Lesson 4: Number Systems
Unplugged | Concept Invention
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).
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.
Getting Started (5 mins)
Activity (30 mins)
Students will be able to:
- Reason about permutations and symbols as arbitrary abstract concepts that can be used to represent numbers.
- Invent their own “number system” with symbols and rules for getting from one permutation to the next.
- Print and cut out (or have students cut out) the shapes from the Teacher Resource - Number Systems: Circle-Triangle-Square - Resource or from colored paper.
- Each group of students needs at least 3 of each shape (a full set would be 10 of each shape).
For the Teacher
- Teacher Resource - Number Systems: Circle-Triangle-Square - Resource
- U1L04 - Teaching Tips & Tricks Video - Video (download)
For the Students
Getting Started (5 mins)
How many ways can you represent "7"?
This discussion aims to prepare students to distinguish between the familiar symbols we use to represent numbers and the quantities we are actually representing.
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.
"How many different ways can you represent the quantity “7”? Take one minute to write your ideas down before sharing with your neighbors."
- Give students about one minute for silent work and then use a share-out activity to share ideas from the class. We recommend just writing them somewhere the whole class can see them.
Do a whip-around and have students suggest the different symbols used to represent numbers
Once you've got a decent list up that everyone can see ask a few directed questions:
"If we kept going how many ways of representing “7” do you think we could come up with?"
- There exist an infinite number of representations.
"Why do you think we use the symbols we do use to represent numbers? Who decided that?"
- To learn more about numbering systems read Roman and ‘Arabic’ Numerals.
"If we were going to design a new system for representing numbers, what features would this system need to have?"
- The answer to this is the subject of today's lesson.
In the previous lessons you 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 (30 mins)
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 students 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.
Circle-Triangle-Square Activity - Create a number system using symbols
- Form teams of 2 or 3 students each.
- Distribute Activity Guide - Number Systems: Circle-Triangle-Square - Worksheet
- Each group should be provided paper shapes (at least seven 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.
There are two major "beats" to this activity:
- Discover all the 3-place patterns1
- Figure out a way to order them so that the sequence is predictable.
Students should discover a total of 27 unique patterns.
When students record all of the patterns that they come up with on paper and number them it foreshadows assigning a numeric value to a distict set of symbols.
You may need to emphasize that the 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 students might quickly recognize that there are 27 distinct groupings. However, ordering them is frequently a challenge for students because outside the context of math class they might not immediately apply what they 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 students towards thinking in this way include:
- Could you always tell me which permutation comes next?
- Could a classmate easily follow your rules to generate the same order?
- Would your rules still work if I only asked you to make all the permutations of length 2? What if I asked you instead to make all the permutations of length 4 or 5?
1 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 CSP.
Instructions: (from the activity guide: Activity Guide - Number Systems: Circle-Triangle-Square - Worksheet)
Given 3 places to work with, make as many unique patterns as you can using only circles, triangles and squares.
The diagram on the right shows a few examples of some 3-place patterns. NOTE: Order matters -- so, for example: Circle-Triangle-Square is a different pattern than Square-Circle-Triangle, even though both have one of each possible shape.
Challenge 1 - Find all of the 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.
- Jot down the rules of your system below.
- Suggestion: to test your rules, have someone follow them to see if they can recreate your organized list above.
Give students a fair amount of time to get into groups and start to arrange the shapes. They should be trying to discover patterns and rules in an effort to find all the possible unique configurations.
Students will need to do three things:
- Use the cut-out shapes 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.
Take a look at the next lesson on binary numbers to guage how deeply you need to go into number systems for this wrap up. You might be able to blend the end of this lesson with the beginning of the next more seamlessly.
The diagram on the right (click to expand) shows a technique for generating all the unique patterns with 3 shapes. The strategy mimics how we typically count in most number systems. Then, instead of shapes, if you just say circle = 0, triangle = 1, and square = 2 you can see how you might represent any pattern with 3 digits.
Present Student Circle-Triangle-Square Number Systems
Use a share out activity to allow students to share their systems with their classmates. Either in groups or as a class, students should read through their classmates’ rules, assess whether they are clear, and test them out to see how they are similar to or differ from their own rules and the patterns they generate.
You just made a number system!
