Discrete Mathematics (UM160) & Geometry of Numbers (UM170) Course Tips

Are you taking Discrete Mathematics (UM160) or Geometry of Numbers (UM170) this year, looking for strategies to succeed? As options in the math division broaden at the university level, are you wondering if these courses would be right for you? We interviewed an instructor for this course, and here’s their advice about success in UM160 and UM170, along with information about their content and format.

Course Overviews

Discrete Mathematics is a single-semester fall course. It’s a core course for anyone interested in majoring in math or computer science in college. It builds on students’ prior experience with proofs such as proof by induction or by inclusion/exclusion. Topics covered include graph theory and discrete probability.

Geometry of Numbers (previously called “lattice point geometry") discusses Pick’s Theorem and its implications. This includes counting the number of lattice points in a circle, some exploration of fractals, and what kinds of shapes are impossible to draw with vertices sitting on an integer lattice. This is a very unique course which you are unlikely to find at a university or anywhere else.

These courses do not have to be taken together. However, Discrete Mathematics is a prerequisite for Geometry of Numbers, so they form a natural fall and spring pairing.

Course Topics

The topics for Discrete Mathematics include:

• Enumerative combinatorics

  • Sets, functions, injectivity, and surjectivity

  • Sum and product rules for sets

  • Principle of inclusion exclusion

  • The pigeonhole principle

  • Binomial coefficients

  • Combinations and permutations

  • Combinatorial proofs and identities

  • Sequences

  • Mathematical induction

  • Polynomial fitting

  • Recurrence relations

• Graph theory

  • Trees

  • Planar graphs

  • Coloring

    • Vertex coloring

    • Edge coloring

  • Euler Paths

  • Hamilton Paths

  • Euler Circuits

  • Hamilton Circuits

  • Enumeration of trees

• Discrete probability

  • Independence

  • Bayes' Theorem

  • The probabilistic method

The topics for Geometry of Numbers include:

• Lattice polygons

• Basic properties of lattice points in the plane

• The algebraic structure of the integer lattice

• Pick's theorem — we present three different proofs.

• Applications of Pick’s theorem.

  • Regular lattice n-gons

  • Farey sequences

  • Ford circles

• Lattice points in and on a circle

• Minkowski's theorem

Course Formats

For Discrete Mathematics, the course format is pretty standard. For class preparation, UM160 has lectures in addition to this textbook. There are no quizzes, but there is a midterm exam and final exam, both proctored. To prepare for the exams, the most important thing is to completely understand all problems on the weekly homework. The homework is designed to be challenging, but it’s good preparation for the exams. The homework is extra challenging for students without much prior proofs experience. 

In Geometry of Numbers, lectures are not available, only a textbook. There is neither a midterm nor a final exam, and no quizzes. Instead, students have around 120 problems which they must complete over the course of the semester and write up in detail with LaTeX. Weekly homework requires students to submit a few of the problems they’re currently working on, to ensure they stay on pace. Students also present a problem to the class five times per semester, and peers give feedback.

Frequent Challenges

Both courses require students to read and write many proofs. Students without much prior experience with proofs struggle more, especially in Discrete Mathematics since it’s a fall semester course. (Students who take Discrete Mathematics in the fall, even if they didn’t have much proofs experience prior, will have gained a lot of experience by the time they get to Geometry of Numbers in the spring.) 

To help with the proofs, it’s recommended that students first take Advanced Problem Solving and Proof Techniques (OM050). Many students do succeed in UM160 and UM170 without OM050 or other significant proofs experience – it is by no means impossible. But, do know that, if you don’t have much prior proofs experience, the homework will likely take you longer and make sure you know how to take advantage of the available supports.

For Discrete Mathematics specifically, some of the graph theory problems require a lot of experimentation. There’s not just one path or formula to the answer, and it just takes time to play around with it until you solve it. Be patient, start early, and enjoy the process!

For Geometry of Numbers, it’s a much more independent course. Many students enjoy this format, but just be careful to plan your time. You can’t do 120 hard problems in the last week of school.

Available Supports

• For the 2024-2025 school year, there are no peer tutors.

• Teaching assistants (TAs) are students who have previously taken the course and are available to help students. TAs invest more time commitment and work more closely with the instructor than peer tutors.

  • TAs hold two hours of office hours weekly, typically via Pronto. More information can be found on the course Canvas page, or by speaking with the instructor or TA.

  • Sometimes, a fellow student can explain something better than a teacher can, because they know what it’s like to be a student and where another student is likely to struggle to understand something.

• Your instructor is also there to help you

  • Instructors hold office hours once or twice a week, where students can come to discuss course topics.

    • Note that you do not need to have a particular question to attend office hours—you can come to discuss a general topic, explore an idea you have, ask a specific question about homework, and much more! Teachers encourage students to attend office hours to explore content related to course material but beyond the bounds of what is required/what will be tested on. 

      • Office hours are a place to have fun and get to know your teacher better!

    • You are never inconveniencing your teacher by attending office hours—they want to talk to you.

    • If you find it hard or embarrassing to ask questions in class, office hours can be a great place to ask questions, when you can just talk to your instructor 1-on-1. It is very normal to feel hesitant about asking questions—you are not alone! However, since all the material builds on previous material, making sure you find a way to ask any questions you have is very important.

  • Instructors can also be reached by email. You might decide to email your teacher, for example, if you have a (relatively quick) question that needs to be answered before their next office hours.

Time Commitment

For Discrete Mathematics, past students have spent 4-10 hours on coursework outside of class. Time commitment depends greatly on prior proof experience. 

For Geometry of Numbers, past students have spent around 4-5 hours on coursework outside of class.

Why Take These Courses

Discrete Mathematics is a core course for anyone looking to major in mathematics or computer science in college. It’s a great way to test out what these paths would be like. If you love the course, maybe those are majors you want to consider.

Geometry of Numbers is an interesting course with unique content. You’re not going to find it at a university. Think of it as a fun course – no exams, fascinating and special content, and a chance to learn LaTeX.