Drop of Doom!4
Students will use the function for the height of a free falling object to explore functions and their
graphs.
Suggested Grade Range: 7-12
Approximate Time: 1 hour
State of California Standards:
Mathematics Standards Grade 7: Algebra and Functions
1.2: Use the correct order of operations to evaluate algebraic expressions.
1.5: Represent quantitative relationships graphically and interpret the meaning of
a specific part of a graph in the situation represented by the graph.
Mathematics Standards Grades 8-12: Algebra I
23.0: Students apply quadratic equations to physical problems, such as the
motion of an object under the force of gravity.
Science Content Standards Grade 8: Physical Sciences
1. The velocity of an object is the rate of change of its position. As a basis for
understanding this concept:
b. Students know that average speed is the total distance traveled divided by the total
time elapsed and that the speed of an object along the path traveled can vary.
d. Students know the velocity of an object must be described by specifying both the
direction and the speed of the object.
e. Students know changes in velocity may be due to changes in speed, direction, or both.
f. Students know how to interpret graphs of position versus time and graphs of speed
versus time for motion in a single direction.
Science Content Standards Grades 9-12: Physics
1. Newton’s laws predict the motion of most objects. As a basis for understanding this
concept:
e. Students know the relationship between the universal law of gravitation and the effect
of gravity on an object at the surface of Earth.
Relevant National Standards:
Mathematics Common Core State Standards:
4
An early version of this lesson was adapted and field-tested by Katiria Hernandez and Gina Hryze,
participants in the California State University, Long Beach Foundational Level Mathematics/General
Science Credential Program.
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8.EE Understand the connections between proportional relationships, lines, and linear
equations.
8.F Define, compare, and evaluate functions.
Next Generation Science Standards:
5-PS2-1. Support an argument that the gravitational force exerted by Earth on objects is
directed down. [Clarification Statement: “Down” is a local description of the direction
that points toward the center of the spherical Earth.] [Assessment Boundary: Assessment
does not include mathematical representation of gravitational force.]
HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion
describes the mathematical relationship among the net force on a macroscopic object, its
mass, and its acceleration. [Clarification Statement: Examples of data could include
tables or graphs of position or velocity as a function of time for objects subject to a net
unbalanced force, such as a falling object, an object rolling down a ramp, or a moving
object being pulled by a constant force.] [Assessment Boundary: Assessment is limited to
one-dimensional motion and to macroscopic objects moving at non-relativistic speeds.]
Lesson Content Objectives:
 Evaluate functions using the correct order of operations.
 Describe and represent functions using tables and graphs.
 Make interpretations about an object’s motion and position based on its function and
graph.
Materials Needed:
 Lined or graph paper
 One copy per student of the warm-up activity sheet and “Drop of Doom!” activity
sheet (included)
Adapted From:
Rubin, K. (n.d.). Illuminations: Roller Coasting Through Functions. Retrieved from
http://illuminations.nctm.org/LessonDetail.aspx?id=L839.
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Summary of Lesson Sequence
 To introduce the lesson, connect students’ work on graphing functions from the warm up
(included) to their experiences with roller coaster rides by discussing how the motion of a
roller coaster may be described using formulas or graphs.
 Lead students through compiling notes while modeling how to evaluate and graph a
function that represents a falling object’s motion.
 Guide students through their own practice of evaluating a function, completing a table of
values, and graphing the function using the “Drop of Doom!” activity sheet (included).
 Check for students’ understanding by asking the key questions provided.
 To close the lesson, allow students to discuss in pairs how to find the time it takes for a
roller coaster to reach the bottom of a drop.
 For independent practice, students may create and graph their own function to represent
the motion of a roller coaster that they design.
Assumed Prior Knowledge
Prior to this lesson students should know how to use the correct order of operations to
solve equations and be able to graph ordered pairs to represent a function.
Classroom Set Up
Students will be asked to participate in discussions and work in pairs for portions of this
lesson.
Lesson Description
Introduction
Provide students with the “Warm Up: Preparing for the Drop of Doom” (included) to
focus their attention to working with functions. As students are finishing the warm up,
ask students to discuss what mathematics is associated with roller coaster rides.
Ask: How do you think roller coasters and math are related? [speed, height, formulas,
etc.] Explain that engineers use functions to determine a roller coaster’s height above
ground after a certain amount of time.
Input and Model
 Lead students through notes by modeling using the function describing the height of
an object in free fall dropped above Earth. Explain the following:
The height of an object that is dropped from above Earth can be determined using the
formula h = f(t) = ½ (-32)t2 + s, where s is the starting height of the object in feet.
