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Discrete Mathematics
Discrete mathematics is a contemporary field of mathematics that is widely used in business and industry. It is sometimes called the mathematics of computers, or the mathematics used to optimize finite systems. It is an important part of the high school mathematics curriculum. According to the National Council of Teachers of Mathematics (2000), "discrete mathematics should be an integral part of the school mathematics curriculum" (Principles and Standards for School Mathematics, p. 31). Discrete mathematics is used to answer questions like the following:
- What is the most efficient route to plow all the streets in this neighborhood after a snowstorm (or pick up all the trash)?
- What is the best way to schedule 8 committee meetings without any conflicts, given that some people are on more than one committee?
- How can we schedule all the tasks on this large project (like a construction project or a new product launch) so that the entire project is finished in the least amount of time?
- Will there be enough phone numbers available to accommodate all the phones, faxes, and mobile phones in this area?
- What is the optimal medicine dosage for a patient, in order to maintain the right amount of medicine in the body while it is naturally metabolized?
- How can we model and analyze a changing population, or the changing amount of money in an investment program?
There are 4 main discrete mathematics topics in the Core-Plus Mathematics curriculum, which are used to answer questions like those above.
1. Vertex-Edge Graphs - Diagrams consisting of points (vertices)
and line segments (edges) connecting some or all of the points
These mathematical diagrams can be used to solve problems related to networks
and paths. Also, you can use vertex-edge graphs to solve problems related
to relationships, like conflict or prerequisite, among a finite number
of objects. For example, the network might be a road network or a communication
network, and you might want to find a shortest path through the network
or a travel route that visits each designated city exactly once. Core-Plus
Mathematics students also learn the most common application of vertex-edge
graphs in business and industry - using critical path analysis (also called
the PERT technique) to efficiently schedule large projects.
2. Recursive Formulas - Formulas that describe the current state
of a system in terms of previous states
Recursive formulas can be used to solve problems related to sequential,
step-by-step change. For example, you can use recursive formulas to compute
the amount of money in a savings account over time, the monthly payment
for a car loan, the changing chlorine concentration in a swimming pool,
or the predicted population of a whale species ten years from now.
3. Counting - Using mathematical techniques to systematically
count objects
For example, you might want to count the number of possible computer passwords
or ATM personal identification numbers (PINs) to make sure there are enough
possibilities for all the customers, and also enough so that someone can't
break into the system simply by trying all possibilities. Counting also
has important purely mathematical uses, such as determining the coefficients
in the expansion of algebraic expressions like (a + b)12.
4. Matrices - Rectangular arrays of numbers
Matrices are used in algebra, geometry, statistics, and probability, with
many applications in science, business, and industry. For example, a matrix
can be used to store and manipulate data, such as the statistics from
a baseball game or the results of an experiment, to create computer animations,
or to solve systems of linear equations.
These are the main topics of discrete mathematics contained in the Core-Plus Mathematics curriculum. They are included because they are important in mathematics, in real-world applications, and in contemporary life. The Mathematical Strands Chart shows the sequence of discrete mathematics units and provides links to more detailed overviews of the topics studied.