E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Science as Inquiry: As a result of their activities in grades 5–8, all students should develop
• Understanding about scientific inquiry.
• Abilities necessary to do scientific inquiry: identify questions, design and investigation, collect and interpret
data, use evidence, think critically, analyze and predict, communicate, and use mathematics.
Source: National Science Education Standards
National Council of Teachers of
Mathematics (NCTM) Expectations
Geometry
• Specify locations and describe spatial
relationships using coordinate geometry
and other representational systems.
• Apply transformations and use symmetry
to analyze mathematical situations.
• Use visualizations, spatial reasoning, and
geometric modeling to solve problems.
Data Analysis
• Develop fluency in adding, subtracting,
multiplying, and dividing whole numbers.
• Represent and analyze patterns and
functions using words, tables and graphs.
• Model problem situations with objects and
use representations such as graph, tables,
and equations to draw conclusions.
• Investigate how a change in one variable
relates to a change in a second variable.
• Collect data using observations, surveys
and experiments.
• Select and apply appropriate standard
units and tolls to measure length, area,
volume, weight time, temperature, and size
of angles.
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Science Process Skills:
• Observing
• Comparing
• Inferring
• Data Collection
• Communicating
• Interpreting
Objective:
Math Process Skills:
• Observing
• Comparing
• Calculating
• Recording
• Inferring
• Analyzing
• Data Collection
• Problem Solving
• Interpreting
Time: 60 Minutes
• The learner will recognize geometric properties and
relationships and apply them to other disciplines and to
problems that arise in the classroom or in everyday life.
• The learner will collect data using observations and
experiments.
• The learner will represent data using tables and graphs.
Instructor Materials:
• 5 - 8 pre-constructed rockets capable of
reaching approximately 40 feet in altitude
• 100 foot (30 meter) tape measure
• "Table of Tangents"
• AltiTrakTM
• Computer with PowerPoint® installed
• Projection system
• PowerPoint presentation
• Ball (or similar small object you can safely toss
into the air)
Student materials:
Per student
• Calculator
• "Table of Tangents" (one per student or pair
of students depending on how you choose to
complete the lesson)
• Scrap paper for solving math equations
Per tracking station
• Clipboard
• AltiTrak
• Orange safety cone (optional)
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Instructor Background Information:
Key Vocabulary
Adjacent Side—The side
next to the reference angle
in a right triangle. The
adjacent side cannot be the
hypotenuse.
Angular Distance—The
space between two lines
or planes that intersect;
the inclination of one line
to another; measured in
degrees.
Apogee—The farthest or
highest point (reached by a
rocket)
Baseline—In triangulation,
the side of one of a series of
coordinated triangles the
length of which is measured
with prescribed accuracy
and precision and from
which lengths of the other
triangle sides are obtained
by computation.
Indirect Measurement—A
technique that uses
proportions to find a
measurement when
direct measurement is not
possible.
Opposite Side—The side
opposite the reference
angle. The opposite side
cannot be the hypotenuse.
Tangent—A tangent to a
curve at a point is a straight
line that touches the curve
at that point.
Trigonometry—The branch
of mathematics concerned
with the properties of
trigonometric functions
and their application to the
determination of the angles
and sides of triangles. Used
in surveying, navigation, etc.
Hypotenuse—The side
opposite the right angle
in a right triangle. It is the
longest side of a right
triangle.
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Pythagorean Theorem
Pythagoras was a mathematician who lived in ancient Greece.
He developed a theory, known as Pythagorean Theorem, which
compares the sides and angles of a right triangle (a triangle
where one angle measures 90 degrees, and the other two angles,
when added, equal 90 degrees [angles C + D = 90°]).
The Pythagorean Theorem is summed in the equation a2 + b2 = c2
In other words, in a right triangle, if a and b are the
measurements of two of the sides, then the sum of their squares
will equal the square of side c (the hypotenuse or longest side).
(The hypotenuse is always directly across from the right angle. It
never touches or intersects the right angle.)
If we make a square out of each side of a right triangle, the square
on the longest side would have the same area as the other two
squares added together.
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Based on this relationship, Pythagoras found that there are
consistent ratios between the sides and angles of a right triangle.
We can use the application of these ratios to mathematically find
unknown measurements within the triangle.
