Getting Started (5 mins)

How many ways can you represent "7"?

Prompt:

"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.

Discuss

• 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:

Suggested Prompts:

• "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.

Activity (30 mins)

Circle-Triangle-Square Activity - Create a number system using symbols

Preparation:

• 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.

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Wrap-up

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.

Discuss the Rules Created for the Number Systems

Prompts:

• 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.

NOTE:

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.

Assessment

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.

Prompts:

• 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?

Extended Learning

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.