Mathematics at work

Mathematics in the Aerospace Sector

It is projected that by 2016 there will be at least 500,000
jobs in aerospace maintenance and manufacturing that pay
well and offer opportunities for advancement. According
to the Bureau of Labor Statistics’ Occupational Outlook
Handbook, technological advances in the industry require
mechanics to have an especially strong background in
electronics — and mathematics — to keep pace with the
changes in the industry. Additionally, many aerospace firms
recently have begun to bring together teams composed of
customers, engineers and production workers to pool ideas
and make decisions concerning aircraft design at every
phase of its development. The aerospace industry provides
good job opportunities for individuals with teamwork skills
and proficiency in technology and applied mathematics.

Available Aerospace Jobs

Within the aerospace industry, there are a variety of entry-
level jobs that pay well and provide opportunities for
advancement — jobs for high school graduates with
postsecondary training or education but less than a four-
year college degree. High school graduates with 18 to 24
months of specialized postsecondary training leading to FAA
certification are qualified to become aircraft mechanics.

Core Mathematics Knowledge
in Today’s Aerospace Jobs

Developed by secondary, postsecondary, business,
industry and government leaders, the Career Cluster
Pathway Plan of Study for Maintenance, Installation
and Repair was designed to serve as a guide for
students’ educational and career goals. This Plan
of Study for individuals pursuing a career in aerospace
recommends a rigorous set of mathematics courses for
students to take at both the secondary and postsecondary
levels in traditional or vocational settings. This Plan
of Study shows in detail how the foundation provided
by courses such as Algebra I , Geometry, Algebra II,
Trigonometry or Statistics, Physics, and Computer
Applications equips high school graduates with the
mathematical knowledge and skills needed for success
on the job. Until high school graduates understand the
advanced mathematical skills used in the aerospace
sector, they will remain unable to meet the demands of
this high-growth industry.

*Remaining percentage of workers in occupation have a bachelor’s degree or higher
Source: Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2008–09 Edition.

Ensuring College and Career Readiness:
The American Diploma Project

In 2001, Achieve and several partner organizations
launched the American Diploma Project (ADP) to
identify a common core of English and mathematics
academic knowledge and skills, sometimes referred to
as “benchmarks,” that American high school graduates
need for success in college and the workforce. These
ADP benchmarks, released in the 2004 report Ready or
Not? Creating a High School Diploma That Counts, are
the result of two years of intensive research conducted
in colleges and universities as well as workplaces across
the country.

The real-world expectations identified by ADP are signi -
ficantly more rigorous than many current high school
graduation standards — which helps explain why many
high school graduates arrive at college or the workplace
with major gaps in their English or mathematics preparation.

To help pinpoint the academic knowledge and skills
required for future employment, ADP commissioned
 leading economists to examine labor market projections
for the most promising occupations — those that pay
enough to support a family and provide real potential
for career advancement. ADP then surveyed officials
from 22 industries, ranging from manufacturing to
financial services, about the most useful skills for
their employees to bring to the job.

ADP also worked closely with two- and four-year post-
secondary faculty from five partner states to determine
the prerequisite English and mathematics knowledge and
skills required to succeed in entry-level, credit-bearing
higher education courses. These conversations revealed
an unprecedented convergence of the knowledge and
skills employers and postsecondary faculty say are
needed for new employees and freshmen beginning
credit-bearing coursework to be successful.

“Mathematics at Work” Series

Following up on the work of ADP, Achieve has produced a series of “Mathematics at Work” brochures to
examine how higher-level mathematics is used in today’s workplaces. The brochures present case studies
drawn from leading industries nationwide to illustrate the advanced mathematics knowledge and skills
embedded in jobs that offer opportunities for advancement and are accessible to high school graduates.

The series underscores the value of a rigorous high school curriculum in mathematics. All high school
graduates — regardless of whether they enroll in college, join the workforce or enter the military —
benefit from acquiring a comprehensive knowledge base and skill set in mathematics.