One of the ratios is called the tangent (tan) ratio. If you know the
measurement of an acute angle in a right triangle, and know the
measurement of either the opposite or adjacent side of the acute
angle, you can determine the length of the unknown side.
If A is an acute angle (an angle less than 90° but more than
0°) of a right triangle†
tan A =
∠A
measure of the adjacent side to ∠ A
measure of the opposite side to
or
tan A =
A
B
Knowing this ratio, we can solve problems, including the
elevation or altitude of an object, as well as depth and distance
from one point to another. Most mathematicians use the builtin tangent function on a scientific calculator to help solve these
problems. However, when not using a scientific calculator, one
can refer to the "Table of Tangents." This table contains known
tangent values for angles.
There is an angle formed by the hypotenuse and either one of the other sides. The side opposite that angle
is referred to as the “opposite side,” and the side next to the angle is referred to as the “adjacent side.”
†
Acute Angle
Opposite Side
Adjacent Side
Hypotenuse
Hypotenuse
Acute Angle
Adjacent Side
Opposite Side
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
For instance, if we know one acute angle is 30° and the adjacent
side is 150 meters, we can determine the length of the opposite
side, using the equation below.
Opposite Side
?
Hypotenuse
Acute Angle
30°
Adjacent Side
150 meters
tan of 30° =
opposite side
adjacent side (150 m)
Begin by clearing the denominator by multiplying both sides
of the equation with the lowest common multiple of the
denominator. This yields
tan of 30° degrees x 150 m = opposite side
Use a calculator or the "Table of Tangents" (see end of lesson for
table) to determine the tangent of 30°.
tan 30° = .58
Substitute the tangent for 30° in the equation and solve for the
opposite side.
.58 x 150 m = 87 m
Opposite Side
87 meters
Hypotenuse
Acute Angle
30°
Adjacent Side
150 meters
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Instructor Preparation:
üü Familiarize yourself with proper usage of the AltiTrak.
üü For step 13 of the lesson, find a stationary object and a point
from where the students will stand while using the AltiTrak.
(You may want to mark the "tracker" spot with tape or an
orange safety cone.)
üü At the launch site, determine safe locations for the tracker
stations. Set up three to four tracker stations (see PowerPoint
slide "Tracker Stations"). Measure and record the distance
between the tracker stations and launch stand. (It may be
helpful to mark the tracker station locations with an orange
safety cone.)
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Lesson:
Note:
Familiarize yourself with the
AltiTrak, making certain you have
full comprehension of its use,
before beginning this lesson.
Note:
It is important to determine the
students’ knowledge of “degrees”
in the context of geometry. If
they lack an understanding,
you will need to conduct a brief
discussion, with useful visual
aids, to provide the necessary
comprehension.
Pythagorean Theorem
1. With the students, briefly review the independent variables
that affect a rockets’ acceleration (force and mass). (They
should be familiar with these variables from previous
Newton’s Laws of Motion lessons.) Point out that one way
to determine the affects of force and mass on a vertically
launched rocket is to measure the rocket’s highest launch
height, known as “apogee.”
2. Explain that it would be nearly impossible to directly measure
the apogee with standard measuring tools, such as a meter
stick or measuring tape; therefore, they will use "indirect
measurement." Indirect measurement uses proportions, or
how the size of one thing compares to the size of another, to
find a value.
3. In this lesson, the method used to determine height is based
on a geometric relationship known as the “Pythagorean
Theorem” (more specifically, the tangent ratio). Pythagoras,
a mathematician who lived in ancient Greece 2500 years
ago, developed this theorem. His theorem applies to right
triangles.
4. Display the diagram below on the board or projection
system.
Note:
Explain that a right triangle is one where one of its interior
angles measures 90 degrees, and the other two angles, when
added, equal 90 degrees (angles C + D = 90°). The sum of
all the angles equals 180 degrees. The longest side, which is
always opposite the right angle, is the hypotenuse.
To assist in the students'
understanding of a right triangle,
you may want to provide some
examples of triangles that are
not right triangles. This allows
for easy comparison.
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
5. Point out that because of how the Pythagorean Theorem
works, if we know the angle of one of the acute angles (an
angle less than 90°) and the side opposite or adjacent to the
acute angle, we can determine the length of the unknown
side.
