Discrete Mathematics for Pre-College Students
Update (May 16, 2018): DMPC is a UC “a-g” approved course, with approval in the mathematics (“c”) category.
The Discrete Mathematics Pre-Collegiate curriculum has been developed by a team of teachers, mathematicians, and mathematics educators at the Center for Research in Mathematics and Science Education (CRMSE) at San Diego State University. The curriculum includes eight units, and for each unit we have produced a student text as well as accompanying teacher materials. These materials provide an accessible introduction to topics that are at the heart of discrete mathematics. Additionally, these materials were designed to enable teachers of twelfth-grade mathematics courses to engage their students in developing mathematical habits of mind.
This unit poses a series of problems to set the tone and the expectations for problem solving, critical thinking, group collaboration, and writing in math. Problems have been taken from the course’s modules.The emphasis is on helping students see mathematics as a sense-making activity rather than a sequence of algorithms or formulas to be memorized and applied. Selected problems are accessible to students from different levels and allow for multiple entry points and problem-solving approaches. All students can engage with the mathematics in some way, and teachers can differentiate instruction based on students’ readiness, background, and interests. Students will build perseverance, collaborate with others to consider new problem-solving approaches, and communicate their understanding in speaking and writing.
Sample lesson: Trading Places
How can you figure out a winning strategy in a two-player game? What is it about the rules of the game that make it work all the time? Can you identify similarities across games that look different? In this unit, students play and analyze two-player games and develop an understanding of what it takes to describe a complete strategy and how to justify it. Students will be asked to write explanations in full sentences for why their strategies for each game work. We encourage them to go through a writing process in which they revise their arguments multiple times over the course of the unit and write arguments for different games, so that they can hone their justification skills and clarity in their writing.
Sample lesson: Thai 21
An exchange between a teacher and a student about the Games unit:
Student: “I really liked the Games Unit – it was fun!”
Teacher: “What was fun about it?”
Student: “I liked winning! And it was fun to try to win because, at first, I thought it was going to be easy, but then I really had to think in order to figure out the winning strategies. I was never bored and always challenged.”
Graphs: Connectivity & Traceability
How can we represent and analyze connections with vertex edge graphs? Students will learn that (and how) graphs can be useful tools for tracking relationships among objects or people and for modeling a wide variety of situations. A social network is the special case of a graph in which the objects are people and the connections are some form of relationship between them, such as friendship (e.g., “friends” on Facebook, which must be mutually agreed upon). Students will learn what a graph is through creating and interpreting representations of relationships, and develop a heightened awareness of the difference between what is being represented and how we are choosing to represent it.
Sample lessons: Nice to Meet You
Graphs: Planarity & Colorability
How can analyzing the properties of graphs lead to insights about geometry and topology? This unit builds on the previous unit with respect to exploring, modeling, pattern generalizing, and justifying. However, the emphasis in this unit is on algebraic and symbolic reasoning. Students will learn to identify appropriate properties of graphs that can be used to solve a given problem. They will formulate and manipulate algebraic quantities purposefully. Finding two different ways to represent the same quantity is a prominent strategy in proving several conjectures involving the Euler Characteristic for planar graphs and solids. Students will also experience and learn to construct several types of proofs, the most prominent of these being case-by-case proof and proof by construction.
Sample lessons: Gabriela’s Doodles
Iteration and Recursion
How can observing repeated processes lead to powerful generalizations? This unit is focused on motivating this idea and giving students an opportunity to practice thinking in terms of recurrence relations. Students are first introduced to the idea of a recurrence relation through the Tower of Hanoi problem, which is easy to solve for smaller, more concrete cases, but requires recursive thinking to reach the answer for more general cases. They are then introduced to many problems that all result in the Fibonacci sequence, a particular recursively defined sequence, as a way to motivate more rigorous justifications for why certain patterns adhere to recurrence relations.
Sample lesson: The Legend of the Golden Disks
Sequences & Series
This unit focuses on arithmetic and geometric sequences and series, building on the work from the Iteration and Recursion unit. We want to recognize that arithmetic and geometric sequences and series are particular kinds of recurrence relations. In particular, we want to motivate the fact that sequences defined by recurrence relations often have closed-form formulas, and arithmetic and geometric sequences and series are particularly nice in this regard. Students will experience iteration as a powerful tool to use for coming up with an explicit formula for a recurrence relation. Additionally, we approach this from a geometric angle, using geometric problems to approach arithmetic series and questions stemming from fractals – geometric objects defined recursively – to approach geometric series.
Sample lesson: Hidden Figures
How can you send secret messages? If you want to read a message that has been encrypted, how long will it take you to decode the message? This unit aims to engage students in exploring and working with a selection of mathematical concepts utilized in the field of cryptography that are accessible for high school students who have completed algebra. It provides the perfect stage for engaging students to think about math in a way that will require them to explore and uncover the structure that exists in language, but is not necessarily explicitly communicated. Students will be introduced to various transposition and substitution ciphers. These encryption systems will necessitate functional thinking and the understanding of modular arithmetic. Students will explore how language constructs such as word length and letter frequency play a role in cryptography. They will apply frequency analysis and known-plaintext attack strategies to cryptanalyze messages. Furthermore, this unit provides a natural setting to advance students’ counting strategies, particularly the fundamental counting principles, permutations, and combinations.
Sample lesson: ngMaki It Dciuflfti to daeR
In other units of this curriculum, students have naturally encountered many counting problems in a variety of contexts. For this unit, we are demonstrating a serious attempt to unify and formalize the counting principles and combinatorial reasoning skills that have already been infused throughout the course. Most counting problems are not typically solved by applying a set of theorems and formulas. Instead, they must be solved through a careful analysis of possibilities by exploring the structure and the conditions imposed in the given situation, and by applying various counting methods. The primary goal of the unit is to advance students’ set-oriented way of thinking and to develop habits of mind that will be used extensively throughout most quantitative careers.
Sample lesson: Counting FUNdamentals
If these samples have sparked your interest, contact us to discuss using DMPC in your school or district.