Mathematics pilots

Career Preparation for Aircraft
Maintenance Technicians

The more than 100,000 aircraft maintenance technicians
(AMTs) at work in the public and private sector today have
a wide range of talents and skills that keep America’s
airplane fleet flying safely and smoothly. Although they
may have different educational backgrounds and areas
of expertise, all AMTs have one thing in common — a
solid foundation in mathematics, enabling them to earn
rigorous Federal Aviation Administration (FAA) certification
in airframe and power plant maintenance. From fabricating
parts and installing hydraulic systems to testing electrical
circuits and repairing airplane engines, AMTs possess the
highly sought-after mechanical knowledge and skills —
grounded in mathematics — that are valued by the airline
industry.

"Without a strong math and science
background, today’s technician is
not likely to progress far beyond
basic line maintenance."

La Vern Phillips, Regional Director
Aviation Institute of Maintenance

To earn FAA certification, applicants must have at least
a high school diploma and 18–24 months of specialized
postsecondary training, be it on the job, at a community
or technical college, or in a specialty occupation in the
military. Through a combination of oral and practical
exams, the FAA assesses the competency of applicants
in handling equipment such as ignition analyzers, riveters
and compression checkers. It also tests their familiarity
with scientific principles such as basic gas laws and
the properties of metals as well as the underlying
mathematical concepts that cover many core ADP
benchmarks. For example, the FAA requires that all
AMTs understand:

Measurements with units when replacing and
fabricating fuselage parts

Proportions and ratios when assessing the electrical
needs of circuitry

Solving equations with one or more variables when
repairing hydraulic systems

Calculating volume of spheres, parallelepipeds and
cylinders when assessing fuel needs

Modeling with linear equations when computing
the center of gravity (CG) of an aircraft

The FAA certification process tests the understanding
future AMTs have of the array of instruments and
processes used on a daily basis, thereby evaluating
their ability to apply their mathematical skills to the
various challenges they face. Given the vital stakes
attached to aircraft safety, it also is critical that every-
one on the tarmac — pilots, engineers and AMTs — is
able to communicate with other members of the team
by speaking clearly about an airplane’s electronics
and mechanics through the universal language of
mathematics. Because no two aircraft are exactly
the same, AMTs must know when and how to apply
particular solutions to unique problems and modify
approaches from aircraft to aircraft as appropriate.
 

aviation safety

Safety in Numbers: Mathematics
in Action on the Tarmac

America’s future is up in the air — literally — and it
is up to airframe and power plant technicians to keep
America flying safely. The image of a tire-kicking, grease-
covered mechanic has long since been replaced by the
professionalism of today’s FAA-certified AMT. These
mathematically adept craftsmen ply their trade across
every facet of aviation.

Repairing a Crack in the Fuselage

Mathematical Reasoning and
Problem Solving

During the standard preflight check, AMTs are respon-
sible for inspecting the airplane for any suspicious signs
of wear and tear. Faced with a crack in an airplane’s
fuselage, a team of AMTs must first determine the most
effective method of repair . To do this, AMTs depend
on knowledge of the physical properties of metals and
composites — their strength, flexibility and
durability — when put under the stress
of flight. AMTs need to understand the
relationship between the materials used
to construct the aircraft and the parts
needed to make the repair, and know
the best methods for securing these
parts to the aircraft. Calculating
sheer strength also will help the
AMT select the most appropriate
materials for mending
the crack — such as
using metal rivets,
a composite
mixture of

resin and hardener, or the fabrication of a new part. If
a new part is required, the shape of the fabricated part
is another important variable — especially if the crack
has appeared on a part of the airplane with significant
curvature. In that situation, AMTs have little choice but
to create a new part to replace the damaged section.

Fabricating New Parts for Repair
 
Geometry, Algebra and Teamwork

Depending on the size of the crack, the weight
restrictions and the bend allowance of the fabricated
material, AMTs must perform a sophisticated series
of algebraic and geometric calculations to determine
the amount of material needed for the replacement
part, the shape it should take, and the maximum
weight allowance and other tolerances.

For example, creating a patch for a particularly curved
section of the airplane’s fuselage (or wing) requires
calculating the bend allowance of sheet metal — a
dimensional adjustment that must be factored in to
ensure the safety of the replacement part. When AMTs
bend metal to the desired form, the material of the
outside angle is stretched, while the material on the
inside angle is compressed. Determining the proper
bend radius is essential to constructing a strong and
sound part. Too small a radius can cause further cracking,
while too large a radius can result in costly overruns
of material, excess weight and ultimately a disruption
in the airplane’s CG. To properly calculate the bend
allowance, AMTs must measure the length of the
brackets and the material thickness along with the
angle and radius of the bend. They also must determine

the “neutral axis,” or the K-factor — the percent of
the material thickness where there is no stretching or
compressing. Once the bend allowance is calculated,
technicians can fabricate a new part; choose the
appropriate rivet size, rivet pitch and tooling; and
successfully repair the aircraft.

