 Real-World Math Tutorial
For High School Students
Teachers and parents are always looking
for real-world examples of mathematics. We've prepared a math
tutorial that demonstrates how math is really used in the real world. The tutorial
is aimed
primarily at high school students. Topics range from simple logic problems to using calculus to model
complicated systems. Includes several Computer Science and Math programs
to demonstrate these real-world math examples using Microsoft Windows.
Table of Contents
1. Problem Solving
Chapter 1 covers some key points in solving problems - stressing
solution of real, not textbook, problems.
2. Home Heating Mathematics
Chapter 2 reviews the math
behind thermostats and furnace operation. Examples illustrate computing
temperature changes within a home.
This chapter
provides an application in one particular area, that of home heating. We
discuss the mathematics behind a thermostat and provide equations that allow
computation of temperature changes within a home. Sample problems are given, as
well as suggestions for further study.
3. Satellite Orbit Problems
Chapter 3 reviews the dynamics behind satellite
motion illustrating typical orbits, orbit transfers and rendezvous problems.
Satellites in orbit around the earth are commonplace. They are used
to transmit television signals, telephone calls, weather data, airplane
navigational information, and radio programs. And, many of today’s
cars are equipped with GPS (global positioning satellite) locators that
help them get around in traffic! Manned satellites, such as
the Space Shuttle, allow for all types of valuable experimentation and space
repair work. In this chapter, we look at some
of the mathematics and physics behind satellite orbits. We first answer the question of why
satellites orbit by discussing the dynamics of satellites. Then, we examine
typical orbit problems, such as how to establish an orbit and how to move from
one orbit to another. Many example problems and suggestions for further work
are provided.
4. Pendulums and Complex Numbers
 Chapter 4 reviews how
complex numbers are used in a real problem - pendulum motion. Need
familiarity with quadratic equations.
In the study of quadratic equations, students are introduced to the
concepts of imaginary and complex numbers. A common question is:
why do we have to worry about imaginary numbers? After all, if
they're imaginary, they don't exist! This chapter gives a real-world example of the use of complex numbers
and shows how they relate to real quantities. We show how a quadratic equation (and
complex numbers) can be used to model the dynamics of a swinging pendulum.
Equations for both a normal and an upside-down (inverted) pendulum are
developed. We also discuss how we can use mathematics to balance an inverted
pendulum (the same concept used in every rocket and missile launch in history).
The equations presented can provide a springboard for classroom
simulation studies.
Computer
Pendulum Simulation software included.
5. A Look at Real-World Problem Solving
 The example problem in Chapter 5 involves
the trajectory of a projectile. The problem is to reach a desired point by
selecting a launch angle. This problem has a wide variety of real
applications: rocket and missile launching, targeting and intercepts,
satellite orbit transfers and rendezvous, numerical optimization,
polynomial root finding, and solving nonlinear equations. The mathematics
needed to solve the problem are not too involved - only algebra and
trigonometry. The process of solving the problem, however, is at times
detailed and tedious. But, this is a concept that needs to be taught:
real-world problems are not necessarily easy. Before looking at the
solution, we will examine the real-world problem solving process.
In this chapter, a
closed-form expression (an exact equation) to the trajectory problem
(required angle to reach a point) is found. Closed-form solutions
are rarely possible in the real-world. We usually resort to solving our
problems numerically, using a computer. Requires
algebra and trigonometry.
Trajectory Simulation software is included.
6.
Another
Look at Real-World Problem Solving
Chapter 6 reviews a numerical approach to the problem studied in Chapter 5.
In this chapter, we look again at the projectile problem, but the solution
emphasis is on iterative, numerical solutions. Both one-dimensional and
two-dimensional solution methods are discussed, as are some of the checks
that must be made and the pitfalls that should be avoided when using such
methods. For classroom purposes, examples of potential applications of the
trajectory equations are discussed. Requires pre-calculus and
trigonometry.
7. Solving Problems Numerically
 Chapter 7 is a discussion of numerical
methods for solving "unsolvable" problems. Illustrated with rocket
engine example and coordinate conversions. Rocketry software included.
In this chapter, we look at solving two
problems numerically. The problems solved are relatively simple and
actually have closed-form solutions. We use simple examples to illustrate
the steps involved in obtaining numerical solutions without getting too involved
in the mathematics. And, by having closed-form, or exact, solutions
available, it gives us something to check our computer programming techniques.
The first problem (one-dimensional) requires the determination of a model rocket
engine's burn time in order to achieve a desired altitude. The second
problem (two-dimensional) is a numerical implementation of converting from
rectangular to polar coordinates. While solving these problems, we will
address some of common problems encountered when implementing numerical
(computer-based) solutions. Suggestions for using the example problems and
potential expansions of the problems are addressed.
8.
Mathematics of Robot Arms
 Chapter 8 is a study of the mathematics behind the
modes of robot arm operation.
Robotic arms are
commonplace in today's world. They are used to weld automobile bodies,
employed to locate merchandise in computerized warehouses, and used by the
Space Shuttle to retrieve satellites from orbit. They are reliable and
accurate. This reliability and accuracy is due to the computer a robot arm
uses in determining where and how it should move. This control computer is
programmed with some basic mathematics. In this chapter, we will look at the mathematics behind robot arms.
