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 mathematicians make/ask. In the meantime, you get a better sense for what they are thinking.
Concepts: polygons; angle sums in polygons; identifying and extending patterns; expressing patterns as algebraic formulas; equivalent expressions; distributive property
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, and the bottom one has six triangles.
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 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 what the sum of the interior angles in the hexagons is—and if it is the same for all hexagons.
I wonder what the sum of the interior angles in other polygons is and if there are patterns to the sums.
I notice that each time I add a side, the angle sum increases by 180°.
I notice that when the segments come from one vertex, the number of triangles is always 2 less than the number of sides.
I notice that when the segments meet a one point inside the polygon, the number of triangles is always equal to the number of sides.
As they notice and wonder, students may create and analyze more polygon drawings like the ones in this prompt. Many of their creations may flow out of things that they have wondered about. For example, they may
Draw pictures like this for polygons with different numbers of sides.
Try to draw pictures like this for concave polygons.
Draw pictures in which the polygons are split into triangles in other ways. For example:
Students should analyze the drawings that they create: compare and contrast; look for patterns, make predictions, and describe their observations and discoveries algebraically.
Reflecting and Extending
I notice that I can use the patterns I discover to create formulas for the angle sums.
I wonder what causes the sum of the angles to increase by 180° (when the number of sides increases by 1).
I wonder if the patterns I discovered are still true for concave polygons (and how I can prove it).
I wonder what happens if I add the exterior angles of a polygon.
I wonder if I can create different formulas by drawing different kinds of pictures. For example: (1) connecting the point in the middle to a non-vertex, or (2) connecting 2 or more interior points to the vertices.
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—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:
180 • (6 – 2) = 180 • 4 = 720°.
By investigating other polygons, students may discover that the pattern of having 2 fewer triangles than sides than the number of sides holds whenever you draw the segments from a single vertex. Thus, 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 expression! When you join each vertex to a point in the interior of the polygon, the number of triangles is 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 hold for polygons with any number of sides.
180 • n – 360
The two formulas are equivalent. That is:
180 • (n – 2) = 180 • n – 360
Students may recognize 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 angles is greater than 180°. Students may also create even more expressions by thinking of different ways to split the polygon into triangles. For example, if they use the picture shown above in the “Creating” section, they may discover the expression
180 • (n + 1) – 540.
There is an infinite number of expressions like this, and they have some very cool patterns!