Tufts University Discrete Math Workshop

What Is Discrete Mathematics?

Discrete Mathematics is a rapidly growing and increasingly used area of mathematics, with many practical and relevant applications.

  • Because it is grounded in real-world problems, discrete mathematics lends itself easily to implementing the recommendations fo the National Council of Teachers of Mathematics (NCTM) standards. (The recently published Standards and Principles for School Mathematics notes that "As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum.")

  • Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be introduced at all grade levels, even with children who are not yet fluent readers.

Discrete mathematics will make math concepts come alive for your students. It's an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Teachers have found that discrete mathematics offers a way of motivating unmotivated students while challenging talented students at the same time.

Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be easily be introduced at the middle school grade level.

EXAMPLE: Linear Programming

Minimize C = 3x + 2y on the given feasible set.

Students spent a lot of time graphing lines without seeing how it can be useful. Linear programming is a powerful tool for finding the optimal value of a linear function on some feasible set. The feasible set is created by solving a system of linear inequalities. Solutions can be found graphically so even students who have not studied systems of equations can solve these problems.

EXAMPLE: Systematic Listing & Counting

There are 45 creatures here. How many of them are fish?

Systematic listing and counting are crucial analytical skills which play a fundamantal role in many areas of mathematics, in particular probability. There are many nuances of counting which are often missed in elementary courses. One of our goals is to shed light on this topic by exploring many examples and employing a variety of learning styles.

EXAMPLE: How Many Possibilities?

Combinations and permutations can range from simple to highly complex problems, and the concepts used are relevant to everyday life. Problems and solution methods can range so much that these mathematical ideas can be used with students from elementary school to high school.

Even young students with limited reading skills can solve problems with combinations of small numbers of items. For example, given that a classmate has two shirts and three pairs of pants, students can determine that there are six possible outfits. They can reason about this problem and even draw out the different options.

For older students, more advanced solution strategies can allow them to handle more complex problems, such as the following:

I have a 6-CD player in my car and I own 100 CD's.
How many different ways can I load 6 CD's into my player?

1st Slot 2nd Slot 3rd Slot 4th Slot 5th Slot 6th Slot

EXAMPLE: Which pizza place is closest?

Voronoi diagrams allow students and teachers to explore a technique that is used in a variety of applications, while at the same time employing critical thinking skills and geometric concepts.

These types of diagrams allow you to map out the areas in a given space that are closest to one specified point or another. For example, if there are 17 ice cream shops of equal quality in your town, a Voronoi diagram can show you which one is the closest for each region of town. This example is shown in the picture below.

This technique is used in biology, chemistry, geology, forestry, and more, as well as in resource planning and placement. This last topic is easily familiar to students as it can include determining placement for a new cell phone tower. or a new pizza place!

These diagrams are constructed using perpendicular bisectors, but students can approach these either strictly through geometric constructions, or through a more algebraic approach for more advanced students.

You are lucky enough to live in a town with 17 ice cream shops, each as good as the next. On the town map below, the ice cream shops are marked with letters. For each numbered point, which ice cream shop or shops should you frequent?