Thought for the day: It may take students a take a long time to suggest "noticings and wonderings" that lead to the ideas you have in mind. Be patient. Over time, they get a better feel for the kinds of observations and questions that are most productive. In the meantime, you get a better sense for what they are thinking.
Concepts: polygons; interior angle sums in polygons; identifying and extending patterns; expressing patterns as algebraic formulas; equivalent expressions; distributive property (possible linear relations)
Examples of noticing and wondering
I notice two congruent hexagons that have been split into triangles in different ways.
I notice that all of the segments inside the top hexagon start from one vertex.
I notice that all of the segments inside the bottom hexagon meet at one point inside the hexagon.
I notice that the top hexagon has four triangles (two less than the number of sides), and the bottom one has six triangles (the same as the number of sides).
I notice that all of the angles in the top picture's triangles are part of the hexagon's angles, but in the second picture, some are not.
I wonder if there is a pattern to the number of triangles I can draw inside different polygons when all of the segments (1) come from one vertex, and (2) meet at one point inside the polygon.
I wonder if there are connections between the angles in the triangles and the angles in the hexagon.
I wonder what the interior angles in the hexagons add up to—and if the sum is the same for all hexagons.
I wonder what the interior angles in other polygons add up to and if there are patterns to the sums.
I wonder if the patterns I discovered are still true for concave polygons.
You can learn more about using this image and related concepts in Exploration 1: Polygon Perambulations from my book, Advanced Common Core Math Explorations: Measurement and Polygons.
This prompt illustrates two ways to calculate the sum of the interior angles of a hexagon. In the top picture, the hexagon is decomposed into four triangles, which is two less than the number of sides. The sum of all of the angles in the triangles is equal the sum of the interior angles in the hexagon. Therefore, the sum of the hexagon's angles is 180 • (6 – 2) = 180 • 4 = 720°.
By investigating other types of polygons, students may discover that the pattern in which the number of triangles is 2 less than the number of sides persists as long as you draw all possible segments from a single vertex. Consequently, the sum of the interior angles of an n-sided polygon (an
n-gon) is 180 • (n – 2).
The bottom picture leads to a different result. First, when you subdivide the polygon by joining each vertex to a point in the interior of the polygon, the number of triangle is now equal to the number of sides. However, 360° worth of the triangles' angles do not belong to the hexagon! (See the angles surrounding the point in the interior.)
As a result, you multiply 180 by the number of sides (without subtracting 2), but you compensate by subtracting 360° for the angles that are not part of the hexagon. Students should verify that these patterns continue to hold for polygons with any number of sides. Therefore, an alternate formula for the sum of the interior angles is 180 • n – 360.
The two formulas are equivalent. That is:
180 • (n – 2) = 180 • n – 360
Students may notice that this as an example of the distributive property. Both formulas always produce the same result for a given number of sides.
It is interesting to explore these ideas with concave polygons, which have "indentations" where one or more interior angle is greater than 180°.