u07a2 algerbra in the real world and everyday life (5)
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Running head: ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 1
Algebra in the Real World and Everyday Life
Hal Hagood
u07a2
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ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 2
Table of Contents
Page Number
Table of Contents …………………………………………………………………… 2
Introduction ………………………………………………………………………… 3
Ways That Algebra Affects Business or Science ………………………………….. 5
How Algebraic Concepts Can Solve Everyday Problems in Life …………………. 6
Ways Algebra Can Solve Everyday Problems in Business or Science ……………. 9
A Surprising Finding About How Algebra Affects Personal Life, Business, and Science 10
References ………………………………………………………………………….. 14
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ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 3
Introduction
Algebra, some of us fear it while some of us embrace it, algebra is not “arithmetic with
letters” it is better described as a way of thinking. At its most fundamental level, arithmetic and
algebra are two different forms of thinking about numerical issues. Many of these examples have
been taken from our classroom discussions while others are examples I have discovered in my
own research for this paper, several examples of each will be cited.
“Let’s start with arithmetic. This is essentially the use of the four numerical operations
addition, subtraction, multiplication, and division to calculate numerical values of various things.
It is the oldest part of mathematics, having its origins in Sumeria (primarily today’s Iraq) around
10,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction
of money as a means to measure an individual’s wealth and mediate the exchange of goods and
services.
There are several ways to come to an understanding of the difference between arithmetic
and (school) algebra. First, algebra involves thinking logically rather than numerically. In
arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.
Arithmetic involves quantitative reasoning with numbers; algebra involves qualitative reasoning
about numbers. In arithmetic, you calculate a number by working with the numbers you are
given; in algebra, you introduce a term for an unknown number and reason logically to determine
its value” (What is algebra, 2011).
For example, putting numerical values for a, b, c in the familiar formula
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in order to find the numerical solutions to the quadratic equation
is not algebra, it is arithmetic.
“In contrast, deriving that formula in the first place is algebra. So too is solving a
quadratic equation not by the formula but by the standard method of “completing the square” and
factoring. When students start to learn algebra, they inevitably try to solve problems by
arithmetical thinking. That’s a natural thing to do, given all the effort they have put into
mastering arithmetic, and at first, when the algebra problems they meet are particularly simple
(that’s the teacher’s classification as “simple”), this approach works. In fact, the stronger a
student is at arithmetic, the further they can progress in algebra using arithmetical thinking. For
example, many students can solve the quadratic equation x2 = 2x + 15 using basic arithmetic,
using no algebra at all. Paradoxically, or so it may seem, however, those better students may find
it harder to learn algebra. Because to do algebra, for all but the most basic examples, you have to
stop thinking arithmetically and learn to think algebraically” (What is algebra, 2011).
The need for algebra does not make it any easier to learn but in today’s world with the
technology it embraces, it is essential that we develop our thinking skills to match what today’s
world requires. The use a computer is one of those essential skills and being able to use that
computer efficiently to do arithmetic requires algebraic thinking.
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ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 5
Ways That Algebra Affects Business or Science
There are many ways that algebra affects business and science and in several diverse
approaches. These include the fields of astronomy, biology, chemistry, construction, economics,
education, environment, finance, geometry, government, health/life sciences, labor, physics,
sports/ entertainment, statistics/ demographics, technology and transportation.
To highlight a few for examples let us start with astronomy, astronomers use math all the
time. “One way it is used is when we look at objects in the sky with a telescope. The camera that
is attached to the telescope basically records a series of numbers - those numbers might
correspond to how much light different objects in the sky are emitting, what type of light, etc. In
order to be able to understand the information that these numbers contain, we need to use math
and statistics to interpret them. Another way that astronomers use math is when they are forming
and testing theories for the physical laws that govern the objects in the sky” (Real Life
Applications of Algebra, 2013).
Almost all consumer math involves calculating transactions. This ranges from calculating
change to interest on a loan. Algebra can be used to fill in the blanks for more complex equations
such as calculating the total loan and interest given a fluctuating rate, very necessary in today’s
world and markets. Yet another example is the use of algebra in the construction business, it is
widely known for calculating square footage, cubic footage, and angles when building.
“Mathematics in algebra is used by construction workers in many ways. When setting out a site,
mathematics is used to get the dimensions correct. It is also used calculating the amount of
materials to order, and when cutting materials to size. Very few tasks do not involve some use of
mathematics” (Real Life Applications of Algebra, 2013).
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ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 6
How Algebraic Concepts Can Solve Everyday Problems in Life
Everyday examples of how algebraic concepts solve everyday problems are numerous.
One of my favorites are these telephone illustrations, I am constantly searching for the best deal.
They provide a good example, since linear functions show a constant rate of change between 2
variables. For example, If a cell phone company charges a start-up fee of $75 and then $.05 for
every minute used, the function is y = .05 x + 75.