We will study trajectory planning. We
look at three basic problems: kinematics, inverse kinematics and
trajectory planning (getting from one point to another). We examine each of
these problems separately, using the two-link robot arm.
The Robotic ARM software program demonstrates the concepts presented here. Requires algebra
and trigonometry skills.
9. Fractals from Polynomial Solutions
 Chapter 9 is an illustration of the beauty
of math - how solving for the roots of a polynomial can generate beautiful
fractal graphics. Solving a linear
algebraic equation is a simple process and to find the roots of a
second-order polynomial, we use the quadratic equation. There are
also specific procedures for finding the roots of a third-order
polynomial. However, for fourth-order and higher, we need other ways
to find roots. In this chapter,
we study the Newton-Raphson method for finding roots. To use the Newton-Raphson method to
find polynomial roots, we need an initial guess at a root. Every point in the
complex plane is a potential solution, hence a potential guess. If we use each
point in the complex plane as an initial guess, compute the resulting converged
root, and track which guess converged to which root, we obtain a mapping of
initial guesses to final roots. This mapping, when drawn in color on a computer
screen, can provide pretty and surprising results. Such mappings are fractals.
We develop a procedure for generating fractals from the solution of a general
polynomial. For each step in the procedure, the pertinent equations are
provided to help you understand the technique and develop your own computer
routine (with modifications, if desired).
Requires some algebra. Fractals Simulation software included.
10. Chaos in a Real System
 Chapter 10 demonstrates how something as simple as a water
wheel can exhibit chaotic, strange behavior.
In this chapter, we model a four bucket version of the
Lorenz water wheel - a famous system that exhibits a chaotic nature. Equations
modeling the rotational dynamics, frictional effects, and bucket flow
characteristics are developed. Then, implementation considerations for solving
the equations are presented and some typical results presented. This is an
advanced application of mathematics and physics. You should be familiar
with differential equations and physical system dynamics to understand the
material. The Water Wheel
simulation program demonstrates the concepts presented here.
This chapters has detailed dynamics equations
to build a simulation. Water Wheel software included.
11. Computing Airplane Takeoff Speeds
 Chapter 11 demonstrates
how a pilot knows how fast an airplane needs to be going prior to takeoff.
This chapter describes the process performed
by an onboard airplane computer in determining speeds the pilot uses in making
decisions during takeoff. The mathematics involved requires knowledge of
second-order polynomials. As a pilot accelerates
from a stop, he/she must be cognizant of a quantity referred to as the decision
speed. This is the speed at which the pilot must choose to either continue with
the takeoff or abort the takeoff and stop the airplane. The value of this speed
depends on many things: altitude, temperature, airplane configuration, weight,
braking energy, and engine type, to name a few. But, for demonstration
purposes, we can compute this speed using some simple physics and math. The
speed can be determined in two ways: a direct solution and a numerical
solution.
Polynomial Equation software included.
12. Computing Airplane Stopping Distance
Chapter 12 demonstrates
how derivatives can be applied to real-world problems.
This chapter presents two applications of derivatives (a beginning
calculus topic) related to computing the distance required to stop an
airplane. This is an important problem because this distance must be
within the constraints of the runways where a plane might land.
First, we look at using curve-fitting techniques to develop an equation
for brake force that insures a smooth transition from zero braking to full
braking force. Second, we develop equations for the deceleration and
speed of an airplane and show how we can numerically solve these equations
(simple differential equations) to obtain an approximate value for
stopping distance. Example data and results are presented for
several different airplane models. We discuss curve
fitting and braking requirements for a jetliner. Requires basic calculus
skills.
Computer Math & Science Fair Simulation Software Included with The Real World
Math Tutorial:
Trajectories
A simulation of a flying object. The program introduces estimation and
answer refinement skills.
 Quadratic Equation Solver
Enter coefficients for quadratic
equation - see plot, compute roots and evaluate at various x values.
Polynomial Equation Solver
Enter coefficients for Nth order polynomial - curve is plotted, roots are
found.
Polynomial Fractals
See the cool patterns that
result from numerically solving for polynomial roots.
PENDULUM
studies basic
oscillations - see effects of changing lengths, weight, and friction.
Plots are given
 CIRCUITS
lets you to build
electric circuits from batteries, switches, light bulbs. See effects of
different circuit types, burned out bulbs, switch positions.
LEVERS
teaches you about the
three different lever types and forces required to balance them.
WATER WHEEL demonstrates the
exciting world of chaos. Examine wheel motion as different water flow
rates are used.
 MOON
LANDER
gives practice in understanding
concepts of speed and acceleration as you try to land on the moon.
 WEATHER
WATCH
provides tools that allow you
to identify and predict trends in the weather.
ROCKETRY
is a demonstration of
model rocket launches - vary thrust, burn time, and rocket mass.
ROBOTICS
allows you to study the
world of robots using a 2-dimensional, two-link robotic arm.
Real World
Math and Science Fair User
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