Making a Ball Bounce with Scratch: Introducing young students to Discrete Calculus (and the motion of the Planets, the Stars and the Galaxies)

When a ball bounces, it describes an elegant parabolic path. This is the effect of gravity: the force that regulates the motion of all planets, stars and galaxies.

The traditional study of dynamic systems is relegated to high school or even college, when the students have been introduced calculus, the operations of integration and differentiation.

This chapter presents a programming project using Scratch where young students implement the rules of gravitational forces in order to make a ball bounce realistically in a field of obstacles.

The strategy of the project is to address the problem not with the traditional continuous calculus, but rather with Discrete Calculus, where the operation of integration is a sum, and the operation of differentiation is a subtraction.

Four centuries ago, Galileo used rudimentary tools of Discrete Calculus to discover that a ball rolls down a ramp with a velocity that he described as the Law of the Odd Numbers (such as 1, 3, 5, 7, 9 …). That is probably one of the most beautiful laws in the universe. In fact, it rules the motion of all planets, stars, and galaxies in the universe.

Later, Newton noticed that when subtracting (differentiating) the odd numbers, the difference between them was constant (3-1=2, 5-3=2, 7-5=2). That is the constant acceleration of the gravitational field.

He also noticed that the cumulative sum (integration) of these numbers was (1+3=4, 4+5=9, 9+7=16, 16+9=25…) a set of square numbers: a parabola.

In this project, the gravity is represented by a simple Scratch symbol of addition. The fact of creating gravity with such simple block, that any young students understands well, is essential in the process of exploring the motion of objects in the presence of a gravitational field.

Students learn why a ball accelerates and decelerates, what happens when it gets in contact with a rigid body, such as a wall or a floor, how to rebound, and why the ball knows how high to bounce, regardless of the location of the obstacle.

Because we are very familiar with the motion of a bouncing ball in the real world, the students immediately validate the results of their project by watching on the screens of their computers a ball that bounces in such a realistic way.

The introduction to Discrete Calculus in the school at a young age is one of the essential changes in modern education.

Continuous Calculus was the only viable tool when computation was done with paper and pencil, but its complexity limited its access to very few. Discrete Calculus is based on addition and subtraction, two fundamental operations that young students understand well. Today's technology allows us to address any problem where change occurs, a dynamic problem, with tools that all students understand.