Logical Reasoning
By working through logical puzzles, learn to make rigorous arguments. The puzzles give you practice with direct and indirect proofs, as well as with interpreting complicated arguments. Logical reasoning is useful in all disciplines, and is indispensable in any field that relies on mathematical thinking. This topic can be both useful and enjoyable regardless of your prior background and future interests.
Units
When it comes to scientific calculations, units are your friends. Review conversion factors, metric system units, and the use of scientific notation. Practice translating word problems into mathematical expressions, and get a taste of dimensional analysis. This topic is especially useful for students interested in the physical sciences (like Physics, Chemistry, Biology, and Geology).
Logarithms
Logs, their definition, basic properties, logarithmic scale, and their uses are covered in this topic. Mathematical expressions and models for the number and size of earthquakes, pH value of a solution, the spread of the corona virus, and GDP growth rates will illustrate the use of logarithms in the Physical Sciences, in Economics, and in Mathematics.
Trigonometry
Trigonometric functions model periodic behavior, and are very useful in geometric calculations. Review the definitions, basic properties, graphs, and some applications of trigonometric functions. Especially useful for students interested in Physics and Mathematics.
Word Problems
When you want to use mathematics, you first have to translate a real world scenario into mathematical notation. This involves making simplifying assumptions, dealing with ambiguities, and learning to focus on the relevant information. Get better at doing word problems by tackling problems originating in Physics, Astronomy, Economics, Geology, and other sciences.
Derivatives
By focusing on rates of change and slopes, the idea of a derivative, and the concept of linearization will be explored. Applications to Physics (velocity and acceleration) and Economics (marginal revenue and marginal cost) will solidify your understanding of the main idea of differential calculus. This topic is useful for students who have had no calculus as well as those who want to review the idea of a derivative.
Integrals
What does an integral of a function mean? How are integrals used? What is the fundamental theorem of calculus, and why is finding areas related to finding slopes? In this topic you will explore the answers to these questions as well as applications to Physics and Mathematics. This topic is appropriate for students who are familiar with derivatives, and are planning to take further calculus classes.
Taylor Polynomials
The main idea of calculus is that complicated functions can be approximated locally with polynomials. The powerful idea of Taylor polynomials allows us to approximate integrals, to approximately solve differential equations and more. The prerequisite for this topic is a first course in calculus, and the topic is useful for students heading to second or third semester calculus.
Induction
Mathematical induction is a powerful proof method. In this topic, you will explore patterns, then express them in mathematical statements, and finally prove them using induction. There is no prerequisite for this topic, and it is especially useful for students planning to take mathematics and computer science classes.
Calculus Topics
A more in depth exploration of several calculus topics that bring together multiple ideas from across calculus. Topics include linearization, the arithmetic-geometric mean inequality, and using calculus to approximate the factorial function. The prerequisite for this topic is a first course in calculus, and it will benefit students continuing with second or third semester calculus.
Sequences and Series
The ideas of limits, infinity, and convergence will be explored in the context of sequences and series. What does it mean to add an infinite number of numbers? Is that even possible, and if so, how does one carry it out? What role does calculus play? A previous exposure to limits will be helpful for this topic, and a familiarity with integrals is necessary. The topic is beneficial to students continuing with second or third semester calculus.
Working with Data
In this topic, we explore some ideas in Statistic and in Statistical with the goal of gaining insight from data. Linear Regression, Decision Trees, and simulations using Python will be among the topics covered. The topic will be useful for students interested in the Sciences, Economics, or Statistics.
Combinatorics
Combinatorics is the mathematics of combinations and discrete relations. In this topic, you will learn some basic counting techniques, the use of recurrence relations, and the basic ideas of graph theory. Interested students may like puzzles and brain teasers, or they may want to see an introduction to some common themes in Mathematics and Computer Science.
Number Theory
Explore primes, divisibility, and the mysteries of the integers. While no particular prerequisites are needed, you will practice proofs and conceptual thnking. This topic is useful for students interested in Computer Science and Mathematics courses beyond Calculus.
Matrices
Organize data into rectangular arrays of numbers, and use them to model networks as well as to resize, rotate, and reflect computer graphics. In this topic the basics of matrix algebra and its uses are covered. No particular prerequisites and useful for students interested in Computer Science, Physics, Economics, and Linear Algebra.
Probability
The basic concepts of probability and conditional probability including Bayes's theorem, as well as their applications are explored. No particular prerequisites and useful for students interested in the Sciences, Economics, Statistics, and Mathematics.