Overview

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 we 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).

Goals

Students will be able to:

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 quantitative values they represent. As a result, using the digits 0 and 1 can be a distraction when learning binary initially, so we don't use them 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.

Resources

Getting Started

Preparation

Print and cut out (or have students cut out) the shapes from the "Number Systems: Circle-Triangle-Square" resource or find three different shaped blocks or objects . We need at least 3 of each shape (a full set would be 10 of each shape).

How many ways can you represent "7"?

How many different ways can you represent the quantity "7"? Take one minute to write your ideas down before sharing with your neighbors. For example, if there are seven apples on the table we can represent that fact by writing "seven", "7", "VII", "*******", seven tallies, seven apple drawings, and so on. In the previous lessons we all invented ways to represent a set of messages with bits. Today we will focus on representing numbers. By the end of class, you will have invented your own number system.

Activity: Create a number system using symbols

Preparation

Activity: Circle-Triangle-Square

Given 3 places to work with for creating a pattern, make as many unique patterns as you can using only circles, triangles and squares. For example, the first pattern might be Circle-Circle-Square, the second Square-Circle-Triangle, the third Triangle-Square-Square, and so on.

Challenge 1: Find all 3-place patterns

Challenge 2: Make a system for generating all the patterns

Now that you've listed out all of the 3-place patterns of circles, triangles, and squares, let's put them in a systematic order. You can use any system you like, as long as you create and follow a clear set of rules for getting from one line to the next.

Pro Tip

Wrap Up

You just made a number system!

If you have good rules for generating all the patterns, for getting from one pattern to the next, and you have numbered each pattern so you have a symbol-to-number mapping, you have the beginnings of a number system! Recall: how many different ways you could write the number 7? Well, you now have another way using a system you just made up.

Solution(example)

Let's talk about...

Disscussion Goal

The goal of this final discussion is to establish the general properties of all number systems. You can raise the idea that the systems students developed might be just as legitimate as the ones they use every day - just not commonly accepted. The only requirements for developing a number system are:

  1. You must have a set of unique symbols
  2. You must agree on a fundamental ordering of those symbols. For example: circle comes before triangle, triangle before square. (similarly: 0 comes before 1, 1 before 2, and so on.) If you have that, then you can count, and represent any number.

Connection to Binary System

Assessment

1. Peer-assessment: Write the first few permutations of their system on a blank sheet of paper or braille paper. Trade your paper with the peer to see if he/she is able to predict the next two permutations in the system.

2. If you just had a circle and a square, how many 3-shape permutations could you make?

3. In 50 words or less, describe the concept of a number system. Why are rules required for a number system to be useful?

Extended Learning

Extend to a 4 digit numbering system

Peter Denning explains how "representations of information are at the heart of computing" in this article: Computation: A new way of science (link) Suggested activity: Read and summarize the content. Follow with a class discussion.

Standards Alignment