategies to solve the problem. Though this lesson is not directly related to solving a real world problem, this lesson does make the connection between Geometry and Algebra. Depending on the students’ strategy to solving the problem, finding the relationship between the number of sides of a polygon and the interior angle sum can be very algebraically focused.