Thought for the day: If you look through your teacher's materials (or even just look around you in the world!), you may find your own ideas for Creative Math Prompts.
Concepts: interior angles in polygons; finding angles by reasoning (instead of measuring); symmetry; properties of isosceles triangles; analyzing and extending patterns; describing patterns algebraically
I notice a pentagon in the middle of a five-pointed star.
I notice that the angles at the tips of the stars are marked.
I notice that the star is symmetrical (order-5 rotational symmetry).
I notice that the star is made of five congruent isosceles triangles attached to a pentagon.
I notice five larger isosceles triangles that include the pentagon.
I notice that I could extend the picture outward or inward forever.
I wonder if I can calculate the angles at the tips of the star.
I notice that once I know the angles in the pentagon, I can find the other angles without measuring.
I wonder what the angles at the star-tips would be if I built the star from other regular polygons.
I wonder if there is a pattern to the star-tip angles as the number of sides increases.
I wonder if I can create a formula for the star-tip angle in terms of the number of sides of the polygon.
I notice that polygons with seven or more sides have multiple layers of stars.
As they notice and wonder, students may create their own drawings of stars from polygons with more and more sides. Their drawings may be inspired by things that they have wondered about. For example, they may
Add to the drawing in the prompt in order to analyze it and make new discoveries.
Create stars from hexagons, heptagons, octagons, etc.
Extend their drawings inward and outward to create more polygons and more stars.
Try to create stars from irregular polygons.
For each drawing they create, they may calculate angles (or other measurements), look for patterns, develop equations, etc.
Reflecting and Extending
I notice that the star-tip angle for the 5-pointed star is 36°.
I notice that there are many ways to calculate this angle.
I notice that the star-tip angles increase more and more slowly as the number of sides increases.
I notice that the star-tip angles seem to get closer and closer to 180°, but never get there.
I notice that different strategies for finding star-tip angles lead to different formulas.
I wonder if I can use algebra to prove that all of the expressions in these formulas are equivalent.
I wonder if can find patterns and formulas for the angles in the multiple layers of stars (for polygons with 7 or more sides).
I wonder how the different side lengths in the picture compare (what their ratios are).
I wonder how the areas of the pentagon compare to the areas of the triangles.
I wonder how the sides lengths and areas would compare if I extended the picture outwards or inwards by making more pentagons and stars.
This image can inspire endless observations and questions. In these notes, I focus on angles. For a more structured activity built around these ideas, see Exploration 3: Starstruck! in my book Advanced Common Core Math Explorations: Measurement and Polygons.
Students who have learned about sums of interior angles in polygons will know (or be able to figure out) that each interior angle in the pentagon measures 108°. From there, they may use many different strategies (involving vertical angles, supplementary angles, interior angles in triangles, symmetry, etc.) to determine that the angles at the star-tips are 36°. Some may notice that the sum of the star-tip angles is 36 • 5 = 180°—the same as the sum of the interior angles of a triangle, and they may wonder why this happens.
From this point, you may explore stars built by extending the sides of other regular polygons. For example, stars built from hexagons have star-tip angles of 60°, and stars built from regular octagons have star-tip angles of 90°. As the number of sides increases, the star-tip angles increase ever more slowly, gradually approaching—but never quite reaching—180°. Depending on the strategies you use and the observations you make, you can find many different formulas that calculate the angles of the star-tips from the number of sides of the regular polygon. Most often, students find a very complex formula:
Sometimes, they discover simpler equivalent formulas such as
The possibilities for further questions and discoveries are nearly endless! For example, when you extend the sides of polygons that have more sides, you begin to get multiple layers of stars to explore within a single picture.