The context of this PullOut 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 handfit 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 righttriangle
trigonometry to derive a formula
for the moon’s apparent angular
diameter based on its MoontoEarth
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
MoontoEarth distance.
These PullOut 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 (*).
FIF2 Use function notation, evaluate functions for inputs
in their domains, and interpret statements that use function
notation in terms of a context.
FIF4 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.*
FIF6 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.*
FIF7 Graph functions expressed symbolically and show
key features of the graph, by hand in simple cases and
using technology for more complicated cases.*
FBF1 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.
FBF3 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.
BTF5 Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and
midline.*
BTF7 Use inverse functions to solve trigonometric equations
that arise in modeling contexts; evaluate solutions
using technology, and interpret them in terms of the
context.*