Maintaining Center of Gravity

Measurement and Proportions

A team of AMTs working together also must consider the
effect the repair will have on the airplane’s CG. Because
an aircraft is designed to be balanced forward and aft,
the CG is located in a precise range along the chord
of the wing — the distance between the leading and
trailing edge of a wing. If the actual balance point is too far
forward or too far aft, the pilot may experience difficulty
maintaining level flight.

When making adjustments to an airplane’s fuselage,
AMTs must find the CG by weighing the aircraft with
platform scales or by using load cells positioned at
designated points along the fuselage. AMTs must
calculate these specific weights and the dimensions
of those points to determine “moments” along an axis
and determine the range within which the CG can be
located. If, after making the repair, the actual CG falls
beyond the approved envelope, the technicians need
to calculate where and how much ballast they should
add to reposition the CG within the approved range.

Since ballast is useless weight and only wastes fuel,
fabricating durable but lightweight replacement parts
is key to minimizing the need for additional costly
adjustments to the airplane’s CG.

Keeping Airplanes Flying

Keeping an aircraft ready for flight requires AMTs to make
dozens of complex decisions every day as they inspect,
repair and maintain a wide variety of aircraft. Ensuring
an airplane’s airworthiness and the safety of the crew
and passengers is not based on a wing and a prayer —
it is grounded in a sophisticated understanding of how
to apply the principles of algebra, trigonometry and even
calculus to airplane maintenance. Keeping America flying
safe and sound is just another example of mathematics
at work today.

"Whether it is manufacturing a new
component, repairing a damaged
structure, performing an aircraft
weight and balance calculation, or
performing operational checks ,
mathematics is essential."

E. Wayne Lee
Head Aircraft Services Branch
NASA

Mathematics + Teamwork = Success

AMTs must have a strong foundation of mathematical
knowledge and skills to successfully inspect, repair and
maintain the airworthiness of an airplane. AMTs need
to tap into their problem-solving skills to diagnose any
potential issues that arise and make well-informed
decisions to resolve those problems quickly and
efficiently.

When a team of AMTs identifies a crack in the fuselage
of an airplane, for example, they must first hone in on
the best repair solution, using inductive and deductive
reasoning. AMTs also use their knowledge of various
materials’ properties and sheer strength to determine
the appropriate materials and fabrication methods to
repair a crack on a particular part of the aircraft.

The technicians rely on many mathematics-based skills
to fabricate a new part, including using real and rational
|numbers, calculating ratios and proportions, and measuring
plane figures. Computing the bend allowance requires
AMTs to use algebraic and geometric concepts to ensure
that the newly fabricated part fits just right.

AMTs must monitor the consequences of their actions on
the airplane’s overall airworthiness and performance. When
the team of AMTs installs the new part, they may shift the
airplane’s CG. If so, the AMTs will need to calculate a new
CG, requiring the application of such mathematics skills
as converting units of measurement and solving equations
with one or more variables.

Although fabricating a new part to repair a crack in an
airplane relies on AMTs’ mathematical and technological
knowledge, teamwork also is critical. ADP research shows
that both in the college classroom and in the work-
place, professors and employers identify collaboration
and communication as important factors for success.
Since AMTs may have individualized training in particular
systems, such as engine repair, AMTs often work together
and learn from each others’ expertise to ensure that every
airplane is in top-flight condition.

The required postsecondary training — be it in the
field or in the classroom — combined with the rigorous
FAA certification test and the on-the-job instruction, all
underscore the need for future AMTs to leave high school
|having completed a rigorous set of courses aligned with
college and career readiness expectations, especially in
mathematics.

"Mathematics plays a huge role
in the day-to-day activities of
aircraft mechanics. Consciously
or subconsciously, mechanics
utilize their knowledge of math.
The basic principles of algebra,
trigonometry and even calculus
are applied towards ensuring the
airworthiness of the aircraft and
the safety of the crew."

E. Wayne Lee
Head Aircraft Services Branch
NASA