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Once in a Blue Moon

Marsha Davis & Floyd Vest

 Mathematics Topic:Modeling, Various Application Areas:Inverse Functions

| ©2018 Consortium 114 | 16 pages |

The context of this Pull-Out is the changing appearance of the moon from night to night as well as its effects on Earth’s tides. In Activity 1, students are given data on day number and the illuminated fraction of the moon’s apparent disk. Based on these data, students fit by hand an illumination model of the form f(x) = Acos(B(x + C)) + D. They use their models to determine the number of blue moons (the second full moon in a month) for the year 2018. In addition, students use their illumination models to approximate the instantaneous rates of change of the illuminated fraction corresponding to different phases of the moon. To conclude the activity, students fit another illumination model using their calculator’s sinusoidal regression capabilities and compare this model to their hand-fit model.

Activity 2 focuses on the moon’s apparent size as viewed by an observer on Earth. Students discover that the moon’s elliptical orbit causes its apparent size as well as its brightness to vary. Students use right-triangle trigonometry to derive a formula for the moon’s apparent angular diameter based on its Moon-to-Earth distance. They use this formula to determine the percentage increase in the moon’s angular diameter during a supermoon (when the moon is at perigee, closest to the earth) compared to a micromoon (when the moon is at apogee, farthest from the earth). In addition, students compare the brightness of a supermoon to a micromoon using the relationship that a full moon’s brightness is inversely related to the square of the Moon-to-Earth distance.

These Pull-Out activities address the following standards from the Common Core State Standards for High School Mathematics.

It should be noted that making mathematical models is a Standard for Mathematical Practice and specific modeling standards are indicated by a star symbol (*).

F-IF-2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF-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; periodicity.*
F-IF-6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate rate of change from a graph.*
F-IF-7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
F-BF-1 Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF-3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
B-TF-5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*
B-TF-7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate solutions using technology, and interpret them in terms of the context.*