Tessellations Activity - Angles in Polygons

Overview: By way of tessellations, students investigate the interior angles of polygons and generalize to a pattern rule.

Curriculum Expectation(s):

• Measurement and Geometry: Determine, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems

Learning Goal(s):

• To explore/identify the relationship between the sum of the interior angles and the number of sides in a polygon

Timing and sequencing: This activity will take one period. Introduction to geometric properties of polygons.

Classroom construct: Students work in groups of 3

Materials:

• Scissors
• Paper
• Coloured Pattern Tiles
• Protractors

Description:

Show the video Anatomy of an Escher Flying Horse: https://www.youtube.com/watch?v=NYGIhZ_HWfg

Ask Students: "Which shapes will tessellate & why?"

Give them a container of coloured pattern blocks to explore. Since there is no pentagon in the set, they will need to trace and cut out several copies if they also wish to explore a 5 sided shape.

Coloured Pattern Blocks Pentagon to trace and cut out

Students may start placing tiles together to determine if the polygons tessellate or not.

For example, the following shapes tessellate:

Possible teacher prompts:

• What allows the polygon to come together perfectly with no gaps?
• What do you notice here (point as shown in the diagram)

• Are there relationships between the angles in a polygon and the number of sides the polygon has?
• Can you generalize your findings to a polygon with n-sides?
• Can you connect your generalizations (pattern rule or equation) for an n-sided polygon to a diagram or pattern block model?
• Do your generalizations work if the polygon is not regular?
• As students start to come up with ideas, the teacher can suggest students keep track of findings on a table such as the following:

Polygon	Number of Sides	Does it Tessellate?	Measure of Each Interior Angle if Regular	Sum of the Interior Angles
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
n-sided polygon

Possible student thinking about interior angles and tessellation:

• Students may choose to measure some interior angles with protractors or they may reason using the idea that angles in a triangle add to 180o

• If the shapes tessellate their interior angles will be a factor of 360o, for example three hexagons come together when tessellating so each interior angle will be 360o/3 and the sum will be 360o/3*6

Possible student thinking about the sum of interior angles:

• 6 of the green triangles fit on the yellow hexagon which leads to the idea that the interior angles add to (6*180o)-360o this means that the sum of the interior angles will be 720o and each interior angle will be 120o

• The hexagon can be split into 4 triangles which leads to the idea that interior angles add to (4*180o)-360o

Possible student thinking about generalizing to a pattern rule for the sum of the interior angles:

• Each time the number of side increases by 1, another triangle is added. The sum of the interior angles is a linear pattern.
• The sum of the interior angles can be represented using two different equations which are equivalent.

s=180n-360   OR    s=180(n-2)

• If we know the sum of the interior angles we can divide by the total number of sides to determine the measure of each interior angle if the polygon is regular.