Or
Back in the days when you actually had to pay for long distance service shopping for the
best long distance plan was pretty important. For example a telephone long distance company
would cost me $3.00 each month for a $0.03 per minute long distance rate. To calculate the cost
where "x" is the number of minutes and "y" is the cost of the long distance usage I can use the
linear equation y = 0.03x + 3. So for 60 minutes …
y = 0.03x + 3
y = 0.03(60) + 3
y= 0.03 x 60 + 3
y= 4.8
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Another one of my favorite examples has to do with cooking a chicken and since I love to
cook this was very timely. “The time, in minutes, taken to cook a chicken, C, is linearly related
to the weight, in kilograms, of the chicken, k. It takes 40 minutes to cook a 1.4kg chicken and 50
minutes to cook a 1.8kg chicken.
Find the linear equation relating cooking time, C, with the weight of the chicken, k, for a
1.7kg Chicken. This is the same as a line with ordered pairs, (k,C) and two points are given; find
the equation for the line. You are looking for C=mk+b, much like you were finding y=mx+b.
The points on this line are (1.4,40), and (1.8,50).
Slope, m=(50-40)/(1.8-1.4)=25.
From C=mk+b, b=C-mk. Pick a point and find b.
b=50-25*1.8=5
Equation would then seem, C=25k+5.
If k=1.7, C cooking time predicted is C=25*1.7+5=47.5 minutes” (Algebra.com, 2013).
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Another excellent example that most of us deal with at one time or another is also
provided in the use of exponential equations to represent and solve savings or loan situations.
“Fill in the formula A= P (1 +r/n)^nt when dealing with compound interest and A = P(2.71)^rt
when dealing with continuously compounded interest. "A" represents the total amount of money
with which you will end up or will have to pay back, "P" represents the amount of money put
into the account or given in the loan, "r" represents the rate expressed as a decimal (3 percent
would be .03), "n" represents the number of times interest is compounded per year, and "t"
represents the number of years the money is left in an account or the the number of years taken
to pay back a loan. You can calculate any one of these parts by plugging in and solving if you
have the values for all of the others. Time is the exception because it is an exponent. Therefore,
to solve for the amount of time it will take to amass, or pay back, a certain amount of money, use
logarithms to solve for "t" (Algebra in Real Life, 2013).
Linear equations can be used to determine affordability. One can use it to calculate
service fees for both a variable and flat rate. An example would be a membership fee to a local
club or gymnasium. If the charge is $25 a month for X number of months plus an upfront fee to
join one can budget that amount. For example $25/month, there is a $75 flat fee, and you have a
budget of $275, your equation would look like this: 25x + 75 = 275. Solving for x tells you that
you can afford eight months at that club or gym.
As one can see there are numerous examples of the use of algebra in real life situation
and many times we use it without actually realizing we are doing so. Quadratic equations can be
used to find the possible values of something both minimum and maximum. They can also be
used to determine the aspect of situations as they relate to each other. There are many other
examples but suffice it to say we all use algebra in one form or another in our everyday lives.
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ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 9
Ways Algebra Can Solve Everyday Problems in Business or Science
Biology and chemistry are two of the best examples in the use of algebraic methods and
procedures. “Algebraic biology applies the algebraic methods of symbolic computation to the
study of biological problems, especially in genomics, proteomics, analysis of molecular
structures and study of genes. Computations in the field of biology are done in order to simulate
experiments and/or calculate features of a biologic process or structure. Such as for example
calculating mathematical predictions of intercellular features, cellular interaction, body reaction
to chemicals and analysis of heritage. In recent years, methods from algebra, algebraic geometry,
and discrete mathematics have found new and unexpected applications in systems biology as
well as in statistics, leading to the emerging new fields of "algebraic biology" and "algebraic
statistics." (Real Life Applications of Algebra, 2013).
The use of algebra in chemistry is another prime example, it offers many ways: to
manipulate equations and solve for a problem. “A gas equation from chemistry PV=nRT. P is the
pressure (in atm), V is the volume (in L), n is the moles, R is a constant (.082059 L*atm mol-1
K-1), and T is the temperature (in K)” (Real Life Applications of Algebra, 2013). The use of
computer algebra has come into play and is used increasingly in chemistry and for chemistry
education.
In business and the economic sector the use of algebra is as wide as it is varied. Fractions
and decimals are found in all business application as are the calculations for proportions and
percentages. “Measurements are necessary for many different business processes and involve
computations shared by all different aspects. Economic trends are plotted by economists using
matrix algebra in linear equations, regression, linear models, linear programming, and Markov
chains” (Real Life Applications of Algebra, 2013).
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ALGEBRA IN THE REAL WORLD AND EVERYDAY LIFE 10
A Surprising Finding About How Algebra Affects Personal Life, Business, and Science
There are many surprising uses for algebra in science, business and its use in everyday
life. A surprising use I discovered in the field of science was its use in space travel. I found this
to be fascinating. It has to do with algebra, communications and the lunar outpost.