Draw or project the image below.
Opposite Side
?
Hypotenuse
Acute Angle
30°
Adjacent Side
150 meters
6. Explain that to determine the length of the "opposite side,"
we can use a simplified equation.
tangent of angle x known side = unknown side
7. Most students will not be familiar with the term "tangent." It
is not important that they have a full comprehension of it at
this grade level. Simply explain that it is a value determined
by making some calculations that relate to the triangle.
However, they should understand that these values are
represented in a "Table of Tangents," and they can use these
values to solve the above equation.
8. Project the "Table of Tangents" (see PowerPoint slide).
9. Solve the example provided in step 5 by walking the students
through the solution, using the "Table of Tangents" for 30°.
.58 x 150 m = 87 m
Opposite Side
87 meters
Hypotenuse
Acute Angle
30°
Adjacent Side
150 meters
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
10. To help students understand how to use this equation
to determine the rocket apogee, ask them to imagine
that the rocket launch stand is positioned at the point of
the right angle of a right triangle. If we mark off a known
distance to the point located at the tracker, then all we need
to determine the distance of the apogee is the angle as
observed from the known distance.
Apogee
?
?
Distance between stand & tracker (baseline)
150 meters
Note:
Depending on the comprehension
of your students, you may want
to explain that they should also
add to the apogee measurement
the height of the person using the
tracking device. This will provide a
more accurate measurement.
Note:
Before collecting the launch data,
be certain to review the launch
safety procedures as outlined in
the applicable rocketry appendix.
11. Explain that to find the angle, we can use a device called an
altitude tracker. We use the tracker to measure the angle to
the apogee.
Once finding the angle with the altitude tracker, use the table
of tangents and the equation to determine the apogee
Rocket Launch Data Collection
12. Explain the components of the AltiTrak and demonstrate its
proper use (refer to PowerPoint slide of Altitrak with labeled
parts).
• Hold and extend AltiTrak to aim.
• Keep both eyes open.
• Move the device up until you see the top of the “target.”
At that point, release the AltiTrak trigger.
• Read and record the angle.
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Note:
Have students work in tracking
pairs, one student sights the
rocket while the other reads the
angle. The reading should always
be less than 90°.
Note:
When students are using the
AltiTrak to follow a rocket’s
flight, it’s important that they
keep both eyes open. This makes
it easier to track the rocket.
13. Brief the students on the location of the tracker stations and
explain the measurement of the adjacent line (baseline), the
distance between the tracker station and the launch stand
(see PowerPoint slides).
14. To prepare the students for using the AltiTrak, take them
outside and have them practice on the predetermined
stationary object.
Once they are comfortable with the stationary object,
advance their skills by having them track a moving object
(such as a ball tossed into the air).
15. Once the students are comfortable using the AltiTrak,
proceed to the launch site.
Point out the "tracker station" locations.
16. Ask: What is the highest point a rocket reaches called?
(Apogee)
17. Have the students predict how high they think the rockets
will travel.
Note:
The “Table of Tangents” is
not utilized in most current
mathematics curriculum.
Instead, students use graphing
calculators, which calculate the
formula automatically. You may
find it useful to demonstrate this
function for the students using a
graphing calculator to familiarize
them with one of the features of
this technology.
18. Conduct rocket launches, allowing students to track and
record the angle of the rocket at apogee on the launch data
sheet. If possible, you may want to conduct a practice launch,
giving students an opportunity to practice tracking a fastmoving rocket.
19. Once launching is complete, return to the classroom and
have the students determine the mean angle for each launch.
Then, using the table of tangents, have them calculate the
altitude of the launches. (It may be helpful to walk them
through a sample.)
4 Check for Understanding:
Ensure students are using the table of tangents
chart correctly. Monitor and re-teach as needed.
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
20. Data Analysis:
Strategic Question:
Does the Pythagorean
Theorem (tangent ratio)
work to calculate distances
on the ground as well as
determining altitude?
Explain your answer.
(Yes. As long as we know the
value of one acute angle on
a triangle and the length
of either the adjacent or
opposite side of the angle, we
can determine the length of
the unknown line.)
A. Collect the results of each launch to create a bar graph.
B. Compare and discuss the results.
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Activity Log
What's My Angle
Launch Data Sheet
Apogee
(opposite)
?
