In this lesson, Mary and Claire think about how an object’s height changes in relation to its angle of rotation as it moves along a circular path. They also explore what happens to the changes in height as the object moves. They explore how this relates to circular motion along different sizes of circles.
The students get a feel for how the height above a circular midline changes as they trace their fingers along a circular path. They think about when their height changes quickly and when it changes slowly.
The students explore the path a fly takes as it travels along the tip of a fan blade at 8 stops. They think about the fly’s height above the middle of the fan and how that changes with the angle of rotation.
The students explore the path a fly takes as it travels along the tip of a fan blade at 12 stops. They think about the fly’s height above the middle of the fan and how that changes with the angle of rotation.
Mary and Claire reflect on the patterns they notice regarding how the height above the midline changes, and how the angle of rotation changes on any size circle.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Develop an understanding of how the angle of rotation of an object traveling along a circular path and its height above the midline of the circle covary.
Explain why the rate of change in height is not uniform and varies depending on the angle.
Notice that, in circles of different sizes, the patterns in how the height of an object above the midline changes with respect the angle of rotation remains consistent.
Common Core Math Standards
CCSS.MATH.CONTENT.HSF.TF.A.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Mary and Claire’s work in this lesson lays a foundation for working with the unit circle and trigonometric functions. They analyze how the height of an object changes as it travels around a circle. They connect the height to the angle of counterclockwise rotation and identify how the height changes as the angle of rotation changes.
According to the CCSSM, mathematically proficient students “are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.” Mary and Claire create drawings to model how a fly sitting on the blade of a fan moves as the fan turns [e.g., Episode 2, 0:45]. They identify two covarying quantities for the context: angle of rotation and height and discuss how those two quantities are related [e.g., Episode 3, 0:47; Episode 4, 0:46].
According to the CCSSM, “Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.” Throughout this lesson, Claire and Mary examine several different circles and reason about how the height of an object traveling around each circle would change as the angle of rotation changes. As Mary and Claire investigate how height changes with different angles, they notice the regular patterns that occur and express those patterns in general terms. For example, the students explore specific instances of when the height is changing the quickest [e.g., Episode 2, 3:00 and Episode 3, 0:25] and use those observations to make a generalization about how height and angle of rotation covary [Episode 4, 1:32]. They also use their observations to generalize some conclusions regarding how the height changes in predictable
