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 preconstructed 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 11gon. 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.
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.
M6D1