Notice that the height of the object is a function of time and does not depend on the
object’s mass. The coefficient -32 comes from the fact that an object’s acceleration
due to gravity as it falls to Earth is -32 feet per second squared. The formula may be
simplified to f(t) = -16t2 + s, where: t = time in seconds and s = initial height in feet.
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The formula may be used to create a chart of values for the time and height of the object
which may be graphed as ordered pairs (t, f(t)).
For example, an object dropped from 144 feet above Earth, will have an initial height of
144 ft. (see table below).
Copy the following table in your notes:
Time in
Height in
Ordered Pair
2
f(t) = -16(t) + s
seconds
feet
(t, f(t))
2
f(0) = -16(0) + s
144
(0, 144)
0
2
f(1) = -16(1) + s
128
(1, 128)
1
2
3


2
f(2) = -16(2) + s
2
f(3) = -16(3) + s
80
0
(2, 80)
(3, 0)
Ask students to discuss what happens 3 seconds into the roller coaster ride.
Graph the resulting ordered pairs on a height vs. time coordinate plane.
Guide Students Through Their Practice
Distribute the “Drop of Doom!” activity sheet (included) and allow students to work
independently or in pairs. Guide students by moving through the classroom assisting
those who need it and checking that students are using the correct order of operations.
Check for Understanding
Check for students’ understanding by orally guiding them through the key questions
below. These questions may be used to determine whether students are ready to do
similar work on their own.
1. Why are all of the graphical representations in the first quadrant only? [Time is on
the x-axis and we cannot have negative time. Height is on the y-axis, and roller
coasters do not go below ground or “0.”]
2. The equation we are using does not take into account certain things that may have an
effect on the roller coaster as it drops. What are some things that could affect the
drop? [e.g., friction, weight of people in the car, weather]
3. If you saw the ordered pair (3,112) in your data table, what would it mean? [After 3
seconds, the roller coaster is 112 feet above ground.]
4. Let t represent the time in seconds and f(t) represent the height above ground of the
roller coaster. What is a question that could represent the ordered pair (t, 60)? What
is a question that could represent the ordered pair (4.5, f(t))?" [The ordered pair (t,
60) represents the question, "How long will it take the roller coaster to be 60 feet
above ground?" The ordered pair (4.5, f(t)) represents the question: "How high above
ground will the coaster be after 4.5 seconds?"]
5. Which of the following ordered pairs would be unreasonable if it appeared in the
context of this problem?(1,144), (-2, 1000), (8, -124), (0,1)? Why? [The second, third,
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and fourth are unreasonable ordered pairs. The second is unreasonable because time
should not be negative. The third is unreasonable because roller coasters do not travel
124 feet below ground. The last is unreasonable because the initial drop is more than
1 foot tall.]
Independent Practice
Students may practice what they learned independently by designing a roller coaster
drop, writing the function for it’s height, finding values for the height at different times,
and graphing the results. To design their roller coaster, students are choosing the height,
s, of the drop in the formula: f(t) = -16t2 + s .
Closure
Allow students to discuss a way to find out the time it takes for a roller coaster to reach
the bottom of a drop in pairs and write their responses as a “ticket out the door.”
Suggestions for Differentiation
 Students who have difficulty evaluating functions or graphing ordered pairs could be
pulled into a small group to receive assistance while the class is working on their guided
practice.
 To extend the activity, advanced students may be introduced to the formula for the
motion of a roller coaster that has an initial velocity before it begins to fall: f(t) = – 16t2 +
vt + s.
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Warm Up
Preparing for the Drop of Doom
Directions: Complete each function table by finding the values for y. Show your work in the
space provided. Graph the ordered pairs (x, f(x)) for each equation.
1. f(x) = x – 5
x
Show work here:
Graph:
Show work here:
Graph:
f(x)
-2
1
2
4
2. f(x) = 2x
x
f(x)
-2 ½
0
2
3
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Drop of Doom!
Name _________________
A new roller coaster ride at Six Flags Magic Mountain recently opened in Valencia,
California. At the highest point of the ride, Drop of Doom drops thrill seekers from
a record-breaking height!
Use the formula f(t) = -16t2 + 400 to determine the height of the coaster at
several times during the descent and use the data to determine how long it takes
the coaster to reach the bottom.
1. Complete the table to determine how long it takes Drop of Doom to reach the
bottom of its highest drop:
Time t
f(t) = -16t2 + 400
Height (ft)
Ordered Pair (t, f(t))
(sec)
0
1
2
3
4
2. Graph the ordered pairs below:
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3. What is the height of the coaster before it
begins to drop? How do you know?
4. After how many seconds does Drop of Doom
reach the bottom? How do you know?
5. If you did not know the height of a specific drop on a roller coaster, how could
you find out the height without measuring it directly? Write a plan for how you
would collect the data you would need to determine the height of a drop on a
roller coaster you were watching or riding.
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