Mary and Claire explore another student’s reasoning who thought that you could use arc length to determine the angle of rotation. They compare two arcs with different lengths.
Mary and Claire find the distance traveled by the tips of the blades of two wind turbines. They relate those arc lengths to the angles of rotation made by the blades.
Mary and Claire explore the path traveled by two people who live in different locations, Michigan and the Caribbean, as the earth makes a full rotation. They figure out how much of a rotation the earth needs to make for each person to travel 1,000 miles.
The students use their conclusion from the previous episode to figure out how many degrees and gips the person in the Caribbean rotates as they travel 1,000 miles.
Mary and Claire revisit their thinking from the first episode. They consider what else you would need to know to determine the angle of rotation, if you know that length of the arc the object traces out while traveling along a circular path.
Claire and Mary create several circles of various sizes. On each of the circles, they trace out an arc whose length is the radius of the circle. They create angles at the centers of the circles that cut out these arcs. They compare those angles. A new unit of angle measure is defined: the radian.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Evaluate the claim that an angle whose vertex is at the center of a circle can be measured by the length of the arc it subtends.
Analyze scenarios where arc lengths are portions of circles of various sizes.
Articulate the idea that the same angle will subtend a larger arc on a larger circle.
Solve problems involving arc length and angle measure in practical contexts.
Develop strategies to find the angle of rotation based on arc length measurements.
Explain that the length of the arc an angle subtends will always be the same, regardless of the size of the circle, if the arc length is measured in radii.
Compare and contrast radians with degrees and other units of angle measure.
Understand the definition of a radian.
Common Core Math Standards
CCSS.MATH.CONTENT.HSF.TF. A.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Mary and Claire examine the relationship between radius and circumference of three circles of varying sizes. On each circle, they mark off arcs with length equal to the circle’s radius. Through their investigation, they grasp the concept that regardless of a circle’s size, approximately six radians fit into its circumference. Additionally, the students revised their earlier ideas that angle measure cannot be determined by arc length, and instead come to understand radians as a way of measuring angles of rotation using arc length.
CCSS.MATH.CONTENT.HSG.C.B.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Mary and Claire examine the relationships between a variety of angle measures and the lengths of the arcs the angles subtend. They initially explore wind turbines and the distance a blade travels given an angle of rotation. They extend their thinking by analyzing the distances traveled by two people at different latitudes on Earth. These contexts help Claire and Mary to understand that arc length and angle of rotation are closely related, and given the radius of the circle, one can be used to find the other.
Common Core Math Practices
CCSS.MATH.PRACTICE.MP3. Construct viable arguments and critique the reasoning of others.
According to the CCSSM, “Mathematically proficient students…can recognize and use counterexamples… justify their conclusions, communicate them to others, and respond to the arguments of others. They…distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.” Mary and Claire develop strategies to find the angle of rotation based on arc length measurements and explain why arc length can be used to determine angle measure, engaging in mathematical argumentation and critiquing the reasoning of a hypothetical student. In Episode 1, Claire argues that arc length cannot be used to find the angle of rotation, while Mary seems to think that is possible. They each offer explanations grounded in their prior understandings and diagrams to support their claims. In Episode 2, they analyze the argument of a fictional student who claims that an arc that is longer than another represents an angle of rotation that is larger as well. They explain why this isn’t true and create drawings to support their counterargument. In Episode 6, Claire and Mary revisit their initial claims regarding angles of rotation and arc length, and revise their thinking based on their explorations in this lesson.
