In this lesson, Mary and Claire develop a graph of the sine function. They grapple with how to scale the x- and y-axes so they can show the relationship between an object’s angle of rotation and its height above the midline of a circle as it travels along the circumference of that circle.
The students mark in an object’s height above the midline of a circle at several stops along its circular path. They coordinate these heights with the object’s angle of rotation and mark these in as well.
The students attempt to develop a graph that relates the angle of rotation of a fly and its height above the midline as it travels on the blade of a fan in a circular path. They begin by measuring the fly’s heights above the midline at several stops along the its circular path and plotting these heights on a graph.
The students revise the graph they created in the previous episode. They plot several points that relate the height of an object above the midline of a circle and its angle of rotation as it moves around a circular path.
Mary and Claire begin to create a graph that relates an object’s angle of rotation, measured in radians, with its height above the midline, measured in radii, as it travels along a circular path.
The students reflect on how to use the sine button on a calculator to find the height of an object at various stops as it travels along a circular path.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Understand that the sine function relates an object’s height above the midline of a circle with its angle of rotation as it travels along the circumference of that circle.
Describe the angle of rotation in terms of 𝜋 radians.
Interpret the output of the sine function as the object’s height above the midline measured in radii (as opposed to cm, inches, feet, etc.).
Attend to the scale of x- and y-axes. In particular, this lesson can help students to see 1 unit on the y-axis as representing a height equivalent to the length of 1 radius of the circle and 1 unit on the x-axis as representing a rotation of 1 radian.
Common Core Math Standards
CCSS.MATH.CONTENT.HSF.IF.C.7.E. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Claire and Mary work throughout this lesson to develop an accurate graph that relates the height of an object as it rotates around a fan to the angle of rotation of that object. They use pipe cleaners to measure the distance on a circle, cut the pipe cleaners, and lay them on a graph. In doing so, they create a physical representation of the graph, which they then use to discuss more general features of the sine function.
CCSS.MATH.CONTENT.HSF.IF.B.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
In the previous lesson, Claire and Mary discussed ideas about how the height of an object rotating around a fan changes. They continue these conversations in this lesson and explore intervals on which the height is increasing the fastest and slowest. They identify these intervals, both on the circle they work with to generate their graph, and on the graph itself. They also identify points on the graph of the sine function where the function reaches its maximum and minimum values and connect that to the movement around the circle. They use the same reasoning to identify places on the graph where the height would be zero.
According to the CCSSM, “Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” Tasks in this lesson require students to graph a relationship between the height of a fly on a rotating fan blade and the angle of rotation of the fan blade. Mary and Claire work hard to understand the quantities and quantitative relationships in this situation. They start by describing the relationship using a diagram of a circle marked with radians [Episode 1]. They continually return to the context as they annotate the diagram with heights at various positions around the circle, and they quantify distances in terms of absolute height as well as change in height [e.g., Episode 1, 2:10]. In Episodes 2 and 3, they extend their work with the circle by translating the heights they found into points on a graph using pipe cleaners [e.g., Episode 2, 2:00 and Episode 3, 0:50]. For Claire and Mary, the pipe cleaners represent the actual distance as measured on the circle that represents the fan as well as abstracted points on the graph representing the relationship between height and angle of rotation.
According to the CCSSM, “Mathematically proficient students try to communicate precisely to others…. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently.” This lesson features several opportunities for Mary and Claire to attend to precision. For example, in Episode 2[e.g., 5:15], the students label the x-axis of the graph by stop number, which essentially ignores the quantity angle of rotation, but they fix this in Episode 3 by redrawing the graph with the x-axis in radians [0:39]. However, they realize that their intervals along the axis are not precise, with the physical space between 2 radians and 2.28 radians being the same as the physical space between 2.28 radians and 3.28 radians. They once again relabel their axis to make it more precise [1:25]. In Episode 3 they create a graph using pipe cleaners to represent heights and points [4:53] and revise the graph in Episode 4 to be more precise by adding additional pipe cleaner points [e.g., 1:05]. Finally, as they draw a graph of the sine function using digital tools, they realize they are not sure of the exact height at certain points along the circle. To remedy this, they rescale their circle so that the radius of the circle matches the height of 1 radius on their graph [6:23]. This allows the students to find heights using the circle and then transfer those directly to their graph to create a more precise representation of the sine function.
