Precalculus through Coding and Animation

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Calculus and the Modern World

Calculus is at the foundation of modern world. In our daily life, we enjoy help from many modern gadgets (electric bulbs, cars, computers, cell-phones, etc.). Humans could not have invented them without using calculus. Therefore, you need to understand calculus (and precalculus) to study science and engineering.

Our Approach

In our modules, we teach precalculus and calculs based on coding, visualization and hands on exercises.

In general, there are two aspects of learning mathematics. First you master a set of abstract rules, and then you learn to apply those rules to solve real life problems. For example, a kid learns subtraction as an algorithm that applies concepts like borrow on decimal numbers. On the other hand, the algorithm helps in determining the change out of ten dollars after paying eight dollars and sixty seven cents at a grocery store.

Often it is not obvious to a kid that he needs to apply the rules of subtraction to determine change out of ten dollars. That ability to make connection with practical problems comes after extensive practices beyond learning the rules.

When it comes to precalculus and calculus, most high-school students spending time learning the rules. They do not get enough practice to make connection with real life. This is suboptimal, because they tend to forget the rules due to not applying them.

Our project-based approach helps students understand and remember the rules, because they apply them to solve problems like animation).

Project

In this class, you need to solve at least one and preferably two animation problems. In the first project, you will make an ball bounce around the walls of a box. In the second project, you will make a triangular or rectangular shape move around a plane, where the movement will include both translation and rotation.

R Language

We use R language in our class. Although R is not the most efficient platform for animation, it has two advantages. First, it is based on vectors and second it is easy to learn.

Here are the basics of R. We covered them in the first class.

Outline of the Topics Covered in this Module

Topic 1. Basics of R

We will use R programming language for all of our coding including the animation project. In this class, you learn about core commands in R, vectors, plotting, and solving linear equations.

Topic 2. Mathematical Induction

Mathematical induction is a versatile tools in mathematics that is used in areas as diverse as algebra, geometry, combinatorics (counting) and number theory. In this class, you learn about using mathematical induction to prove results from sum of series. This class also gives you enough practice in using R so that you can continue with the later projects.

Topic 3. R Plot in Finer detail

Many of our adventures in mathematics and animation need controlling individual points on a plot. In this class, you will learn about the finer details of the “plot” function in R.

Topic 4. Matrix Algebra

This class introduces you to the concepts like matrices and determinants, which are used widely in mathematics, physics and engineering. You already got to use them in R in solving linear equations, but here you learn them formally.

Topic 5. Connection between Matrix Algebra and Geometry

For animation, we need to move points on the screen. You will learn that matrices taught in the previous topic are key to accomplish that task.

Topic 6. Trigonometry Rules

In this session, we will learn about the rules of trigonometry and the trigonometric functions.

Topic 7. Trigonometry, Geometry and Matrix Algebra

We can form special matrices based on trigonometric functions, which will simplify our task of moving points on the screen.

Topic 8. Complex Numbers (Optional session)

Although complex numbers come from the attempts to define square root of  − 1, they are also intricately linked to geometry and matrix algebra. In fact, the rules of quantum mechanics are stated in two ways - based on matrices and based on complex numbers. In this class, we will introduce you to complex numbers and explain their connections with geometry.

We may not have time for this session and the topic is not directly related to our animation project.

Topic 9. Animation Project

We will spend two to three sessions on the animation project and accomplish the tasks based on concepts learned in the previous classes.