Students will explore the properties of number systems by effectively inventing a base-3 number system using circles, triangles and squares as the symbols instead of Arabic numerals. Students are asked to create rules that explain how each arrangement of symbols can be generated or predicated as an orderly, logical series. The objective is to understand that you can represent any number with any agreed-upon set of symbols that appear in an agreed-upon order. This is as true for circles, triangle and squares as it is for the digits 0-9, or the number systems we commonly see in computer science (binary and hexadecimal).

Purpose
In computer science, it’s common to move between different representations of numbers. Typically we see numbers represented in decimal (base-10), binary (base-2), and hexadecimal (base-16). The symbols of the decimal (base-10) number system - 0 1 2 3 4 5 6 7 8 9 - are so familiar that it can be challenging to mentally separate the written symbols from the abstract values they represent. As a result, using the digits 0 and 1 can be a distraction when learning binary initially, so we don't in this lesson.

We want to expose the fact that numbers themselves (quantities) are laws of nature, but the symbols we use to represent numbers are an arbitrary, man-made abstraction. Sometimes students memorize conversions from one number system to another without really understanding why. By effectively inventing their own base-3 number system in this lesson the goal is for students to see that all number systems have similar properties and function the same way. As long as you have 1) a set of distinct symbols 2) an agreement about how those symbols should be ordered, then you can represent any number with them.