“This problem is part of a series of problems that apply Algebra and Geometry principles
to U.S. Space Exploration policy. Exploration provides the foundation of our knowledge,
technology, resources, and inspiration. It seeks answers to fundamental questions about our
existence, responds to recent discoveries and puts in place revolutionary techniques and
capabilities to inspire our nation, the world, and the next generation. Through NASA, we touch
the unknown, we learn and we understand. As we take our first steps toward sustaining a human
presence in the solar system, we can look forward to far-off visions of the past becoming realities
of the future. The vision for space exploration includes returning the space shuttle safely to
flight, completing the International Space Station, developing a new exploration vehicle and all
the systems needed for embarking on extended missions to the Moon, Mars, and beyond. By
2020, NASA astronauts will again explore the surface of the Moon. This time, we're going to
stay, building outposts and paving the way for eventual journeys to Mars and beyond. The
crewed and robotic return to the Moon requires robust and reliable communications. It will be
important to maintain constant communications with Earth. Therefore, 24 hours per day/7 days
per week coverage at the outpost is a requirement. This will likely be accomplished by a
combination of communication satellites in orbit around the Moon and communication
equipment on the lunar surface” (NASA, 2013).
A theoretical problem is given here, “Suppose a lunar outpost has been established on the
rim of Shackleton crater. Shackleton crater is small, about 19 kilometers in diameter, and is not
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in range of any satellite. The rim is slightly raised and in an area of near-permanent sunlight that
provides access to solar power. The crater’s permanently dark interior is a cold trap that may
contain water ice. This site is within the South Pole-Aitken (SPA) basin, the oldest and biggest
impact feature on the Moon. The SPA basin is about 12 kilometers deep and exploration of its
geologic features may provide useful information about the lunar interior. Whenever astronauts
travel away from the lunar outpost for scientific study or exploration, constant communications
will be provided by surface to surface communication towers. Receivers within a certain radius
of the communication tower antenna can pick up signals. Because the Moon is a sphere, the
surface to surface signals cannot be received beyond the point of tangency, P, of the line of sight
distance, d (Figure 2). Precision in these calculations will be critical to the communication
signal, therefore carry all calculations to two decimal places. Also, if you are using a calculator,
make sure it is set to degrees (not radians)” (NASA, 2013).
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In the business environment the most interesting thing I found was how algebra can be
applied to the stock market. There are a lot of people who have tried the mathematical approach
and most use some form of pattern matching. That is to say, mathematical methods to predict the
future Forex rates and stock moves. These are based on closing and opening candle patterns in
the charts. Unfortunately they are also open to a wide range of interpretation and thus lack the
specific knowledge necessary for successful trading. There is however a system of linear
equations that can be used. It is bit more of a complex approach to mathematical market
predictions. To do this one can assume that each time point in a chart = n has a price Pn and its
own parameter Rn. To make it work as such use Rn * Pn + Rn-1 * Pn-1 + … + R1 * P1 + R0 *
P0 = Current Price. From this one can assemble a system of the linear equations to find all the
Rn values. It can also be used to extrapolate the future value/targets by knowing previous prices.
Of course in all of this there is one small problem, the markets by their very nature are
highly unpredictable in nature. If they were purely mathematical in their approach things would
be different but the markets are driven by emotional, political and psychological factors that
mathematical formulas cannot take into account. Bottom line in all of this, don’t quit your day
job unless you really know what you are doing.
On a personal level it is surprising that we all use a form of what is called mental algebra.
We use it in various ways to determine cost of transportation, options for purchasing gifts, and of
course having money to eat and we use it every day. “Almost every math problem which
involves money, time, distance, perimeter of a fence or skate ramp, volume of something,
comparing prices, rent something – cost versus time, and similar situations are solved using
algebra” (Suite101, 2013).
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References
Algebra.com, (2013). Retrieved May 5, 2013 from
http://www.algebra.com/algebra/homework/Linear-equations/Linear-
equations.faq.question.707072.html
Algebra in Real Life, (2013). Retrieved May 5, 2013 from
http://www.ehow.com/how_5714133_use-algebra-real-life.html
NASA, (2013). Retrieved May 5, 2013 from
http://www.nasa.gov/pdf/514479main_AL_ED_Comm_FINAL.pdf
Real Life Applications of Algebra, (2013). Retrieved April 30, 2013 from
http://wiki.answers.com/Q/What_are_some_real_life_applications_of_algebra
Suite101, (2013). Retrieved May 5, 2013 from http://suite101.com/article/children-use-
algebra-everyday-to-solve-problems-a59021
What is algebra, (2011). Retrieved April 30, 2013 from
http://profkeithdevlin.org/2011/11/20/what-is-algebra/