Length of course: Full Year
Subject area: Mathematics (‘C’) / Advanced Mathematics
Prerequisites: IM 2 or Algebra I and Geometry
Grade levels: 11th, 12th
In this student-centered course, students engage in problem solving experiences that allow for productive struggle and mathematical knowledge construction. The goal of the course is to advance students’ thinking as defined by the Standards for Mathematical Practice, while Discrete Math content is used as a context in which students develop these ways of thinking. Each lesson presents students with phenomena from within Introductory: Game Theory, Graph Theory (Connectivity and Traceability), Combinatorics, Cryptography, and the Study of Sequences and Series (i.e., Functions defined explicitly or recursively over the set of natural numbers). From these experiences students will engage in many forms of mathematical reasoning, including inductive and deductive reasoning, reasoning with recursion, definitional and structural reasoning. Definitions, concepts and skills are necessitated through problem-solving. Students are responsible for defining, modeling, pattern generalizing, conjecturing and justifying. They are expected to collaborate and communicate mathematical ideas regularly in verbal and written forms.
See below for the list of topics of study and sample lessons for this curriculum.
This introductory module poses up to four problems teachers can use to establish the tone and the expectations for problem solving, critical thinking, group collaboration, student agency and writing in mathematics. Each lesson introduces students to the mathematical domains they will study during the year: Games, Graph Theory, Iteration/Recursion, Cryptography and Counting.
This module largely explores a class of combinatorial games called two-player impartial games, where the same set of moves are available to both players, regardless of whose move it is, each game has a finite number of possible moves, and games must end with a win (or loss). No ties or draws are possible.
Students will repeatedly explore, observe, model, conjecture, generalize and prove.
This module is an introduction to vertex-edge graphs and their properties. Through a process of guided reinvention, students (re)create the concept of graph in three canonical forms; as a drawing, a table and a list (commonly referred to as vertex-edge, adjacency matrix and set). 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. The main focus of study is the mathematical structures of undirected and unweighted planar graphs for modeling pairwise relations.
Using graphs as a context, the overarching goal is to advance students’ ability to reason deductively (SMP3), communicate ideas (SMP6), and make connections (SMP7). In keeping with our assumptions about teaching and learning the module appeals to situations students can imagine, asking them to model problems arising in everyday life, society, and the workplace. In this way, we aim at necessitating graphs as a tool for problem-solving. Students will learn what a graph is through creating and interpreting representations of relationships. Another learning goal entails developing a heightened awareness of the difference between what one chooses to represent and how one chooses to represent it. By the end of the module, students will have had substantial experiences solving problems using graphs.
This module engages students in pattern generalizing, leveraging numerical and geometric situations. Students are 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 reasoning to reach the answer for more general cases. We ask students to justify why their recursive relations hold for all cases by explaining what it is about the rules of the game that causes the result. The module is themed by introducing problems which can be modeled by a Fibonacci sequence. Students will search for, generalize, and justify patterns in the Fibonacci sequence. In addition, they will construct and apply recursive patterns including the Golden Ratio and the Golden Rectangle found in nature, architecture, and other real-life contexts. This module also focuses on arithmetic and geometric sequences and series, building on work from the previous lessons. We want students to recognize that arithmetic and geometric sequences and series are particular kinds of recurrence relations. 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. Again, we ask students the familiar question, what is it about the way the structure is defined that causes your conjecture to hold for all cases?
This module aims at engaging 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 to engage students in thinking about math in a way that will require them to explore and uncover structures that must be precisely communicated in order to generate and decrypt coded communication. In this module, students will be introduced to various transposition and substitution ciphers. These encryption systems will necessitate reasoning with functions and the development of basic modular arithmetic. Students will explore how language constructs such as word length and letter frequency play a role in cryptography. Furthermore, this module provides a natural setting to advance students’ counting strategies, particularly the Fundamental Counting Principle, permutations and combinations.
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.