Introducing angle measures (and using arcs) without taking seriously the quantification of angle measure likely sends students the message: use these numbers to perform calculations and find other numbers, but do not worry about what the numbers, arcs, and calculations mean. (p. 244)
Citation
Moore, K. C. (2013). Making sense by measuring arcs: A teaching experiment in angle measure. Educational Studies in Mathematics, 83(2), 225-245.
Inspiration
Gaining insights into students' meanings for angle measure including its quantification via conceiving the arc that subtends an angle's openness as a measurable attribute.
Methods and Participants
A teaching experiment with three undergraduate precalculus level students.
Takeaway(s)
Traditional approaches to angle measure are absent of mathematical operations (per Piaget's definition). They focus on learned facts of associations and benchmark values, the use of ready-made protractors, and calculations involving numbers. Hence, students' primary meanings for angle measure are learned facts and associations, in which angle measures are little more than names to give known angles and geometric objects.
In this work, I illustrate that an arc approach to angle measure—which foregrounds any angle measure unit as based on a fractional amount of the circle's circumference that subtends an angle with that unit openness—provides students a coherent meaning for angle measure. An angle measure of one degree (of 360) means that 1/360ths of any circle's circumference centered at the vertex of the angle subtends the angle's openness. An angle measure of 1 radian (of 2π) means that 1/(2π)ths of any circle's circumference centered at the vertex of the angle subtends the angle's openness.
Need to create a protractor? Construct a circle, construct a diameter, determine the circumference of the circle, choose your angle measure (e.g., degrees), determine the fractional amount of the protractor's circumference for one unit (e.g., 1/360th of the circumference), and use a piece of waxed string to measure along and partition the circumference using that unit-arc length.
The fractional amount approach also provides a generative understanding for understanding angle conversion in terms of quantitative operations. As an example, consider an angle measure of 36 degrees. To determine the angle measure in radians, we understand that 36 degrees means that 36/360ths (10%) of a circle's circumference subtends the angle's openness. Thus, the corresponding radian measure is (36/360)*2π, or 10% of 2π (i.e., 0.2π or ~0.628). Similarly, we could find the arc length that subtends this angle's openness by determining 10% of the corresponding circle's circumference. Generalizing, we can write:
(d/360) = (𝜃/(2π)) or 𝜃 = (d/360)*2π or (d/360) = (s/(2πr)) and so on;
d being a measure in degrees, 𝜃 being a measure in radians, and s and r being measures of arc length and the radius in the same unit for the same circle.
Each reflects using the underlying basis of angle measure as the fractional amount of a circle's circumference that subtends the angle's openness.
Similarly, the arc approach supports a meaning for radian measure that foregrounds a multiplicative relationship between the arc that subtends an angle and an arbitrary circle's radius. To say a radian measure is 𝜃 is to say that the measure of the arc length that subtends the angle's openness is 𝜃 times as large as the measure of circle's radius r. For example, if the radian measure is 2.5, we are saying that for any circle centered at the vertex of the angle, the arc length that subtends the angle's openness is 2.5 times as large as the containing circle's radius. Hence, we can write:
𝜃 = s/r or s = 𝜃*r.
We can also note that the angle measure in radians is in an inversely proportional relationship with r. If the arc length remains a fixed length, to obtain an angle measure that is c times as large requires that the radius of the containing circle becomes 1/c times as large.
Instructional Implications
As a teacher, ask yourself, "Can my students construct a protractor displaying integer values using a waxed piece of string, a blank protractor, and a ruler?" If the answer is no, then there is work to do in supporting their quantification of angle measure.
As a teacher, ask yourself, "Do my students view radians and degrees as merely scaled versions of each other?" If the answer is no, then there is work to do in supporting their quantification of angle measure.
As a teacher, ask yourself, "Can my students generate angle conversion formulas merely through leveraging their angle measure meanings?" If the answer is no, and they instead need to rely on memorized formulas, then there is work to do in supporting their quantification of angle measure.
