LESSON PLAN #1: Discover a Relationship

Transcription

1 LESSON PLAN #1: Discover a Relationship Name Alessandro Sarra Date 4/14/03 Content Area Math A Unit Topic Coordinate Geometry Today s Lesson Sum of the Interior Angles of a Polygon Grade Level 9 NYS Mathematics, Science, and Technology Learning Standards Addressed Standard 1: Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions. Standard 3: Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry. Objectives: The student will generalize a formula for the sum of the interior angles of a polygon with n sides, with adequate guidance (cognitive). Materials: 1 task sheet per student (to record experiment) 1 piece of loose-leaf paper 1 overhead of task sheet Anticipatory Set: To reintroduce the fact that a triangle has interior angles that sum to measure 180 degrees I will: Ask the students to take out a sheet of loose-leaf. Say, Could you all please take out a sheet of loose-leaf? Instruct them to fold over a corner of the paper (preferably one without holes) so that the sides of the triangle they are creating are perpendicular to the edges of the paper. Now, what I want you to do is fold over a corner of the paper so that the sides of the triangle you are creating are perpendicular to the edges of the paper. If you ll watch me, I ll show you. Demonstrate if necessary. Tear the triangle off the paper neatly by creasing it. Fold the triangle both ways, and tear it away from the paper like this. Show the class how to crease and tear the paper. You may also use scissors if you know it wont be a problem. Review the kinds of angles we have by asking the students to draw them on the board (acute, obtuse, reflexive, straight) along with their degree measures. Now, you guys remember the different kinds of angles, right? Give them to me and we ll write them down on the board. Now ask the students to imagine that the base side of the triangle (teared edge) is their straight angle (this can be lead into by asking what other angles in the triangle are), with

2 the vertex being the midpoint of that side. What I want you to do now is to imagine that the base side of the triangle, which is the teared edge here, is a straight angle. Instruct the students to fold the three principal vertices of the triangle so that they meet at the midpoint. Now, fold the three vertices of the triangle so that they meet at the midpoint, or the vertex of the straight angle. Ask the students if they have recognized the relationship, or the point that was about to be made. Can anyone see the relationship here? Now draw a diagram on the board and label the three principal vertices angle 1, angle 2 and angle 3, sum them, and set them equal to 180 degrees (the students may realize this themselves). Here we have angle 1, angle 2, and angle 3, and they all make up our straight angle. We also know that the measure of a straight angle is 180 degrees, right? So angle 1, plus angle 2, plus angle 3 is equal to 180 degrees. Conclusion: The sum of the interior angles of a triangle measure 180 degrees. If we apply this formula to our triangle, we see that the interior angles of our triangle sum to 180 degrees. Isn t that neat! Experimenting: Now that the students know the interior angles of a triangle equal 180 degrees, have them pair up, and give them the handout (attached). I would like you all to please pair up. Let s take a look at this first line here. On the overhead, go through the first line with them, since they have basically already completed it, and record the information. Now explain each column of the worksheet thoroughly, making sure that the students realize the connection between the sum of the interior angles of each polygon and the number of triangles each polygon can be divided into (multiply 180 degrees by the number of triangles). Now I want you to work together and complete this table. Start by indicating the number of sides in each polygon. Then divide each polygon into the smallest number of triangles possible, and record this value for each polygon. Then, and listen to this because it gets tricky, take 180 degrees and multiply it by the number of triangles in each polygon. Record this value as the sum of the interior angles of each polygon. The sum of the exterior angles of each polygon is always going to be 360 degrees. We ll take a look at a proof of this later on in the year. When they are done generating data, ask each group to contribute a few lines, and copy them over to the overhead so all students can check their answers. Ok, now lets get all of this information down and revise it. Reflecting and Explaining: Tell the pairs to discuss the data they have generated, and formulate a conjecture to relate the sum of the interior angles to the sides of a general n-sided polygon (give an ample but not extraneous amount time, and refer to the relationship between the number of triangles and the sum of the interior angles). Tell the students to see if they can notice patterns while examining their columnar data. Now what I want you to do is look at your data, and see if you can formulate a hypothesis about the sum of the interior angles of a polygon. Try and relate this hypothesis to the number of sides of each polygon. Some questions to ask: What is the difference between the number of sides and the number of triangles for each entry? Is there a pattern in the sum of the interior angle entries? Relate it directly to the number of sides (n).

3 Hypothesizing and Articulating: Instruct each group to have one person come up and write their conjecture on the board. Now, I would like one person from each group to come up to the board and record their answer. Then filter out all incorrect assumptions by politely asking the students what they did to get them, and then pointing out any flaws. Would the other student of each group please tell us how they came up with their hypothesis? Show the students the correct form of the equation (n-2)(180) after completing the n-gon row of the table with them. Here is the correct equation for the sum of the interior angles of a polygon with n sides. Then go further by showing the class that when you divide this formula by n the result is the measure of one interior angle. Does anyone know what you get when you divide this result by n? If time allows, you may ask the students this question. Verifying and Refining: With the time left in class, allow the students to work on the rest of the sheet to make sure they understand these principles, while checking their work and answering any questions they may have. They will finish the worksheet for homework. Now if you would all work on the questions that would be great. I you have any questions feel free to ask me, and this will be due tomorrow for homework. Closure: Ask the students to make sure they do their HW because they will all be called up to the board to contribute answers. You will all be called up to the board to contribute answers tomorrow, so make sure you do it! They should also be prepared to show how they got the answers they did. To wrap up, ask the students, What did you learn today? and, How did you go about formulating a conjecture? Make sure that the students say something about finding a pattern in the sum of the interior angles, and relating that pattern to the number of sides of a polygon. Go with the student s responses and give affirmative support, closing with something along the lines of, And you guys did it all by yourself, now wasn t that fun? Have a nice day and I ll see you tomorrow! Be proud of yourselves! Homework/Assessment: Finish worksheet (attached) Extensions: Go into detail on how to find the measure of one angle of an n-sided polygon (see above).

4 Name Polygons What is a polygon? Remember the sum of the interior angles of a triangle Let s investigate the sum of the angles of the following polygons: Polygon Drawing # of sides # of triangles sum of interior sum of exterior Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon DoDecagon n-gon Hypothesis: 1) What can you say about the sum of the interior angles of a polygon (related to the number of sides n)? 2) What happens if we divide that result by the number of sides (n)?

5 Practice: 1)What is the sum of the angles in a decagon? 2)What is the sum of the angles in a heptagon? o 3)What is the name of the polygon whose interior angles sum to 5760? Find the sum of the measures of the interior angles of a regular polygon with the given number of sides: 4) 9 5) 11 6) 10 7) 18 For a regular polygon with the given number of sides, find the sum of the measures of the exterior angles: 8) 15 9) 40 10) 22 11) 100 Find the measure of one interior angle of a regular polygon with the given number of sides: 12) pentagon 13) dodecagon 14) octagon 15) heptagon 16) 20 sided 17) 24-sided 18) What is the name of the figure whose interior angles all equal 150 0? Find the number of sides of a polygon whose interior angles have the given number as the sum of their measures: 19) ) ) )