Angles in polygons inquiry
Mathematical inquiry processes: Explore cases; find rules; conjecture and generalise; reason and prove. Conceptual field of inquiry: Types of polygons; interior and exterior angles of polygons; improper fraction as a multiplier; algebraic proof.
The prompt originated in a discussion about angles in regular polygons. The year 8 class had worked out the interior and exterior angles of an equilateral triangle, square, pentagon, and hexagon. After recording their results on the board, the teacher invited students to notice and wonder.
A pair of students observed that in the case of the triangle and hexagon the number of sides had doubled, the interior angle had doubled but the exterior angle had halved.
The teacher wondered aloud, "Would this be true for a square and regular hexagon? What about a pentagon and a decagon?"
The prompt is the generalisation of the students' observation presented as a conditional statement. It turns out that the first part about interior angles is only true for the first case, although there is a relationship that holds in all cases (see 'Lines of Inquiry' below). However, the statement about exterior angles is true in all cases.
Students' responses to the prompt
The inquiry can be used to introduce new knowledge to the class. The teacher does not need to 'pre-teach' the concepts in the prompt. Indeed, students are often skilled at making inferences from the information available in the statement. Moreover, the teacher can often draw out and build upon what the students do know.
Their questions and observations will, at the most rudimentary level, focus on polygons, and interior and exterior angles. A misconception that often arises at this early stage, which the teacher should correct, is that the exterior angle is the whole 'outside' angle.
Students often want to speculate about the truth of the statement even without a clear understanding of the terms. They easily accept its plausibility because, as one year 7 student explained, "the inside angle gets bigger as the number of sides increases and the exterior angle gets smaller."
Lines of inquiry
1. Finding interior and exterior angles
As part of a structured inquiry students draw triangles from one vertex of a polygon to the others, work out the sum of the interior angles and, after doing this for different polygons, derive the formula for each interior angle (see the PowerPoint).
The teacher should draw out the structure of the diagrams to support students' reasoning later in the inquiry. There are always three fewer diagonals than sides because three vertices cannot be used (the original vertex and the two either side). Then, there will always be one fewer triangle than vertices.
2. Finding relationships
The conditional statement in the prompt is false for interior angles after the first case. However, there is a relationship that holds in all cases:
The relationship could also be expressed as a ratio - 1:2 for the triangle and hexagon, 2:3, 3:4, 4:5, ..., (n - 2):(n - 1).
Why is the statement false for interior angles, yet true for exterior angles?
The fact that the sum of an interior angle and an exterior angle is 180o means that both statements cannot be true simultaneously except in one case. If a is the interior angle and b is the exterior angle, then a + b = 180o and 2a + b/2 = 180o. Solving the equations simultaneously gives 3a = 180o and a = 60o, b = 120o. So, both contentions in the prompt are true only in the case of an equilateral triangle and a hexagon.
However, the prompt is true for exterior angles. That is because an exterior can be calculated using 360o/n (where n is the number of sides). If n is doubled, then 360o/n is halved. For example,
Regular pentagon: 360o/5 = 72o
Regular decagon: 360o/10 = 36o
The exterior angle (b) and the number of sides (n) are inversely proportional: b = 360o/n.
This is not the case with interior angles. The interior angle and number of sides are not proportional.
As the number of triangles that can be drawn inside a polygon is two fewer than the number of sides, there cannot be a constant ratio between the interior angle and number of sides for all polygons.
Regular pentagon: ((5-2) x 180o)/5 = 108o
Regular decagon: ((10-2) x 180o)/10 = 144o
The number of triangles has increased from three to eight, while the divisor has doubled. If the interior angle were to double, the number of triangles would have to be 12, which is an increase by a factor of four. However, it is impossible to have an interior angle of 216o when the limit is 180o.
Therefore, as the interior angle and number of sides are not proportional and as it is impossible to double the interior angle of a polygon after the equilateral triangle, the contention in the prompt is false.
Students proficient in the use of algebra might derive general terms for the interior and exterior angles of a regular n-sided polygon and a regular 2n-sided polygon. The teacher might then guide them to prove the results.
See the mathematical notes for the proof of the relationship between the interior angles and the proof that the size of the exterior angles double when the number of sides also double.