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).
Students will be able to:
- Students will be able to reason about permutations and symbols as arbitrary abstract concepts that can be used to represent numbers.
- Students will be able to invent our own "number system" with symbols and rules for getting from one permutation to the next.
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.
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
- Form teams of 2 or 3 students each.
- Each team should have "Number Symbols: Circle-Triangle-Square" worksheet.
- Each group should be provided paper shapes (at least three of each). Alternately, provide students materials for making their own, or use tiddlywinks, poker chips, or any other little doo-dads you can find, as long as there are three different types of objects.
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
- Record all of the unique 3-place patterns you can find in the template started below.
- How many are there? Number each one to keep track. (Note there may be more or fewer total patterns than spaces provided)
- Suggestion: try to find the permutations in some kind of organized or systematic way, rather than just randomly.
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.
- Write down the rules of your system
- Suggestion: to test your rules, have someone follow them to see if they can recreate your organized list above.
- Use the symbols to explore and generate all the possible patterns.
- Organize the set of patterns in an ordered system of their own design.
- Write down the rules of their ordering system; a good set of rules will allow someone else to predict or generate each subsequent permutation is in the list.
- The purpose of using these three symbols (as opposed to digits or letters of the alphabet) is to ensure that the activity becomes a more true problem-solving exercise or puzzle. The shapes are enough in most cases to jolt students out of the context of math class and truly invent a number system of their own without realizing it at first.
- Even though you might not come up with systems that we would think of as "common", creativity should be encouraged. It's possible to invent all kinds of rules to get from one pattern to the next.
- The point is that number systems are man-made sets of symbols and rules.
- There are two major "beats" to this activity: 1. Discover all the 3-place patterns, 2. Figure out a way to order them so that the sequence is predictable.
- You should discover a total of 27 unique patterns.
- When you record all of the patterns that you come up with on paper and number them, it foreshadows assigning a numeric value to a distinct set of symbols.
- Goal of the activity is not merely to list all 27 permutations, but to develop a set of rules that could be followed to generate all of them.
- Some of you might quickly recognize that there are 27 distinct groupings. However, ordering them is frequently a challenge because outside the context of math class you might not immediately apply what you already know about number systems, especially place values.
- Nevertheless, creativity should be encouraged. It's possible to invent all kinds of rules to get from one pattern to the next.
- Good questions to direct yourself towards thinking in this way include: Could I always tell me which permutation comes next? Could a classmate easily follow my rules to generate the same order? Would my rules still work if I only make all the permutations of length 2? What if I instead make all the permutations of length 4 or 5?
- We're using the phrase "unique 3-place patterns" rather than the common mathematical word permutation. You may use permutation if you like, but it is not necessary vocabulary for Computer Science Principles.
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.
Let's talk about...
- Were some sets of rules easier to use than others? If so what do you think led to this difference?
- Do you think there are any limits to the number of the symbols we could use to represent numbers?
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:
- You must have a set of unique symbols
- 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
- "What if we only had two symbols: a circle and square? Could we still make a number system?"
- "What if we had 10 symbols: a circle, a triangle, a square, a star, and so on...Could we still make a number system?"
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?
Extend to a 4 digit numbering system
- Cut out one or more additional shapes, or find objects with another shape. Extend your number systems to account for this additional shape.
- Extend your number systems to include 4 shapes or more.
- Try to identify which number a random permutation represents without counting all of the permutations that appear before it. Can you develop any rules?
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.
- CSTA K-12 Computer Science Standards (2011): CT.L2:7, CT.L2:8, CT.L2:9, CT.L2:14
- Computer Science Principles: 2.1.1 (A, B, C, D, E)
- Computer Science Principles: 2.3.1 (A, B)
- Computer Science Principles: 2.3.2 (A, B, C)