If you have good rule for generating all the patterns, and 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.
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.
Discuss the Rules Created for the Number Systems
- 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?
Connect to Number Systems and Binary Numbers
At the end of the lesson a connection to number systems in general can be made and binary numbers in particular.
For example, after demonstrating a rule for the circle, triangle, square number system you could ask:
- "What if we only had two sybmols: a circle and square? Could we still make a number system?"
- The answer is yes, of course, and this is how binary numbers work (we'll see more in the next lesson)
You might then ask the related question:
- "What if we had 10 symbols: a circle, a triangle, a square, a star, and so on...Could we still make a number system?"
- This question should drive the point home - if you had 10 different shapes you could just make them operate like the digits 0-9.
If it would be useful, you might show the Binary Odometer - Code Studio widget that appears in the next lesson.
- The Binary Odometer - Code Studio shows a rolling count for 5 different number systems at once.
By the end of this lesson students do NOT need to know the binary number system or be able to convert between decimal and binary. We will address binary numbers in the next lesson much more specifically, including the idea of place value. Only a general understanding of the concept of number systems must be established.
Peer-assessment: For a student peer-assessment, give students note cards or blank sheets of paper and direct them to write the first few permutations of their system on it. They will then trade their papers with another group, and see if the other group is able to predict the next two permutations in the system.
- If you just had a circle and a square, how many 3-shape permutations could you make?
- Reflection: 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
- Have one or more additional shapes cut out to provide to students. Ask students to extend their number systems to account for this additional shape.
- Ask students to extend their 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. Suggested activity: assign students to read and summarize the content. Follow with a class discussion.
Unit 1: Lesson 4 - Number Systems
There are many different number systems, and you've probably heard of a couple of them! The number system we typically use for things like expressing the cost of something is called base-10, but there are lots of other systems that are used for different purposes. All of these systems share two things: first, they all use symbols or markings to represent values; and second, they all have rules for how to move from one value to the next. Today we're going to explore different patterns that can be represented with three symbols (a circle, triangle, and square), and develop our own rules for moving from one set of patterns to the next.
- Count using circles, triangles, and squares.
- Create a number systems with non-numeric symbols.
- Write the rules for counting with your number system.
In 50 words or less, describe the concept of a number system.
Why are rules required for a number system to be useful?
CSTA K-12 Computer Science Standards (2011)
CT - Computational Thinking
- CT.L2:14 - Examine connections between elements of mathematics and computer science including binary numbers, logic, sets and functions.
- CT.L2:7 - Represent data in a variety of ways including text, sounds, pictures and numbers.
- CT.L2:8 - Use visual representations of problem states, structures and data (e.g., graphs, charts, network diagrams, flowcharts).
- CT.L2:9 - Interact with content-specific models and simulations (e.g., ecosystems, epidemics, molecular dynamics) to support learning and research.
Computer Science Principles
2.1 - A variety of abstractions built upon binary sequences can be used to represent all digital data.
2.1.1 - Describe the variety of abstractions used to represent data. [P3]
- 2.1.1A - Digital data is represented by abstractions at different levels.
- 2.1.1B - At the lowest level, all digital data are represented by bits.
- 2.1.1C - At a higher level, bits are grouped to represent abstractions, including but not limited to numbers, characters, and color.
- 2.1.1D - Number bases, including binary, decimal, and hexadecimal, are used to represent and investigate digital data.
- 2.1.1E - At one of the lowest levels of abstraction, digital data is represented in binary (base 2) using only combinations of the digits zero and one.
2.3 - Models and simulations use abstraction to generate new understanding and knowledge.
2.3.1 - Use models and simulations to represent phenomena. [P3]
- 2.3.1A - Models and simulations are simplified representations of more complex objects or phenomena.
- 2.3.1B - Models may use different abstractions or levels of abstraction depending on the objects or phenomena being posed.
2.3.2 - Use models and simulations to formulate, refine, and test hypotheses. [P3]
- 2.3.2A - Models and simulations facilitate the formulation and refinement of hypotheses related to the objects or phenomena under consideration.
- 2.3.2B - Hypotheses are formulated to explain the objects or phenomena being modeled.
- 2.3.2C - Hypotheses are refined by examining the insights that models and simulations provide into the objects or phenomena.