Lesson 1: Diagonals in a Polygon (90 minute class) Erin Gotel-Bundrige
A diagonal is a line segment that connects non-adjacent vertices in a polygon. Consider the number of diagonals in a triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon. What pattern do you notice? Use this pattern to predict the number of diagonals in a dodecagon (12-sided polygon).
Hint: Consider comparing the number of vertices and the number of diagonals that can be drawn at each vertex.
Essential Questions:
1. What is a polygon? 2. Are there different types of polygons? If so, name some of them and how many sides they have. 3. What are diagonals? 4. How does the number of sides relate to the number of diagonals in a polygon?
Objectives:
Students will make predications using a chart of how many diagonals they think a polygon will have.
Students will learn how to determine the number of diagonals in a polygon using technology.
Students will use a pre-constructed chart to show their final observations about the relationship between the number of sides and the number of diagonals.
Intermath Alignment: Having students to explore the relationship between the number of sides and diagonals provides them with the opportunity to use technology to explore mathematical concepts. Using technology will help students better understand the concept and help them remember the concept better; which will result in them transferring the knowledge to other math concepts.
Task Appropriateness: To make the original task more appropriate for sixth grade, I constructed the polygons and the tables for the students. The main focus of the lesson for sixth grade should be for students to understand the properties of polygons and diagonals.
Assessment:
Informal- teacher observation of student progress and support will be given if needed.
Formal- Students will use a pre-constructed chart to display their final observations using GSP and the formula that they were taught. They will also print the polygons and diagonals they constructed using GSP.
Reflection: I will use an 11-gon as a warm up for the students. This will be the first time that many of the students have worked with figures bigger than an octagon. I will have the students draw diagonals to connect each vertex. I realize that for many students this will be harder than it is for others. However, overall, I expect the students to give up easily because they have never had to do this before. I will use their frustration to discuss the lesson. To discuss the lesson I will use GSP with the following pre-constructed polygons:
Name Sides Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8
I will also have each students’ computer on to save time and the GSP figures will be saved in a file already. Based on students’ prior knowledge, they should already know what 3, 4, 5, 6, 7, & 8 sided polygons look like. Students will use GSP to construct colored diagonals in the polygons. Students will use a pre-constructed chart to keep up with the number of diagonals that each polygon has while using GSP. After students have done all the polygons, students will be shown the formula for finding the number of diagonals in a polygon.
Multiple Representation: As a challenge, students will also find the diagonals in a nonagon, decagon, dodecagon, and a twenty sided figure. These figures will be pre- constructed in another GSP file and the students will also construct colored diagonals in the polygons as well as using the formula they were taught. (This will be a follow up activity for the day after the original lesson).
Procedure:
1. Pass out warm up sheets with 11-gon for students to try to connect the diagonals in the polygon. Students will be given time to try to find all the diagonals in the polygon. (5 minutes) 2. I will teach students how to draw diagonals in the polygons and then we will work together (using overhead projector) to find all the diagonals in the 11- gon. I will also answer any questions that the students will have. (10mins.) 3. After teaching the lesson, Students will use the computer to find the GSP sketches as I talk them through the process (computer is already running and GSP sketches are saved in a file). I will teach students how to draw diagonals in the polygons using GSP and how to change the colors of the lines. Within this time, we will do two polygons together- the triangle and the quadrilateral. (20-25 minutes) 4. After instruction has been given, students will construct diagonals in the 5,6,7, and 8 sided polygon. (30 minutes). I will observe students to make sure they are doing it right. 5. I will teach students how to use a formula for finding diagonals in a polygon. Students will then go back and use the formula to make sure their construction of diagonals are correct. (20 minutes) They will put their final answer in a pre-constructed table. The table will consist of the polygon, number of sides, a place to work out their formula, and a column for the final number of diagonals in a polygon. 6. Students will print out their GSP constructions and turn in their sketches as well as their chart. (5-7 minutes) 7. Students will perform challenge activities on the following day for extra practice. They will also use GSP and the chart for the remedial practices.
GPS Alignment: Even though polygons are not included in the GPS for sixth grade, the skills that are needed to find the diagonals of the polygons can be used for this lesson.
M6P1 Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. Students will communicate mathematically.
M6P3 a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. M6D1 Students will pose questions, collect data, represent and analyze the data, and interpret results. a. formulate questions that can be answered by data. Students should collect data by using samples from a larger population or by conducting experiments.