?
Baseline (adjacent) =
tangent of angle x known side (baseline) = unknown side (apogee)
Rocket
Launch #
1
2
3
4
5
6
7
8
Tracker
Station A
Angle
Tracker
Station B
Angle
Tracker
Station C
Angle
Tracker
Station D
Angle
Mean Angle
and
Tangent
Apogee
(rounded to whole
number)
Activity Log
What's My Angle
Launch Data Sheet
Apogee
(opposite)
?
?
Baseline (adjacent) =
12 meters
tangent of angle x known side (baseline) = unknown side (apogee)
Rocket
Launch #
Tracker
Station A
Angle
Tracker
Station B
Angle
Tracker
Station C
Angle
Tracker
Station D
Angle
Mean Angle
and
Tangent
1
70°
62°
74°
67°
68°
tan = 2.48
2
3
4
5
6
7
8
Apogee
(rounded to whole
number)
30 meters
What’s My Angle? Assessment
Suggested Final Assessment Questions
Comprehension
1. Why is the tangent ratio useful when launching rockets?
a. It can help you figure out the speed or velocity of a rocket.
b. It can help you decide where to best launch your rocket.
c. It can help you determine the best amount of mass to
build your rocket.
d. It can help you determine the distance your rocket has
traveled.
e. All of the above.
Application
2. On which of the following triangles would you be able to use
the tangent ratio to find the length of side C?
a. Triangles A and B
b. Triangles A and C
c. Triangles B and C
d. Triangle B only
e. Triangle C only
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What’s My Angle? Assessment
Suggested Final Assessment Questions
1. Why is the tangent ratio useful when launching rockets?
a. It can help you figure out the speed or velocity of a rocket.
b. It can help you decide where to best launch your rocket.
c. It can help you determine the best amount of mass to
build your rocket.
d. It can help you determine the distance your rocket has
traveled.
e. All of the above.
2. On which of the following triangles would you be able to use
the tangent ratio to find the length of side C?
a. Triangles A and B
b. Triangles A and C
c. Triangles B and C
d. Triangle B only
e. Triangle C only
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
References:
Adjacent Sides. (n.d.). iCoachMath.com. Retrieved from
http://www.icoachmath.com/math_dictionary/adjacent_sides.html
Beginner’s Guide to Rockets. (n.d.). National Aeronautics and Space Administration. Retrieved from
http://www.grc.nasa.gov/WWW/k-12/rocket/guided.htm
Estes Rockets. (n.d.). Retrieved from http://www.estesrockets.com/
Estes, V. (n.d.). Rocket Stability. National Association of Rocketry. Retrieved from
http://www.nar.org/NARTS/TR13.html
Exploratorium: The museum of science, art and human perception. (n.d.). Retrieved from
http://www.exploratorium.edu/
iCoachMath.com. (n.d.). Retrieved from http://www.icoachmath.com/
NASA Glenn Learning Technologies Project. (n.d.). NASA: Pythagorean Theorem. The Physics Front.
Retrieved from http://www.thephysicsfront.org/items/detail.cfm?ID=8390&Relations=1
Pythagoras’ Theorem. (n.d.). Math Is Fun. Retrieved from
http://www.mathsisfun.com/pythagoras.html
The Tangent Ratio. (n.d.). XP Math. Retrieved from http://www.xpmath.com/
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E 3.1.1.5. Math Operations & Applications: C. Geometry, D. Data Analysis
Activity: What’s My Angle?
Table of Tangents
Angle
Tangent
Angle
Tangent
Angle
Tangent
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
.02
.03
.05
.07
.09
.11
.12
.14
.16
.18
.19
.21
.23
.25
.27
.29
.31
.32
.34
.36
.38
.40
.42
.45
.47
.49
.51
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
.53
.55
.58
.60
.62
.65
.67
.70
.73
.75
.75
.81
.84
.87
.90
.93
.97
1.00
1.04
1.07
1.11
1.15
1.19
1.23
1.28
1.33
1.38
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
1.43
1.48
1.54
1.60
1.66
1.73
1.80
1.88
1.96
2.05
2.14
2.25
2.36
2.48
2.61
2.75
2.90
3.08
3.27
3.49
3.73
4.01
4.33
4.70
5.14
5.67
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