Polygons and Triangles in the ClassroomLori Brugger, Mike Kaminski , and Brianna Keeney

Dieruff/Harrison-Morton

Posing Question: Using function notation, can we write the function T(n), which gives the number of triangles T that can be formed from a polygon with n sides?

Notice the convex polygons below with different number of sides. What is similar about your polygons and what is different?

Each convex polygon can be divided into triangles by drawing straight lines (diagonals) from one vertex (corner) to other vertices.

*Note: Elementary students could take their favorite polygon or one given by the teacher, and find the number of triangles in their particular shape.

Recall that the sum of the interior angles of a triangle = 180 degrees. Complete the table below.

Number of sides n Number of Triangles T Sum of the interior angles S

3 1 180◦4 2 360◦5 3 540◦6 4 720◦7 5 900◦

DOK #2. Describe in words the pattern between the number of triangles that can be formed, and the sum of the interior angles. Explain why this pattern will continue as n increases .

We noticed that the sum of the interior angles is the number of triangles times 180.

*Note: Middle school students will be able to extend the table and create a graph of the relationships found in the table.

We can show the relationship between T and n using function notation. In general, f(input) = output. Note that f is the function, while f(input) is an output value.

Notice that the number of triangles = number of sides minus 2. Therefore T(n) = (n – 2).

DOK #3. Using supporting ideas and examples, interpret the equation T(n) = 6.Looking at the equation, the number of triangles is 6. Since we know T(n) = (n – 2), we can say that n – 2 = 6.

Solving for the unknown variable gives us n = 8. The 8 is the number of sides of the polygon.

DOK #4. Apply the concepts from this experiment to create a function to show how to find the sum of the interior angles of the polygon, S, from the number of sides n.

When referring to the table, we notice that when we multiply 180 times the number of triangles, we get the sum of the interior angles of any polygon. Therefore, S(T) = T * 180. Making the connection between

the number of triangles and the number of sides, we have T(n) = (n – 2). By using substitution, we can find the sum of the interior angles of the polygon, S, from the number of sides n with the equation S(n) = (n – 2) * 180.

*Note: High school students will make connections to calculus using function notation.