PURPOSE:

This explainer will present information to acquaint visitors with a two-digit number system. This will also help give visitors some background for the coding computer interactive in the exhibit - particularly the graphic showing a communications dish receiving 1's and 0's.

OBJECTIVES:

1.The visitor will observe place value in a number system.
2. The visitor will learn of examples of the uses of a two-digit number system.
3. The visitor will gain an awareness of how a two-digit number system enables a computer to work.

MATERIALS:

• Number disks (disks can be downloaded beginning at this link)
• 4 Cuisenare rods:  1 (natural), 2 (red), 4 (purple), 8 (black)
• Cuisenarie rod worksheet (download this linked page on legal size paper)
• marked cards for "paper computer" (download this linked page)
• 2 digit balance scale and weights (there is one specially constructed at the Maryland Science Center)
• labels ("I am __________ years old (in a two-digit number system)")

DEFINITIONS:

binary (BYE-na-ree) - having to do with two parts, the binary number system uses only two digits

digit - a symbol used in writing numbers

place value - the value of the symbol in a number system is determined by its place.
Zero is a place holder.
In the decimal system number

402

10², 10¹ and 10⁰

400 and 0 and 2
(4 X 10²) (0 X 10¹) (2 X 10⁰)

two-digit number system - the binary number system, using two symbols 0 and 1.
 

BACKGROUND:

Numbers from a two digit number system are frequently called binary numbers or base two. Older people may shy away from discussions about base two or base ten because it sounds like the "new Math" of the 60's and 70's. Younger children will get confused with the terminology when the word "decimal" is used to refer to base ten. It's simplest to stick to the words "two?digit number system" and "ten?digit number system".

Unlike Roman or Egyptian numerals, present day number systems have place value. Notice that, no matter how many digits are used in a system, the place value rules have to do with the same pattern of exponents.
1011 in base 10 means: (1 x 10³) + (0 X 10²) + (1 x 10¹) + (1 x 10⁰)
1011 in base 2 means: (1 x 2³) + (0 x 2²) + (1 x 2¹) + (1 x 2⁰)
1011 in base n means: (1 x n³) + (0 x n²) + (1 x n¹) + (1 x n⁰)
 

EXHIBIT REFERENCE:

The topics in this explainer are related to the African Math Games demo (the binary sorting activities), and the Error Correction Coding component of the Beyond Numbers Exhibit where:
 

Visitors watch animated  sequences that explain how satellite information is sent to  earth in coded form and later decoded. The visitor gets a chance to be the computer, detecting and correcting errors in a sample code that has been degraded with noise.  In another game, visitors, experience how some codes can correct far more errors than others.

PROCEDURES:

1. Check all materials and equipment to ensure that they are there, functioning, and in good condition.

2. Invite the visitor to think of a number between 1 and 31 and keep that number secret from you.

3. Show each disk and each time ask: Is your number on this disk?
For every "yes" response, mentally add the number that appears in the top left position.
Say: Your number is [the sum]. Let me show you the mathematics behind this trick.
Describe how you added the numbers.

4. Use the Cuisenaire rods to show how any number between 1 and 31 can be found with a combination of 1, 2, 4, 8, and/or 16 without repeating any number.

5. Point out that these numbers are the place values of a two digit number system - a system that has only two symbols, 0 and 1. Electronic computers operate with combinations of "on" and "off" currents - which is like using only two digits. Tell the visitor that you have made a paper computer that can do the trick you just did. Show the stack of cards and point out that the "ones" are broad black rectangles and the "zeroes" are the shorter empty rectangles.

The "ones" can be seen in a pattern when the card is viewed from the side.

6. Starting with the "16" disk and progressing in reverse order ask: "Is your number on this disk?" For a "yes" response, save the black marked cards correspondingto the place value , for a "no" response save the cards that are blank in that place value. Discard the unsaved cards. By the time you are finished you will have discarded all but one card.
Here's an example:
 

	"Is your number on this disk?" "Yes" Then keep all the cards that have a black mark in the left column (the 16's place)
	"Is your number on this disk?" "No" Keep all the cards that do NOT have a black mark in the second from the left column (the 8's place).
	"Is your number on this disk?" "Yes" Keep all the cards that have a black mark in the middle column (the 4's place)
	"Is your number on this disk?" "No" Keep all the cards that do NOT have a black mark in the fourth column (the 2's place)
	"Is your number on this disk?" "Yes" Keep he card that  has a black mark in the last column (the 1's place) ... "Your number is 21." "Wow!"

7. Direct the visitor's attention to the balance scale. (The pointing finger on the end of the scale is used to indicate the region which shows balance.)

8. Point out that the scale can be used to find the two digit version (on the left) of any ten-digit version of a number between zero and 15. For children, offer to find their age in the two-digit system. On the right side of the scale, plastic weights may be placed more than one on a post as long as it is understood that each weight represents an addend of the complete sum. (see picture) On the left side, weights labelled "1" are placed to cover weights labelled "0", so only one weight may be used on each post. When the ten-digit system number on the right is balanced by placing weights on the left, the two-digit equivalent can be read. The number can be written on a sticker and worn by the visitor.

BIBLIOGRAPHY:

I never cracked a book for this one. Maura Hurst showed me the number disk trick and I was eager to find a way to show WHY it worked.

COPYRIGHT: 1995 Maryland Science Center

AUTHOR:
Cathy Brady, Math Specialist Maryland Science Center