In this lesson, Mary and Claire transform the sine function using sliders to model temperature data. They use their models to predict future patterns in temperature.
Mary and Claire reflect on what the differences in these models might suggest for future temperatures.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Notice that in the function y = A • sin(2𝜋/P(x – h)) + k, varying A stretches the graph vertically, varying P stretches the graph horizontally, varying h shifts the graph horizontally, and varying k shifts the graph vertically.
Realize that sinusoidal functions can model phenomena beyond heights on a circle.
Use mathematics to model data that describe an important phenomenon.
Use models to interpret the world around them.
Common Core Math Standards
CCSS.MATH.CONTENT.HSF.TF.B.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
As they model temperature data, Mary and Claire try to determine which sinusoidal function will fit. They use an applet with sliders for each parameter, to make conjectures for how changing A, P, h, and k affect the graph and relate it to their previous understanding of circular motion. They formalize their ideas by creating functions to model temperature data.
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 explore parameters that influence the shape of the graph of the sine function, including amplitude, period, and the midline.
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.
Mary and Claire create functions for different data sets and compare their models for different time periods. They use what they’ve learned to make predictions for what graphs might look like for new data.
According to the CCSSM, “mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace…They 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 analyze climate data and use an applet to model that data with a sinusoidal function. They experiment with changing the height of the midline to match the average temperature [e.g., Episode 2, 1:00]. They explore changing the amplitude to match the difference between the maximum and average temperatures [e.g., Episode 3, 0:47]. They also use their model to predict the shape of graphs that would represent new data [Episode 4].
